Topology of Infinite-Dimensional Manifolds [1 ed.] 9789811575747, 9789811575754

Table of contents :
Preface
Contents
1 Preliminaries and Background Results
1.1 Terminology and Notation
1.2 Banach Spaces in the Product of Real Lines
1.3 Topological Spaces
1.4 Linear Spaces and Convex Sets
1.5 Cell Complexes and Simplicial Complexes
1.6 Simplicial Subdivisions
1.7 The Metric Topology of Polyhedra
1.8 PL Maps and Simplicial Maps
1.9 Derived and Regular Neighborhoods
1.10 The Homotopy Type of Simplicial Complexes
1.11 The Nerves of Open Covers
1.12 Dimensions
1.13 Absolute Neighborhood Retracts
1.14 Locally Equi-Connected Spaces
1.15 Cell-Like Maps and Fine Homotopy Equivalences
Notes for Chapter 1
2 Fundamental Results on Infinite-Dimensional Manifolds
2.1 Remarks on the Model Spaces and Isotopies
2.2 The Toruńczyk Factor Theorem
2.3 Stability and Deficiency
2.4 Negligibility and Deficiency
2.5 The Collaring and Unknotting Theorems
2.6 Classification of E-Manifolds
2.7 The Bing Shrinking Criterion
2.8 Z-Sets and Strong Z-Sets in ANRs
2.9 Z-Sets and Strong Z-Sets in E-Manifolds
2.10 Z-Sets in the Hilbert Cube and Q-Manifolds
2.11 Complementary Basic Results on Q-Manifolds
Notes for Chapter 2
3 Characterizations of Hilbert Manifolds andHilbert Cube Manifolds
3.1 (Strong) Universality and U-Maps
3.2 The Discrete (or Disjoint) Cells Property
3.3 The Discrete F.D. Polyhedra Property
3.4 Characterization of 2(Γ)-Manifolds
3.5 Fréchet Spaces and the Countable Product of ARs
3.6 The Function Space C(X,Y)
3.7 Cell-Like Images of Q-Manifolds
3.8 Characterization and Classification of Q-Manifolds
3.9 Keller's Theorem and the Countable Product of ARs
Notes for Chapter 3
4 Triangulation of Hilbert Cube Manifolds and Related Topics
4.1 Simple Homotopy Equivalences
4.2 Covering Spaces and Algebraic Preliminaries
4.3 The Procedure for Killing Homotopy Groups
4.4 The Splitting Theorem for Q-Manifolds
4.5 An Immersion of a Punctured n-Torus into Rn
4.6 The Handle Straightening Theorem
4.7 The Triangulation Theorem for Q-Manifolds
4.8 Further Results on Q-Manifolds
Notes for Chapter 4
5 Manifolds Modeled on Homotopy Dense Subspacesof Hilbert Spaces
5.1 Cap Sets and F.D.Cap Sets in 2 and Q
5.2 Manifold Pairs Modeled on the Pair of 2 (or Q) andits (F.D.)Cap Set
5.3 Absorption Property and Absorption Bases
5.4 Absorption Bases and Strong Universality
5.5 Four Types of Absorption Bases for 2(Γ)
5.6 C-Universality and Cσ-Universality
5.7 Manifolds Modeled on Absorption Bases for 2(Γ)
5.8 Homotopy Dense Embedding into an 2(Γ)-Manifold
5.9 Four Types of Infinite-Dimensional Manifolds
5.10 Absorbing Sets
5.11 Absolute Borel Classes
5.12 Universal Spaces for the Absolute Borel Classes
Notes for Chapter 5
6 Manifolds Modeled on Direct Limits andCombinatorial ∞-Manifolds
6.1 Preliminaries for Direct Limits
6.2 The Bounded Weak-Star Topology
6.3 Embedding Neighborhood Extension Properties
6.4 Characterization of R∞- and Q∞-Manifolds
6.5 Classification of R∞- and Q∞-Manifolds
6.6 Simplicial Approximations of PL Embeddings
6.7 Retriangulating the Simplicial Neighborhood
6.8 Flattening Simplicial Subdivisions
6.9 Combinatorial ∞-Manifolds and PL ∞-Manifolds
6.10 Characterization of PL ∞-Manifolds
6.11 Characterization of Combinatorial ∞-Manifolds
6.12 Simplicial Complexes with Contractible Links
6.13 Metric Combinatorial ∞-Manifolds
6.14 Bi-topological Infinite-Dimensional Manifolds
Notes for Chapter 6
LF-Spaces
Piecewise Linear R∞-Manifolds
Products of Absorbing Sets and R∞
Appendix: PL n-Manifolds and Combinatorial n-Manifolds
A.1 Characterizations of Combinatorial n-Manifolds
A.2 The Boundary of a Combinatorial n-Manifold
A.3 Regular Neighborhoods in PL n-Manifolds
A.4 PL Embedding Approximation Theorem
Epilogue
Bibliography
Books and Texts
References
Index

Citation preview

Springer Monographs in Mathematics

Katsuro Sakai

Topology of Infinite-Dimensional Manifolds

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Coventry UK and Seoul, South Korea Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A. Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

More information about this series at http://www.springer.com/series/3733

Katsuro Sakai

Topology of Infinite-Dimensional Manifolds

Katsuro Sakai Department of Mathematics, Faculty of Engineering Kanagawa University Yokohama, Kanagawa, Japan

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-981-15-7574-7 ISBN 978-981-15-7575-4 (eBook) https://doi.org/10.1007/978-981-15-7575-4 Mathematics Subject Classification: 57-01, 57-02, 57N17, 57N20, 57N35, 57Q05, 57Q10, 57Q12, 57Q15, 57Q25, 57Q35, 57N99, 54C35, 54C55, 54E55, 54F65, 54H05 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

There is no limit for our thoughts and imaginations since we are made “in God’s image” (Genesis 1:27). Mathematical ability is a gift from God, which enable us to comprehend and to create various abstract concepts. I always thank and praise Jehovah, our Creator. I agree with the following words in the Scriptures: “The fear of Jehovah is the beginning of wisdom, And knowledge of the Most Holy One is understanding.” —Proverbs 9:10 Note: Scripture quotations are from New World Translation of the Holy Scriptures—Revised 2013 (Watchtower Bible and Tract Society of New York, Inc.)

Preface

An infinite-dimensional manifold is a (topological) manifold modeled on a given infinite-dimensional (homogeneous) space E (called a model space), that is, a paracompact space covered by open sets that are homeomorphic to open sets in E. A manifold modeled on E is simply called an E-manifold. Hilbert space 2 and the Hilbert cube Q = [−1, 1]N are typical examples of infinite-dimensional model spaces, where an 2 -manifold and a Q-manifold are also called a Hilbert manifold and a Hilbert cube manifold, respectively.1 We can also take any infinitedimensional topological linear space as a model space. The direct limit R∞ = lim Rn of Euclidean spaces is one of them. It is known that the direct limit Q∞ = − → lim Qn of Hilbert cubes is homeomorphic to some topological linear space. Thus, − → there are various kinds of infinite-dimensional manifolds, which are research objects of Infinite-Dimensional Topology.2 Infinite-dimensional manifolds are more than just generalizations of usual manifolds modeled on Euclidean space Rn . There are many unexpected special phenomena different from finite-dimensional manifolds. From research of infinitedimensional manifolds, various useful tools, techniques, and ideas have been developed, which are attractive and exciting. In addition, there have been many applications to other fields of Topology.3 Let us give some examples of outstanding results related to other fields. In 1928, Fréchet asked whether Hilbert space 2 is homeomorphic to the countable product of lines RN . This question was positively answered in 1966 by R.D. Anderson. It was the dawn of Infinite-Dimensional Topology.4 In 1939, 1 We

consider not only 2 but also non-separable Hilbert spaces as model spaces. Topology is a branch of Geometric Topology, which studies infinitedimensional spaces arising naturally in Topology and Functional Analysis. 3 For the history of Infinite-Dimensional Topology, refer to the article of T. Koetsier and J. van Mill [96, Sect. 4]. 4 Due to Anderson’s essay [9], the starting point is when he answered in [3] affirmatively to a question posed by V. Klee, that is, he proved in 1964 that the product of a triod, T , and the Hilbert cube Q is homeomorphic to Q. 2 Infinite-Dimensional

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M. Wojdysławski conjectured that the hyperspace of a Peano continuum is homeomorphic to Q. This conjecture was finally proved in 1978 and it is known as the Curtis–Schori–West Hyperspace Theorem. Simple Homotopy Theory had been established by J.H.C. Whitehead during 1939–1952, where the Whitehead group and Whitehead torsion were introduced. The topological invariance of Whitehead torsion had been a longstanding problem, but T.A. Chapman proved it in 1973 by using Q-manifolds. In 1954, K. Borsuk conjectured that a compact absolute neighborhood retract (ANR) has the homotopy type of a finite simplicial complex. This conjecture was proved in 1977 by J.E. West, where Q-manifolds were applied. Shape Theory was founded by K. Borsuk in 1968 as the homotopy theory for spaces without a good local behavior. In 1972, Chapman showed that two compacta X and Y in the pseudo-interior (−1, 1)N of the Hilbert cube Q have the same shape type if and only if their complements Q \ X and Q \ Y are homeomorphic. Meanwhile, the theory of infinite-dimensional manifolds was developed and outstanding results had been obtained. In 1969, D.W. Henderson proved the Open Embedding Theorem, that is, every separable 2 -manifold can be embedded into 2 as an open set. In 1970, he joined with R.M. Schori and J.E. West in establishing the Classification and the Triangulation Theorems for Hilbert manifolds, respectively. Namely, it was shown that two Hilbert manifolds are homeomorphic if they have the same homotopy type and that every Hilbert manifold is homeomorphic to the product of Hilbert space and a locally finite-dimensional simplicial complex with the metric topology. For (compact) Q-manifolds, the Triangulation and the Classification Theorems were established in 1973 by Chapman, that is, every compact Q-manifold is homeomorphic to the product of Q and a finite simplicial complex, and two compact Q-manifolds are homeomorphic if and only if finite simplicial complexes triangulating them have the same simple homotopy type. In 1980 and 1981, H. Toru´nczyk succeeded in characterizing Hilbert cube manifolds and Hilbert manifolds topologically. This book is designed as a textbook for graduate students and researchers in various branches related to Topology to acquire the fundamental results on infinitedimensional manifolds and various techniques treating infinite-dimensional spaces. This can also be used as a reference book. Up to now, the following six books have been available for the same purpose: (1) C. Bessaga and A. Pełczy´nski, Selected Topics in Infinite-Dimensional Topology, MM 58 (Polish Sci. Publ., Warsaw, 1975) (2) T.A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28 (Amer. Math. Soc., Providence, 1975) (3) J. van Mill, Infinite-Dimensional Topology: Prerequisites and Introduction, North-Holland Math. Library 43 (Elsevier Sci. Publ. B.V., Amsterdam, 1989) (4) A. Chigogidze, Inverse Spectra, North-Holland Math. Library 53 (Elsevier Sci. Publ. B.V., Amsterdam, 1996) (5) T. Banakh, T. Radul, and M. Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds, Math. Studies, Monog. Ser. 1 (VNTL Publ., Lviv, 1996)

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ix

(6) J. van Mill, The Infinite-Dimensional Topology of Function Spaces, NorthHolland Math. Library 64 (Elsevier Sci. Publ. B.V., Amsterdam, 2002) We selected materials from the above books and compiled them into the present book and also included the new results that are not presented in those books. In addition, we have covered the manifolds modeled on the direct limits R∞ = lim Rn − → and Q∞ = lim Qn . As an infinite-dimensional version of a combinatorial n− → manifold, we have defined a combinatorial ∞-manifold, which triangulates an R∞ -manifold. This is the first book presenting such manifolds. This book is written to be fairly self-contained if combined with the following book, which is cited as [GAGT]: • K. Sakai, Geometric Aspects of General Topology, Springer Monog. in Math. (Springer, Tokyo, 2013) To read [GAGT], the readers are required to have the fundamental knowledge on General Topology. For example, it is enough to finish Part I of the following popular textbook: • J.R. Munkres, Topology, 2nd ed. (Prentice Hall, Inc., Upper Saddle River, 2000) The book [GAGT] also contains some outstanding results in Infinite-Dimensional Topology different from applications mentioned above, that is, the existence of the following spaces or maps: a hereditary infinite-dimensional compact metrizable space (a compactum containing no subspaces of finite-dimension except zero-dimension); an infinite-dimensional compact metrizable space with finite cohomological dimension (Alexandroff’s Problem); a cell-like map of a finite-dimensional compactum onto an infinite-dimensional compactum (Cell-Like Mapping Problem); a separable metrizable topological linear space that is not an absolute retract (AR). The first one is contained in Chapter 4 and the other three are in Chapter 7. Those are not necessary for reading the present book.

Almost all required background knowledge is listed in Chap. 1, whose detailed information is founded in [GAGT]. Besides, we need some additional results, for example, some fundamental results in PL Topology, which are also contained with their proofs. Taking a brief look at this first chapter, the reader can recognize what knowledge he or she should take in and where he or she can. Possibly skipping the first chapter, one can start with the second chapter and go back over necessary parts of Chap. 1 when needed. For Chap. 4, we need some results in Simple Homotopy Theory and Wall’s Obstruction Theory, covering spaces and Algebraic Topology,5 which are written in the preliminary part of Chap. 4, not in the first chapter. Chapter 2 is devoted to the fundamental results on manifolds modeled on an infinite-dimensional normed linear space, the Stability Theorem, the Unknotting Theorem, the Open Embedding Theorem, the Classification Theorem, etc., which are discussed in Bessaga and Pełczynski’s book (1). Here, are also proved the fundamental results on Hilbert cube manifolds, which are discussed in van Mill’s book (3) and Chapman’s lecture notes (2). Furthermore, we prove the Toru´nczyk

5 It is not required for the reader to be familiar with these objects. Only elementary knowledge of Algebraic Topology is necessary.

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Factor Theorem, that is, for each completely metrizable ANR X with weight  τ , the product of X and Hilbert space of weight τ is a Hilbert manifold, which has not been proved in any other book. Combining this theorem with the Classification Theorem, we can easily obtain the Triangulation Theorem for Hilbert manifolds. Toru´nczyk’s characterizations of Hilbert manifolds and Q-manifolds are proved in Chap. 3. For the characterization of compact Q-manifolds, a readable proof is provided in van Mill’s book (3), but the non-compact version is not easily derived from the compact case. However, we discuss the non-compact case too. For Hilbert manifolds, we treat not only 2 -manifolds but also non-separable Hilbert manifolds. In this chapter, some applications of the characterization of Hilbert manifolds and Q-manifolds are also given. In particular, it is proved that every Fréchet space (= locally convex completely metrizable topological space) is homeomorphic to Hilbert space with the same weight and that every infinite-dimensional compact convex set in a metrizable topological space is homeomorphic to the Hilbert cube Q if it is an AR. It is also proved that the space of all continuous map from a nondiscrete compactum to a separable completely metrizable ANR is an 2 -manifold. As mentioned above, Chap. 2 contains the Triangulation Theorem for Hilbert manifolds but not for Q-manifolds. Chapter 4 is devoted to proving the Triangulation Theorem for Q-manifolds, which is contained in Chapman’s lecture notes (2) but is not in van Mill’s book (3). To prove this theorem, we use some algebraic results concerning the Whitehead group and the Wall’s finiteness obstruction. They are contained in the first section, but their proofs are not given. From the Triangulation Theorem for compact Q-manifolds, we have derived the Borsuk conjecture mentioned above. The topological invariance of Whitehead torsion is also proved in this chapter. In Chap. 5, we discuss f.d.cap sets and cap sets for 2 -manifolds and Qmanifolds, which are manifolds modeled on the following incomplete normed linear spaces:    2f = (xi )i∈N ∈ 2  xi = 0 except for finitely many i ∈ N ,    2Q = (xi )i∈N ∈ 2  supi∈N 2i |xi | < ∞ , which are the linear spans of the orthonormal basis of 2 and the Hilbert cube Q =  −n −n 2 n∈N [−2 , 2 ] ⊂  , respectively. We also consider non-separable versions of f.d.cap sets and cap sets, named absorption bases. As a generalization of (f.d.)cap sets, introducing absorbing sets in Hilbert manifolds, M. Bestvina and J. Mogilski gave characterizations of manifolds modeled on the universal spaces of absolute Borel spaces. Since then, there have been many works on absorbing sets. In this chapter, we also intend to generalize the work of Bestvina and Mogilski to nonseparable spaces. Although there are many related results, we treat only a small number of them. For results on absorbing sets, the book of Banakh, Radul, and Zarichinyi (5) can be referred to but non-separable absorbing sets are not treated. Related results also treated in the second book of J. van Mill (6).

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Chapter 6 is devoted to manifolds modeled on R∞ and Q∞ , which are the direct limits of Euclidean spaces and Hilbert cubes, respectively. Here, we prove Heisey’s Theorem, that is, Q∞ is homeomorphic to a locally convex linear topological space. Characterizations of these manifolds are given and their Classification Theorems are proved. We also discuss simplicial complexes triangulating R∞ -manifolds. Such a simplicial complex is an infinite-dimensional generalization of combinatorial manifolds, which is called an infinite-dimensional combinatorial manifold (or a combinatorial ∞-manifold). We prove the so-called Hauptvermutung6 for them. Furthermore, a combinatorial ∞-manifold is characterized as a simplicial complex K, such that every simplex σ ∈ K is a Z-set in |K|, equivalently every σ ∈ K has the non-empty contractible link. We also prove that a countable simplicial complex K is a combinatorial ∞-manifold if and only if |K| is an R∞ -manifold. In the last section, we introduce bi-topological infinite-dimensional manifolds modeled on ∞ N ∞ N N ∞ (R∞ , RN f ) and (Q , Qf ), which are called (R , Rf )-manifolds and (Q , Qf )manifolds, respectively. Every combinatorial ∞-manifold with the weak topology and the metric topology is a manifold modeled on (R∞ , RN f ). Acknowledgments The author would like to express his sincere appreciation to his teacher Professor Yukihiro Kodama, who introduced him to Infinite-Dimensional Topology and had been encouraging him to study in this field. He truly thanks his graduate students Yutaka Iwamoto, Yuji Akaike, Shigenori Uehara, Masayuki Kurihara, Masato Yaguchi, Kotaro Mine, Atsushi Yamashita, Atsushi Kogasaka, Katsuhisa Koshino, and Hanbiao Yang for their careful reading of his lecture notes and helpful comments. He appreciates Taras Banakh for his valuable suggestions. He also feels grateful to Tatsukhiko Yagasaki and Yukinobu Yajima for encouraging him to publish this book. The encouragement of Mr. Masayuki Nakamura, the editor at Springer Japan, has been a great help in completing this book. Moreover, he has many people to thank: his wife, Tomoko, his children, his friends, and others. Most of all, he would like to give thanks to our Maker, Jehovah.

Tsukuba, Japan April 2020

6 For

meaning of this word, see p. 44.

Katsuro Sakai

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Note In the text, numbers in brackets [ ] and [( )], respectively, refer to papers in the References and books or texts in the Bibliography at the end of the book. However, the author’s first book [(15)] is cited as [GAGT]. Moreover, for a theorem (or proposition, etc.) quoted from [GAGT], its corresponding theorem number in [GAGT] is indicated in a frame box at the end of the statement. For example, 2.9.4 means (Theorem) 2.9.4 in [GAGT].

Contents

1

Preliminaries and Background Results . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Banach Spaces in the Product of Real Lines . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Linear Spaces and Convex Sets . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Cell Complexes and Simplicial Complexes . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Simplicial Subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 The Metric Topology of Polyhedra . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 PL Maps and Simplicial Maps . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.9 Derived and Regular Neighborhoods . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.10 The Homotopy Type of Simplicial Complexes.. .. . . . . . . . . . . . . . . . . . . . 1.11 The Nerves of Open Covers .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.12 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.13 Absolute Neighborhood Retracts . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.14 Locally Equi-Connected Spaces . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.15 Cell-Like Maps and Fine Homotopy Equivalences . . . . . . . . . . . . . . . . . . Notes for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 2 10 15 20 27 33 37 39 46 53 57 61 65 73 76 80

2 Fundamental Results on Infinite-Dimensional Manifolds . . . . . . . . . . . . . . . 2.1 Remarks on the Model Spaces and Isotopies .. . . .. . . . . . . . . . . . . . . . . . . . 2.2 The Toru´nczyk Factor Theorem.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Stability and Deficiency . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Negligibility and Deficiency . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 The Collaring and Unknotting Theorems.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Classification of E-Manifolds.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 The Bing Shrinking Criterion . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8 Z-Sets and Strong Z-Sets in ANRs . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9 Z-Sets and Strong Z-Sets in E-Manifolds . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.10 Z-Sets in the Hilbert Cube and Q-Manifolds .. . .. . . . . . . . . . . . . . . . . . . . 2.11 Complementary Basic Results on Q-Manifolds .. . . . . . . . . . . . . . . . . . . . Notes for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

81 82 93 108 121 132 145 153 162 174 182 191 200 xiii

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 (Strong) Universality and U-Maps . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 The Discrete (or Disjoint) Cells Property . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 The Discrete F.D. Polyhedra Property .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Characterization of 2 ()-Manifolds . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Fréchet Spaces and the Countable Product of ARs . . . . . . . . . . . . . . . . . . 3.6 The Function Space C(X, Y ). . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Cell-Like Images of Q-Manifolds .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8 Characterization and Classification of Q-Manifolds . . . . . . . . . . . . . . . . 3.9 Keller’s Theorem and the Countable Product of ARs. . . . . . . . . . . . . . . . Notes for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

203 203 216 229 239 250 260 267 277 285 288

4 Triangulation of Hilbert Cube Manifolds and Related Topics . . . . . . . . . . 4.1 Simple Homotopy Equivalences . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Covering Spaces and Algebraic Preliminaries . . .. . . . . . . . . . . . . . . . . . . . 4.3 The Procedure for Killing Homotopy Groups .. . .. . . . . . . . . . . . . . . . . . . . 4.4 The Splitting Theorem for Q-Manifolds . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 An Immersion of a Punctured n-Torus into Rn . .. . . . . . . . . . . . . . . . . . . . 4.6 The Handle Straightening Theorem . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 The Triangulation Theorem for Q-Manifolds.. . .. . . . . . . . . . . . . . . . . . . . 4.8 Further Results on Q-Manifolds .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Notes for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

289 290 295 300 310 319 325 346 351 352

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Cap Sets and F.D.Cap Sets in 2 and Q . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Manifold Pairs Modeled on the Pair of 2 (or Q) and its (F.D.)Cap Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Absorption Property and Absorption Bases . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Absorption Bases and Strong Universality . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Four Types of Absorption Bases for 2 () . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 C-Universality and Cσ -Universality . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 Manifolds Modeled on Absorption Bases for 2 () . . . . . . . . . . . . . . . . . 5.8 Homotopy Dense Embedding into an 2 ()-Manifold . . . . . . . . . . . . . . 5.9 Four Types of Infinite-Dimensional Manifolds . .. . . . . . . . . . . . . . . . . . . . 5.10 Absorbing Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.11 Absolute Borel Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.12 Universal Spaces for the Absolute Borel Classes . . . . . . . . . . . . . . . . . . . . Notes for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

368 376 387 399 408 418 429 434 447 458 467 476

6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Preliminaries for Direct Limits. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 The Bounded Weak-Star Topology . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Embedding Neighborhood Extension Properties.. . . . . . . . . . . . . . . . . . . .

479 479 483 488

353 354

Contents

xv

6.4 Characterization of R∞ - and Q∞ -Manifolds . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Classification of R∞ - and Q∞ -Manifolds . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Simplicial Approximations of PL Embeddings . .. . . . . . . . . . . . . . . . . . . . 6.7 Retriangulating the Simplicial Neighborhood .. . .. . . . . . . . . . . . . . . . . . . . 6.8 Flattening Simplicial Subdivisions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.9 Combinatorial ∞-Manifolds and PL ∞-Manifolds . . . . . . . . . . . . . . . . . 6.10 Characterization of PL ∞-Manifolds .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.11 Characterization of Combinatorial ∞-Manifolds .. . . . . . . . . . . . . . . . . . . 6.12 Simplicial Complexes with Contractible Links . .. . . . . . . . . . . . . . . . . . . . 6.13 Metric Combinatorial ∞-Manifolds .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.14 Bi-topological Infinite-Dimensional Manifolds ... . . . . . . . . . . . . . . . . . . . Notes for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

492 497 500 506 517 529 535 542 548 554 567 579

Appendix: PL n-Manifolds and Combinatorial n-Manifolds . . . . . . . . . . . . . . . A.1 Characterizations of Combinatorial n-Manifolds . . . . . . . . . . . . . . . . . . . . A.2 The Boundary of a Combinatorial n-Manifold .. .. . . . . . . . . . . . . . . . . . . . A.3 Regular Neighborhoods in PL n-Manifolds .. . . . .. . . . . . . . . . . . . . . . . . . . A.4 PL Embedding Approximation Theorem .. . . . . . . .. . . . . . . . . . . . . . . . . . . .

581 581 590 592 594

Epilogue . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 597 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 601 Books and Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 601 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 603 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 609

Chapter 1

Preliminaries and Background Results

In this chapter, we first introduce the terminology and notation, and then list the background results. The reader may skip this chapter and can read necessary parts later when needed. Several results might be learned in graduate courses, but others are advanced and special. Almost all results are listed without proofs, which can be found in the following book: • K. Sakai, Geometric Aspects of General Topology, Springer Monog. in Math. (Springer, Tokyo, 2013) — [(15)] That is cited as [GAGT] instead of [(15)], by which the reader can confirm proofs and details. When a theorem (or proposition, etc.) is quoted from [GAGT], each corresponding theorem number in [GAGT] is indicated in a frame box at the end of statement. Sections 1.1 and 1.2 are almost identical to the same sections of [GAGT]. Contents of Sects. 1.3, 1.4, 1.12, 1.13, and 1.15 come from Chapters 2, 3, 5, 6, and 7 of [GAGT], respectively. Almost all contents of Sections 1.5–1.11 except Sect. 1.9 are contained in Chapter 4 of [GAGT]. Sections 1.9 and 1.14 are supplements for Chapters 4 and 6 of [GAGT], respectively.

Fundamental results on simplicial complexes are described in Chapter 4 of [GAGT]. Besides, additional preliminary results in PL Topology (Combinatorial Topology) are required for Q-manifolds and R∞ -manifolds, which are not covered by [GAGT]. In Sect. 1.8, a version of the PL embedding approximation theorem 1.8.11 is added together with its proof. Section 1.9 is devoted to regular neighborhoods that are effectively used in PL Topology. Although we refer to them too little, PL n-manifolds (or combinatorial n-manifolds) are main subjects in PL Topology. So we provide an appendix for basic results on them at the end of the book. The following is a good textbook: • C.P. Rourke and B.J. Sanderson, Introduction to Piecewise-Linear Topology, Springer Study Edition (Springer-Verlag, Berlin, 1972, 1982) — [(17)]

© Springer Nature Singapore Pte Ltd. 2020 K. Sakai, Topology of Infinite-Dimensional Manifolds, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-15-7575-4_1

1

2

1 Preliminaries and Background Results

1.1 Terminology and Notation With respect to terminology and notation, we follow the book [GAGT]. For the standard sets, we use the following notation: • • • • • • • •

N — the set of natural numbers (i.e., positive integers); ω = N ∪ {0} — the set of non-negative integers; Z — the set of integers; Q — the set of rationals; R = (−∞, ∞) — the real line with the usual topology; C — the complex plane; R+ = [0, ∞) — the half (real) line; I = [0, 1] — the unit closed interval.

A (topological) space is assumed to be Hausdorff and a map is a continuous function. A singleton is a space consisting of one point. A space is said to be nondegenerate if it has at least two points. A compact metrizable space is called a compactum and a connected compactum is called a continuum.1 For a space X and A ⊂ X, we use the following notation: • • • •

clX A (or cl A) — the closure of A in X; intX A (or int A) — the interior of A in X; bdX A (or bd A) — the boundary of A in X; idX (or id) — the identity map of X.

For a metrizable space X, • Metr(X) — the set of all admissible metrics of X. The cardinality of a set  is denoted by card . The weight w(X), the density dens X, and the cellularity c(X) of a space X are defined as follows: • w(X) = min{card B | B is an open basis for X}; • dens X = min{card D | D is a dense set in X}; • c(X) = sup{card G | G is a pairwise disjoint open collection}. As is easily observed, c(X)  dens X  w(X) in general. In the case where X is metrizable, all these cardinalities coincide (cf. p. 2 of [GAGT]). For spaces X and Y with subspaces X1 , . . . , Xn ⊂ X and Y1 , . . . , Yn ⊂ Y , • X ≈ Y means that X and Y are homeomorphic; • (X, X1 , . . . , Xn ) ≈ (Y, Y1 , . . . , Yn ) means that there exists a homeomorphism h : X → Y such that h(X1 ) = Y1 , . . . , h(Xn ) = Yn ; • (X, x0 ) ≈ (Y, y0 ) means (X, {x0 }) ≈ (Y, {y0 }), where (X, x0 ) is called a pointed space and x0 its base point.

1 Their

plurals are compacta and continua, respectively.

1.1 Terminology and Notation

3

  For the product space γ ∈ Xγ , the γ -coordinate of each point  x ∈ γ ∈ Xγ is denoted by x(γ ),  that is, x = (x(γ ))γ ∈ . We can regard x ∈ γ ∈ Xγ as a function x :  → γ ∈ X γ such that x(γ ) ∈ Xγ for each γ ∈ . For each γ ∈ , the projection prγ : γ ∈ Xγ →  Xγ is defined by prγ (x) = x(γ ). For  ⊂ , the projection pr : γ ∈ Xγ → λ∈ Xλ is defined by pr (x) = x|  (= (x(λ))λ∈ ). In the case Xγ = X for every γ ∈ , we write γ ∈ Xγ = X . In particular, XN is the product space of countable infinite copies of X. When  = {1, . . . , n}, X = Xn is the product space of n copies of X. For the product space X × Y , prX : X × Y → X and prY : X × Y → Y denote the projections. Now, let X = (X, d) be a metric space, x ∈ X, ε > 0, and A, B ⊂ X. We use the following notation:    • Bd (x, ε) = y ∈ X  d(x, y) < ε — the ε-neighborhood of x in X (or the open ball with center x and radius ε);    • Bd (x, ε) = y ∈ X  d(x, y)  ε — the closed ε-neighborhood of x in X (or the closed ball with  x and radius  ε);  center • diamd A = sup d(x, y)  x, y ∈  A — the diameter of A;  of x from A; • d(x, A) = inf d(x,   y) y ∈ A — the distance  x ∈ A, y ∈ B — the distance of A and B; • distd (A, B) = inf d(x, y)    • Nd (A, ε) = x ∈ X  d(x, A) < ε — the ε-neighborhood of A in X;    • Nd (A, ε) = y ∈ X  d(x, A)  ε — the closed ε-neighborhood of A in X. It should be noticed that d(x, A) = dist d ({x}, A), Nd ({x}, ε) = Bd (x, ε), Nd ({x}, ε) = Bd (x, ε), and Nd (A, ε) = x∈A Bd (x, ε). For a collection A of subsets of X, the mesh of A is defined as follows:    • meshd A = sup diamd A  A ∈ A . When there are no possible confusions, we can drop the subscript d and write B(x, ε), B(x, ε), N(A, ε), diam A, dist(A, B) and mesh A. The standard spaces are listed below: • Rn — Euclidean n-space with the norm x =

• • • • • •



x(1)2 + · · · + x(n)2 ,

0 = (0, . . . , 0) ∈ Rn — the origin, the zero vector or the zero element, ei ∈ Rn — the unit vector = 1 and ei (j ) = 0 for j = i;  defined by  ei (i)n−1  0 = R Rn+ = x ∈ Rn  x(n) × R+ — Euclidean half n-space;    n−1 n  = S  x ∈ R x =1 — the unit (n − 1)-sphere; Bn = x ∈ Rn  x  1 — the unit closed n-ball;    n = x ∈ (R+ )n+1  n+1 i=1 x(i) = 1 — the standard n-simplex; Q = [−1, 1]N — the Hilbert cube; s = RN — the space of sequences.

4

1 Preliminaries and Background Results

Note that Sn−1 , Bn , and n are not product spaces even though the same notations are used for product spaces, where the indexes n − 1 and n represent their dimensions. A separable metrizable space M is called an n-manifold (or an n-dimensional manifold)2 if each x ∈ M has a neighborhood that is homeomorphic to (an open set in) In , where In can be replaced with Rn+ . We call x ∈ M an interior point if it has a neighborhood homeomorphic to (an open set in) (0, 1)n (≈Rn ). The set Int M of all interior points of M is called the interior of M, which is open in M. We also call x ∈ M a boundary point if it is not an interior point, that is, any neighborhood of x is not homeomorphic to (an open set in) Rn . In other words, x ∈ M is a boundary point if and only if x has a neighborhood N such that (N, x) ≈ (In , 0) (or (N, x) ≈ (Rn+ , 0)). The set ∂M consisting of all boundary points of M is called the boundary of M, which is an (n − 1)-manifold and closed in M. When ∂M = ∅, M is called an n-manifold without boundary. A closed n-manifold is a compact nmanifold without boundary. The closed n-ball Bn is an n-manifold with ∂Bn = Sn−1 and the n-sphere Sn is a closed n-manifold. Let A and B be collections of subsets of X and Y ⊂ X. We define • • • •

A ∧ B = {A ∩ B | A ∈ A, B ∈ B}; A|Y = {A ∩ Y | A ∈ A}; A[Y ] = {A ∈ A | A ∩ Y = ∅}; Acl = {cl A | A ∈ A}.

The star of Y with respect to A is defined as follows:   • st(Y, A) = Y ∪ A[Y ] (= Y ∪ A∈A[Y ] A). When each A ∈ A is contained in some B ∈ B, it is said that A refines B, which is denoted by A ≺ B or B  A.   It is said that A covers Y (or A is a cover of  Y in X) if Y ⊂ A (= A∈A A). When A is a cover of Y in X, st(Y, A) = A[Y ]. A cover of X in X is simply called a cover of X. A cover of Y in X is said to be open (resp. closed) in X depending on whether its members are open (resp. closed) in X. If A is an open cover of X, then A|Y is an open cover of Y and A[Y ] is an open cover of Y in X. When A and B are open covers of X, A ∧ B is also an open cover of X. For covers A and B of X, it is said that A is a refinement of B if A ≺ B, where A is an open (resp. closed) refinement if A is an open (resp. closed) cover. For a space X, we write • cov(X) — the collection of all open covers of X.

2 The

(topological) dimension of M is equal to n.

1.1 Terminology and Notation

5

For U, V ∈ cov(X), we define st(U, V) = {st(U, V) | U ∈ U}. In the case of V = U, st(U, U) is denoted by st U. When st U ≺ V, we call U a star-refinement of V. We inductively define stn U, n ∈ N, as follows: stn U = st(stn−1 U, U), where st0 U = U (so st1 U = st U).  Let (Xγ )γ ∈ be a family of (topological) spaces and X = γ ∈ Xγ . The weak topology on X with respect to (Xγ )γ ∈ is the topology defined as follows: U ⊂ X is open in X ⇔ ∀γ ∈ , U ∩ Xγ is open in Xγ

A ⊂ X is closed in X ⇔ ∀γ ∈ , A ∩ Xγ is closed in Xγ .3 Suppose that X has the weak topology with respect to (Xγ )γ ∈ and the topologies of Xγ and Xγ  agree on Xγ ∩ Xγ  for any γ , γ  ∈ . If Xγ ∩ Xγ  is closed (resp. open) in Xγ for any γ , γ  ∈ , then each Xγ is closed (resp. open) in X and the original topology of each Xγ is a subspace topology inherited from X. In the case Xγ ∩ Xγ   = ∅ for γ = γ  , X is called the topological sum of (Xγ )γ ∈ and denoted  , X is called the by X = γ ∈ Xγ . In the case Xγ ∩ Xγ  = {x0 } for γ = γ wedge sum (or wedge) of (Xγ )γ ∈ at x0 and denoted by X = γ ∈ Xγ .4 When  = {1, . . . , n}, we write as follows: n

i=1

Xi = X1 ⊕ · · · ⊕ Xn and

n 

Xi = X1 ∨ · · · ∨ Xn .

i=1

Let f : A → Y be a map from a closed set A in a space X to another space Y . The adjunction space Y ∪f X is the quotient space (X ⊕ Y )/∼, where X ⊕ Y is the topological sum and ∼ is the equivalence relation corresponding to the decomposition of X ⊕ Y into singletons {x}, x ∈ X \ A, and sets {y} ∪ f −1 (y), y ∈ Y (the latter is a singleton {y} if y ∈ Y \ f (A)). When Y is a singleton, Y ∪f X ≈ X/A. One should note that the adjunction spaces are not Hausdorff in general. It is necessary to require some conditions for the adjunction space to be Hausdorff.

the finest (or largest) topology such that each inclusion Xγ ⊂ X is continuous. (The term “weak topology” is used with a different meaning by functional analysts, etc.) 4 The wedge (sum) is used for a family of pointed spaces with the common base point. 3 I.e.,

6

1 Preliminaries and Background Results

Let f : X → Y be a map. For A ⊂ X and B ⊂ Y , we write       f (A) = f (x)  x ∈ A and f −1 (B) = x ∈ X  f (x) ∈ B . For collections A and B of subsets of X and Y , respectively, we write       f (A) = f (A)  A ∈ A and f −1 (B) = f −1 (B)  B ∈ B . The restriction of f to A ⊂ X is denoted by f |A. It is said that a map g : A → Y extends over X if there is a map f : X → Y such that f |A = g. Such a map f is called an extension of g. Let [a, b] be a closed interval, where a < b (∈ R). A map f : [a, b] → X is called a path (from f (a) to f (b)) in X, where it is said that two points f (a) and f (b) are connected by the path f in X. An embedding (i.e., an injective path) f : [a, b] → X is called an arc (from f (a) to f (b)) in X, and the image f ([a, b]) is also called an arc. Namely, a space is called an arc if it is homeomorphic to I. A space X is path-connected (or arcwise connected) if each pair of distinct points x, y ∈ X are connected by a path (or an arc). It is said that X is locally pathconnected (or locally arcwise connected) if any neighborhood U of each point x ∈ X contains a neighborhood V of x such that each pair of distinct points in V are connected by a path (or an arc) in U (i.e., for each y, z ∈ V , there is a path (or an arc) f : I → U such that f (0) = y and f (1) = z). In this definition, as is easily observed, V may be a path-connected (or an arcwise connected). The (local) arcwise connectedness looks to be stronger than the (local) path-connectedness, but they are the same concepts, that is: Proposition 1.1.1 An arbitrary space X is path-connected if and only if X is arcwise connected. Moreover, X is locally path-connected if and only if X is locally arcwise connected. 5.14.7 For spaces X and Y , we write • C(X, Y ) — the set of (continuous) maps from X to Y . Given subspaces X1 , . . . , Xn ⊂ X and Y1 , . . . , Yn ⊂ Y , a map f : X → Y is said to be a map from (X, X1 , . . . , Xn ) to (Y, Y1 , . . . , Yn ) and is written f : (X, X1 , . . . , Xn ) → (Y, Y1 , . . . , Yn ) if f (X1 ) ⊂ Y1 , . . . , f (Xn ) ⊂ Yn . We write • C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )) — the set of maps from (X, X1 , . . . , Xn ) to (Y, Y1 , . . . , Yn ); • C((X, x0 ), (Y, y0 )) = C((X, {x0}), (Y, {y0 })). For maps f, g : X → Y (i.e., f, g ∈ C(X, Y )), • f  g means that f and g are homotopic (or f is homotopic to g),

1.1 Terminology and Notation

7

that is, there is a map h : X × I → Y such that h0 = f and h1 = g, where ht : X → Y , t ∈ I, are defined by ht (x) = h(x, t), and h is called a homotopy from f to g (between f and g). When g is a constant map, it is said that f is nullhomotopic, which is denoted by f  0. For a homotopy h : X × I → Y , we call h({x} × I), x ∈ X, the tracks of h, where each h({x} × I) is the track of x ∈ X by h. For spaces X and Y , • X  Y means that X and Y are homotopy equivalent, that is, there are maps f : X → Y and g : Y → X such that gf  idX and fg  idY , where f is called a homotopy equivalence and g is a homotopy inverse of f . Then, we also say that X and Y have the same homotopy type or X has the homotopy type of Y . For each f, f  ∈ C(X, Y ) and g, g  ∈ C(Y, Z), we have the following: f  f  , g  g  ⇒ gf  g  f  . A homotopy h between maps f, g ∈ C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )) requires the condition that ht ∈ C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )) for every t ∈ I, that is, h is regarded as the map h : (X × I, X1 × I, . . . , Xn × I) → (Y, Y1 , . . . , Yn ). When there are maps f : (X, X1 , . . . , Xn ) → (Y, Y1 , . . . , Yn ), g : (Y, Y1 , . . . , Yn ) → (X, X1 , . . . , Xn ) such that gf  idX and fg  idY , we write • (X, X1 , . . . , Xn )  (Y, Y1 , . . . , Yn ); • (X, x0 )  (Y, y0 ) means (X, {x0 })  (Y, {y0 }). For A ⊂ X, a homotopy h : X × I → Y is called a homotopy relative to A if h({x} × I) is a singleton for every x ∈ A. When a homotopy from f to g is a homotopy relative to A (where f |A = g|A), it is said that f and g are homotopic relative to A, which is written as follows: f  g rel. A. Let f, g : X → Y be maps and U a collection of subsets of Y (as usual, U ∈ cov(Y )). It is said that f and g are U-close (or f is U-close to g) if       {f (x), g(x)}  x ∈ X ≺ U ∪ {y}  y ∈ Y ,

8

1 Preliminaries and Background Results

which implies that U covers the set {f (x), g(x) | f (x) = g(x)}. A homotopy h is called a U-homotopy if the collection of non-degenerate tracks of h refines U, that is,       h({x} × I)  x ∈ X ≺ U ∪ {y}  y ∈ Y . In this case, U covers the set    h({x} × I)  h({x} × I) is non-degenerate . When a homotopy from f to g is a U-homotopy, it is said that f and g are Uhomotopic (or f is U-homotopic to g), which is written as follows: f U g. When Y = (Y, d) is a metric space, we can define a metric d called the supmetric on the set C(X, Y ) as follows: d(f, g) = sup min{d(f (x), g(x)), 1}.5 x∈X

The metric space (C(X, Y ), d) is denoted by Cd (X, Y ). The topology of Cd (X, Y ) is called the uniform convergence topology, where each f ∈ Cd (X, Y ) has a neighborhood basis consisting of the following:    Bd (f, ε) = g ∈ C(X, Y )  d(f, g) < ε , ε > 0. For ε > 0, it is said that f and g are ε-close or f is ε-close to g if d(f, g) < ε. A homotopy h is called an ε-homotopy if mesh{h({x} × I) | x ∈ X} < ε. When a homotopy from f to g is an ε-homotopy, it is said that f and g are ε-homotopic, which is written as follows: f ε g.

The compact-open topology on C(X, T ) is generated by the sets    K; U  = f ∈ C(X, Y )  f (K) ⊂ U ,

5 In the case where Y is bounded or X is compact, we can employ the definition d(f, g) = supx∈X d(f (x), g(x)). But, in general, the case d(f, g) = ∞ might occur for this definition.

1.1 Terminology and Notation

9

where K is any compact set in X and U is any open set in Y . With respect to the compact-open topology, we have the following: Proposition 1.1.2 (Properties of the Compact-Open Topology) (1) Every map f : Z × X → Y induces the map f˜ : Z → C(X, Y ) defined by f˜(z)(x) = f (x, y). 1.1.1 (2) For each f ∈ C(Z, X) and g ∈ C(Y, Z), the following are continuous: f ∗ : C(X, Y ) → C(Z, Y ), f ∗ (h) = h ◦ f ; g∗ : C(X, Y ) → C(X, Z), g∗ (h) = g ◦ h.

1.1.3(1)

(3) When Y is locally compact, the following composition is continuous: C(X, Y ) × C(Y, Z)  (f, g) → g ◦ f ∈ C(X, Z).

1.1.3(2)

(4) When X is locally compact, the following (evaluation) is continuous: ev : C(X, Y ) × X → Y, ev(f, x) = f (x). So, every map f : Z → C(X, Y ) induces the map f˜ : Z × X → Y defined by 1.1.3(4) f˜(z, x) = f (z)(x). (5) When X is locally compact, the following inequalities hold: w(Y )  w(C(X, Y ))  ℵ0 w(X)w(Y ). In particular, if X is separable locally compact and Y has infinite, then 1.1.3(5) w(C(X, Y )) = w(Y ). (6) When X is compact and Y = (Y, d) is a metric space, the sup-metric on C(X, Y ) is admissible, that is, the compact-open topology is induced by the sup-metric. 1.1.3(6) Regarding C(X, Y ) as a subspace of the product space Y X , we can introduce another topology on C(X, Y ), which is called the pointwise convergence topology. The space C(X, Y ) with the pointwise convergence topology is written as Cp (X, Y ). For each x ∈ X, the evaluation evx : C(X, Y ) → Y is defined by evx (f ) = f (x), which is the restriction of the projection prx : Y X → X. Thus, the pointwise convergence topology is the coarsest topology such that the evaluations evx , x ∈ X, are continuous and it is generated by the sets    x; U  = f ∈ C(X, Y )  f (x) ∈ U , where x is any point of X and U is any open set in Y . Hence, every open set in Cp (X, Y ) is open in C(X, Y ), that is, the pointwise convergence topology is not finer than the compact-open topology.

10

1 Preliminaries and Background Results

Remark 1.1 Proposition 1.1.2(4) does not hold for the space Cp (X, Y ) even if X is compact and Y = R. For example, let X = {0, 1/n | n ∈ N} and c0 : X → R be the constant map with c0 (X) = {0}. For any neighborhood U of c0 in Cp (X, R) and any neighborhood Vof 0 in X, we can choose a finite set F ⊂ X, 0 < ε < 1, and x0 ∈ V \ F so that x∈F x; (−ε, ε) ⊂ U. Let f0 : X → R be the map defined by f0 (x0 ) = 1 and f0 (x) = 0 for any x ∈ X \ {x0 }. Then, (f0 , x0 ) ∈ U × V but ev(f0 , x0 ) = f0 (x0 ) = 1 ∈ (−1, 1).

1.2 Banach Spaces in the Product of Real Lines Throughout this section, let  be an infinite set. Here, we review Banach spaces6 being linear subspaces of the product R . We write • Fin() — the set of all non-empty finite subsets of . Then, note that card Fin() = card . The product space R is a linear space with the following scalar multiplication and addition: R × R  (x, t) → tx = (tx(γ ))γ ∈ ∈ R ; R × R  (x, y) → x + y = (x(γ ) + y(γ ))γ ∈ ∈ R . These operations are continuous with respect to the product topology of R . Namely, R with the product topology is a topological linear space.7 Note that w(R ) = ℵ0 card Fin() = card . For each γ ∈ , we define the unit vector eγ ∈ R by eγ (γ ) = 1 and eγ (γ  ) = 0 for γ  = γ . It should be noticed that {eγ | γ ∈ } is not a Hamel basis for R and its linear span8 is the following:    Rf = x ∈ R  x(γ ) = 0 except for finitely many γ ∈  , N which is a dense linear subspace of R . The subspace RN f of s = R is also denoted by s f , which is consisting of all finite sequences. As is easily observed, the following are equivalent:

(a) (b) (c) (d)

R is metrizable; Rf is metrizable; Rf is first countable; card   ℵ0 .

6A

Banach space is a complete normed linear space. to p. 23. 8 The linear span of B is the linear subspace generated by a set B. 7 Refer

1.2 Banach Spaces in the Product of Real Lines

11

Thus, when  is uncountable, every linear subspace L of R with Rf ⊂ L is nonmetrizable. Moreover, R (or Rf ) is metrizable only when  is countable. In the case card  = ℵ0 , R is linearly homeomorphic to the space of sequences s = RN , that is, there exists a linear homeomorphism between R and s. On the other hand, we have the following proposition: Proposition 1.2.1 Let  be an infinite set. Then, any norm on Rf does not induce the topology inherited from the product topology of R . Consequently, every linear subspace L of R with Rf ⊂ L is not normable. 1.2.1 We can consider various norms defined on linear subspaces of R , which are not compatible with the product topology as in Proposition 1.2.1 above. In general, the unit closed ball and the unit sphere of a normed linear space X = (X,  · ) are denoted by BX and SX respectively, that is,       BX = x ∈ X  x  1 and SX = x ∈ X  x = 1 . The zero vector (the zero element) of X is denoted by 0X , or simply by 0 if there is no possible confusion. The Banach space ∞ () and its closed linear subspaces c() ⊃ c0 () are defined as follows:    • ∞ () = x ∈ R  supγ ∈ |x(γ )| < ∞ with the sup-norm x∞ = sup |x(γ )|; 9 γ ∈

  • c() = x ∈ R  ∃t ∈ R such that ∀ε > 0, |x(γ ) − t| < ε  except for finitely many γ ∈  ;    • c0 () = x ∈ R  ∀ε > 0, |x(γ )| < ε except for finitely many γ ∈  . These are linear subspaces of R but not topological ones as seen above. The space c() is linearly homeomorphic to c0 () × R by the following correspondence: c0 () × R  (x, t) → (x(γ ) + t)γ ∈ ∈ c(). This correspondence and its inverse are Lipschitz with respect to the norm (x, t) = max{x∞ , |t|}.  Furthermore, ∞ f () denotes Rf with this norm. Then, ∞ ∞ f () ⊂ c0 () ⊂ c() ⊂  ().

9 In

some literature, this space is denoted by m().

12

1 Preliminaries and Background Results

For the weight of these spaces, we have the following: w(∞ ()) = 2card  but w(c()) = w(c0 ()) = w(∞ f ()) = card  (cf. Proposition 1.2.2 in [GAGT]). The topology of ∞ f () is different from the topology inherited from the product topology. Indeed, {eγ | γ ∈ } is discrete in ∞ f () but 0 is a cluster point of this set with respect to the product topology. In the case  = N, we write: • ∞ (N) = ∞ — the space of bounded sequences;10 • c(N) = c — the space of convergent sequences; • c0 (N) = c0 — the space of null-sequences (= sequence tending to 0). ∞ ∞ We also write ∞ f (N) = f , where f = s f as sets (linear spaces) but they have different topologies. It should be noted that c and c0 are separable but ∞ is non-separable. When card  = ℵ0 , the spaces ∞ (), c() and c0 () are linearly isometric to these spaces ∞ , c and c0 , respectively.   Here, we regard Fin() asa directed set by ⊂. For x ∈ R , we say that γ ∈ x(γ ) is convergent if γ ∈F x(γ ) F ∈Fin() is convergent and define



x(γ ) =

γ ∈

lim



F ∈Fin()

x(γ ).

γ ∈F

 When x(γ )  0 for all γ ∈ , in order that γ ∈ x(γ ) is convergent, it is necessary   and sufficient that γ ∈F x(γ ) F ∈Fin() is upper bounded, and then 

x(γ ) =

γ ∈

sup



F ∈Fin() γ ∈F

x(γ ).

  Thus, by γ ∈ x(γ ) < ∞, we mean that γ ∈ x(γ ) is convergent.   x(i) should be distinguishedfrom ∞ For x ∈RN , i∈N i=1 x(i). When the  n ∞ sequence x(i) is convergent, we say that x(i) is convergent and i=1 i=1 n∈N define ∞  i=1

x(i) = lim

n→∞

n 

x(i).

i=1

 ∞ Evidently, if  i∈N x(i) is convergent i=1 x(i) is also convergent ∞  then ∞ and x(i) = x(i). However, x(i) is not convergent even if i=1 i=1 x(i) i∈N i∈N is convergent. In fact, due to Proposition 1.2.3 in [GAGT], the following equiva-

10 In

some literature, this space is denoted by m.

1.2 Banach Spaces in the Product of Real Lines

13

lence holds: 

∞ 

x(i) is convergent ⇔

|x(i)| is convergent.

i=1

i∈N

For each p  1, the Banach space p () is defined as follows:    • p () = x ∈ R  γ ∈ |x(γ )|p < ∞ with the norm xp =



1/p |x(γ )|p

.

γ ∈  11 Like ∞ f (), the space Rf with this norm is denoted by f (). p Similarly to c0 (), we have w( ()) = card . When card  = ℵ0 , the Banach space p () is linearly isometric to p = p (N), which is separable. The space 2 () is the Hilbert space with the inner product p

x, y =



x(γ )y(γ ),

γ ∈

which is well-defined because 

|x(γ )y(γ )| 

γ ∈

x22 + y22 < ∞. 2

For 1  p < q < ∞, we have p ()  q ()  c0 ()

as sets (or linear spaces).

These inclusions are continuous because x∞  xq  xp for every x ∈ p (). When  is infinite, the topology of p () is distinct from the one induced by the norm  · q or  · ∞ (i.e., the topology inherited from q () or c0 ()). In fact, the unit sphere Sp () is closed in p () but not closed in q () for any q > p nor in c0 (). — Refer to [GAGT, p. 17]. For 1  p  ∞, we have Rf ⊂ p () p

as sets (or linear spaces). p

Let f () denote the subspace of p () with f () = Rf as sets.

triangle inequality for xp is known as the Minkowski inequality. The proof can be found on pp. 16–17 in [GAGT].

11 The

14

1 Preliminaries and Background Results p

p

p

When  = N, we write f (N) = f . By Proposition 1.2.1, we know f () =  Rf as spaces for any infinite set . In the above, the sequence (xn )n∈N is contained p q in the unit sphere Sp () of f (), which means that Sp () is not closed in f , p

q

f

f

hence f = f as spaces for 1  p < q  ∞. Note that Sp () is a closed subset f

q

of f for 1  q < p. Concerning the convergence of sequences in p (), we have the following: Proposition 1.2.2 For each p ∈ N and x ∈ p (), a sequence (xn )n∈N converges to x in p () if and only if xp = lim xn p and ∀γ ∈ , x(γ ) = lim xn (γ ). n→∞

n→∞

1.2.4

Remark 1.2 It should be noted that Proposition 1.2.2 is valid not only for sequences but also nets, which means that the unit spheres Sp (), p ∈ N, are subspaces of the product space R , whereas neither R nor Rf is metrizable if  is uncountable. Therefore, if 1  p < q  ∞, then Sp () is also a subspace of q (), while, as mentioned above, Sp () of p () is not closed in the space q (). The unit sphere p Sp () of f () is a subspace of Rf (⊂ R ) and also a subspace of q () for f 1  q  ∞. Remark 1.3 The “if” part of Proposition 1.2.2 does not hold for the space c0 () for any infinite set  (but the “only if” part obviously does hold). — Refer to Remark 3 on p. 17 of [GAGT]. Concerning the topological classification of p (), we have the following theorem due to S. Mazur [104]: Theorem 1.2.3 (MAZUR) For each 1 < p < ∞, p () is homeomorphic to 1 (). p p By the same homeomorphism, f () ≈ 1f (), that is, the pair (p (), f ()) is homeomorphic to the pair (1 (), 1f ()). 1.2.5 For each space X, we simply write Cd (X, R) = Cu (X), where the metric d of R is the usual metric induced by the absolute value | · |, that is, Cu (X) is the metric space with the sup-metric   d(f, g) = sup min |f (x) − g(x)|, 1 .12 x∈X

This metric is not induced by a norm. The topology of Cu (X) is the uniform convergence topology. It should be noted that Cu (X) is a linear space but it is not a topological linear space in general. In fact, the scalar multiplication R × Cu ((0, 1]) → Cu ((0, 1]) is not continuous with respect to the uniformly convergence topology.

12 See

Footnote 5 (p. 8).

1.3 Topological Spaces

15

Let f ∈ Cu ((0, 1]) be defined by f (x) = x −1 for each x ∈ (0, 1]. Then, for any t > 0, if x < t, then |tf (x) − 0f (x)| > 1, that is, d(tf, 0f ) = 1.

Among subspaces Cu (X), we have the following Banach space:    • CB (X) = f ∈ C(X)  supx∈X |f (x)| < ∞ with the sup-norm f  = sup |f (x)|. x∈X

As is easily observed, CB (X) is clopen in C(X). Moreover, CB (X) is a component of the space Cu (X) because CB (X) is path-connected as a normed linear space. When X is compact, we have CB (X) = Cu (X). If X is discrete and infinite, then CB (X) = ∞ (X), so CB (N) = ∞ in particular. The space Cp (X) = Cp (X, R) is a topological linear space as a subspace of the product space RX . The topology of Cp (X) is the pointwise convergence topology. The space Cp (N) is none other than the space of sequences s = RN .

1.3 Topological Spaces The following TIETZE–URYSOHN EXTENSION THEOREM is very useful and applied in various fields: Theorem 1.3.1 (TIETZE–URYSOHN) Let A be a closed set in a normal space X. Then, every map f : A → I extends over X. 2.2.2 Let A be a collection of subsets of a space X. It is said that A is locally finite (resp. discrete) in X if each point has a neighborhood U in X which meets only finitely many members (resp. at most one member) of A, that is, card A[U ] < ℵ0 (resp. card A[U ]  1). If A is locally finite (resp. discrete) in X, then so is Acl (= {cl A | A ∈ A}). Moreover, we say that A is σ -locally finite (resp. σ -discrete) in X if A is a countable union of locally finite (resp. discrete) subcollections. Theorem 1.3.2 (A.H. STONE) Every open cover of a metrizable space has a locally finite and σ -discrete open refinement. 2.3.1 Theorem 1.3.3 (BING; NAGATA–SMIRNOV) For a regular space X, the following conditions are equivalent: (a) X is metrizable; (b) X has a σ -discrete open basis; (c) X has a σ -locally finite open basis.

2.3.4

The equivalence of (a) and (b) in the above theorem is called the BING METRIZANAGATA–SMIRNOV METRIZATION THEOREM. Separable metrizable spaces are characterized as follows: TION T HEOREM and the equivalence of (a) and (c) is called the

16

1 Preliminaries and Background Results

Theorem 1.3.4 (URYSOHN METRIZATION) A space is separable metrizable if and only if it is regular and second countable. 2.3.5 For an infinite set , the hedgehog J () is defined as the following closed subspace of 1 (): J () =

 γ ∈

  Ieγ = x ∈ 1 ()  x(γ ) ∈ I for all γ ∈ ,  x(γ ) = 0 at most one γ ∈  .

Theorem 1.3.5 Every metrizable space X can be embedded into J ()N , where w(x)  card . If X is separable, then X can be embedded into the Hilbert cube Q, and hence into s = RN . 2.3.7 2.3.8 A perfect map f : X → Y is a closed map such that f −1 (y) is compact for each y ∈ Y . Concerning metrizability, the following is known: Proposition 1.3.6 (1) The perfect image of a metrizable space is metrizable, that is, if f : X → Y is a surjective perfect map of a metrizable space X, then Y is also metrizable. 2.4.5(1)

(2) A space X is metrizable if it is a locally finite union of metrizable closed subspaces. 2.4.5(2) A map f : X → Y is said to be proper if f −1 (K) is compact for every compact set K ⊂ Y . In the case that X and Y are metrizable, these concepts coincide, that is: Proposition 1.3.7 For a map f : X → Y between metrizable spaces, the following are equivalent: (a) f : X → Y is perfect; (b) f : X → Y is proper; (c) Any sequence (xn )n∈N in X has a convergent subsequence if (f (xn ))n∈N is convergent in Y . 2.1.5 2.1.6 It should be remarked that a map between locally compact spaces is proper if it is close to a proper map, that is: Proposition 1.3.8 Let f : X → Y be a proper map and U be a locally finite open cover of Y such that every U ∈ U has the compact closure in Y (so X and Y should be locally compact). If a map g : X → Y is U-close to f , then g is also proper. 2.9.6

It is said that proper maps f, g : X → Y are properly homotopic (or f is properly homotopic to g) if there is a proper homotopy h : X × I → Y from f to g. Then, we write f p g.

1.3 Topological Spaces

17

In the above, if h is a proper U-homotopy (resp. a proper homotopy relative to A ⊂ X), we say that f and g are properly U-homotopic (resp. properly homotopic relative to A) and write p

f U g



 resp. f p g rel. A .

It is said that X and Y are proper homotopy equivalent if there are maps f : X → Y and g : Y → X such that gf p idX and fg p idY , where f is called a proper homotopy equivalence and g is a proper homotopy inverse of f . Then, it is also said that X and Y have the same proper homotopy type or X has the proper homotopy type of Y , which is written as follows: X p Y. For proper maps f, f  : X → Y and g, g  : Y → Z, f p f  , g p g  ⇒ gf p g  f  . Applying Proposition 1.3.8 above, to a homotopy h : X × I → Y and a constant homotopy h0 prX : X × I → Y , we have the following: Proposition 1.3.9 Let U be a locally finite open cover of Y such that every U ∈ U has the compact closure in Y . Then, a U-homotopy h : X × I → Y is proper if " ! h0 : X → Y is proper. The (metric) completion of a metric space X = (X, d) is a complete metric  = (X,  d) ˜ containing X as a dense set and as a metric subspace (i.e., d is a space X ˜ restriction of d). Theorem 1.3.10 Every metric space has a completion.

2.3.10

A space X is paracompact if each open cover of X has a locally finite open refinement.13 The following is a direct consequence of the Stone Theorem 1.3.2: • Every metrizable space is paracompact. We have the following characterization: Theorem 1.3.11 A space X is paracompact if and only if every open cover of X has an open star-refinement. 2.6.3 A space X is collectionwise normal if, for each discrete collection F of closed sets in X, there is a pairwise disjoint collection {UF | F ∈ F } of open sets in X such that F ⊂ UF for each F ∈ F. Obviously, every collectionwise normal space is normal. In the definition of collectionwise normality, we can take {UF | F ∈ F } so as to be discrete in X.

13 Recall

that spaces are assumed to be Hausdorff.

18

1 Preliminaries and Background Results   Indeed, choose an open set V in X so that F ⊂ V ⊂ cl V ⊂ F ∈F UF . Then, F ⊂ V ∩ UF for each F ∈ F and {V ∩ UF | F ∈ F } is discrete in X.

It is said that X is hereditarily paracompact if every subspace of X is paracompact. Proposition 1.3.12 (Results on Paracompact Spaces) (1) Every paracompact space is collectionwise normal. 2.6.1 (2) A space is paracompact if it is a locally finite union of paracompact closed subspaces. 2.6.7(1) (3) Every Fσ subspace of a paracompact space is paracompact. 2.6.7(2) (4) A space X is hereditarily paracompact if every open subspace of X is paracompact. 2.6.7(3) (5) Every locally (completely) metrizable paracompact space is (completely) metrizable. 2.6.7(4) From (1) above, it follows that: • Every metrizable space is collectionwise normal. By virtue of (5), the metrizability in the definition of an n-manifold can be replaced with the paracompactness. A closed set A ⊂ X is called a zero set in X if A = f −1 (0) for some map f : X → R. The complement of a zero set in X is called a cozero set. A normal space X is perfectly normal if every closed set in X is Gδ in X (i.e., every open set in X is Fσ in X), which is equivalent to the condition that every closed set in X is a zero set in X, i.e., every open set in X is a cozero set in X (cf. Theorem 2.2.6 in [GAGT]). As is easily observed, we have: • Every metrizable space is perfectly normal. Moreover, we have: Theorem 1.3.13 Every perfectly normal paracompact space is hereditarily paracompact. 2.6.8 A real-valued function f : X → R is said to be lower semi-continuous (abbrev. l.s.c.) (resp. upper semi-continuous (abbrev. u.s.c.)) if f −1 ((t, ∞)) (resp. f −1 ((−∞, t))) is open in X for each t ∈ R. Evidently, f : X → R is continuous if and only if f is l.s.c. and u.s.c. Theorem 1.3.14 Let g, h : X → R be real-valued functions on a paracompact space X such that g is u.s.c., h is l.s.c., and g(x) < h(x) for each x ∈ X. Then, there exists a map f : X → R such that g(x) < f (x) < h(x) for each x ∈ X. Moreover, given a map f0 : A → R of a closed set A in X such that g(x) < f0 (x) < h(x) for each x ∈ A, the map f can be an extension of f0 . 2.7.6 A space X is called a Baire space if X satisfies one of the following equivalent conditions: (a) The countable intersection of open dense sets in X is dense; (b) Every countable intersection of dense Gδ sets in X is dense;

1.3 Topological Spaces

19

(c) The countable union of closed sets in X without interior points has no interior point; (d) If a countable union of closed sets in X has an interior point, then at least one of them has an interior point. The following is easily proved: • Every open subspace and every dense Gδ subspace of a Baire space are also Baire spaces. The following BAIRE CATEGORY THEOREM is very useful and important. Theorem 1.3.15 (BAIRE) Every completely metrizable space is a Baire space. Consequently, it cannot be written as a union of countably many closed sets without interior points. 2.5.1 A metrizable space X is said to be absolutely Gδ if X is Gδ in an arbitrary metrizable space which contains X as a subspace. This concept characterizes the complete metrizability, that is: Theorem 1.3.16 A metrizable space is completely metrizable if and only if it is 2.5.2 absolutely Gδ . For the complete metrizability, the following holds: Theorem 1.3.17 Let X = (X, d) be a metric space and A ⊂ X. (1) If A is completely metrizable, then A is Gδ in X. (2) If X is complete and A is Gδ in X, then A is completely metrizable.

2.5.3

The following two extension theorems are due to M. Lavrentieff: Theorem 1.3.18 (LAVRENTIEFF Gδ -EXTENSION) Every map f : A → Y from A ⊂ X to a completely metrizable space Y extends over a Gδ set G in X such that A ⊂ G ⊂ cl A. 2.5.7 Theorem 1.3.19 (LAVRENTIEFF HOMEOMORPHISM EXTENSION) Let X and Y be completely metrizable spaces with A ⊂ X and B ⊂ Y . Then, every homeomorphism f : A → B extends to a homeomorphism f˜ : G → H between Gδ sets in X and Y such that A ⊂ G ⊂ cl A and B ⊂ H ⊂ cl B. 2.5.8 Let P be a property for subsets of a space X. It is said that X has property P locally if each x ∈ X has a neighborhood U in X which has property P. A property P for open sets in X is said to be G-hereditary if the following conditions are satisfied: (G-1) (G-2) (G-3)

If U has property P, then every open subset of U has P; If U and V have property P, then U ∪ V has property P;  If {Uλ | λ ∈ } is discrete in X and each Uλ has property P, then λ∈ Uλ has property P.

20

1 Preliminaries and Background Results

The following is very useful to show that a space has a certain property: Theorem 1.3.20 (E. MICHAEL) Let P be a G-hereditary property for open sets in a paracompact space X. If X has property P locally, then X itself has property P. 2.6.5 We can apply the above theorem even to a property P for closed sets in X by introducing the property P◦ as follows: an open set U has property P◦ ⇔ cl U has property P. def

It is said that P is F -hereditary if it satisfies the following conditions: (F-1) (F-2) (F-3)

If A has property P, then every closed subset of A has property P; If A and B have property P, then A ∪ B has property P;  If {Aλ | λ ∈ } is discrete in X and each Aλ has property P, then λ∈ Aλ has property P.

Evidently, if property P is F -hereditary, then P◦ is G-hereditary. Therefore, Theorem 1.3.20 yields the following corollary: Corollary 1.3.21 (E. MICHAEL) Let P be an F -hereditary property for closed sets in a paracompact space X. If X has property P locally, then X itself has property P. 2.6.6 Proposition 1.3.22 (Refinements by Open Balls) (1) Let X be a metrizable space and U be an open cover of X. Then, X has an admissible bounded metric ρ such that {Bρ (x, 1) | x ∈ X} ≺ U. Moreover, for given d ∈ Metr(X), ρ can be chosen so that ρ  d, which means that if d is complete then ρ is. 2.7.7(1) (2) Let X = (X, d) be a metric space. For each open cover U of X, there is a map γ : X → (0, 1) such that {B(x, γ (x)) | x ∈ X} ≺ U. 2.7.7(2)

1.4 Linear Spaces and Convex Sets Let E be a linear space.14 It is said that finitely many distinct points x1 , . . . , xn ∈ E are affinely (or geometrically) independent provided that, for t1 , . . . , tn ∈ R, n  i=1

14 Here,

ti xi = 0,

n 

ti = 0 ⇒ t1 = · · · = tn = 0,

i=1

we only consider linear spaces over R.

1.4 Linear Spaces and Convex Sets

21

that is, x1 − xn , . . . , xn−1 − xn are linearly independent. A set A ⊂ E is said to be affinely (or geometrically) independent if all finite distinct points of A are affinely independent. We call F ⊂ E a flat15 (resp. a convex set) if the straight line through (resp. the line segment between) every pair of distinct points of F is contained in F , that is, ∀x, y ∈ F, ∀t ∈ R (resp. ∀t ∈ I), (1 − t)x + ty ∈ F. It is easy to see that F ⊂ E is a flat if and only if F − x is a linear subspace of E for some (or any) x ∈ F , where F − x = F − y for any x, y ∈ F (cf. [GAGT, Proposition 3.1.1, Corollary 3.1.2]). The dimension dim F of a flat F is defined as the dimension of the linear subspace F − x for some (or any) x ∈ F . In the infinitedimensional case, dim F is the cardinal of a maximal affinely independent subset. For a subset A ⊂ E, the smallest flat fl A containing A is called the flat hull16 of A and the smallest convex set A containing A is called the convex hull of A. When A = {x1 , . . . , xn }, we write A = x1 , . . . , xn  and x1 , . . . , xn  =

 n

i=1 z(i)xi

   z ∈ n−1 .

The dimension dim C of a convex set C is defined as the dimension of fl C, where fl C can be obtained by the following: Proposition 1.4.1 For a convex set C ⊂ E,    fl C = (1 − t)x + ty  x, y ∈ C, t ∈ R .

3.2.1

For a convex set C ⊂ E, the radial interior rint C is defined as follows:    rint C = x ∈ C  ∀y ∈ C, ∃δ > 0 such that (1 + δ)x − δy ∈ C . When C = x1 , . . . , xn , we have rintx1 , . . . , xn  =

 n

i=1 z(i)xi

   z ∈ n−1 ∩ (0, ∞)n .

See [GAGT, p. 79]. In particular, rint{x} = {x} and rintx, y = x, y \ {x, y} if x = y. The radial interior of C can also be defined as    rint C = x ∈ C  ∀y ∈ C, ∃z ∈ C such that x ∈ rinty, z .

15 A

flat is also called an affine set, a linear manifold, or a linear variety. flat hull is also called the affine hull.

16 The

22

1 Preliminaries and Background Results

The following set is called the face of C at x ∈ C:  Cx = y ∈ C  = y∈C

   ∃δ > 0 such that (1 + δ)x − δy ∈ C    ∃z ∈ C such that x ∈ rinty, z .

Then, rint C = {x ∈ C | Cx = C}. It is said that a point x ∈ E is linearly accessible from C if rintx, y ⊂ C for some y ∈ C. The radial closure rcl C of C is the set of all linearly accessible points from C. It should be noted that rcl C ⊂ fl C by Proposition 1.4.1, hence fl rcl C = fl C. We have the following inclusions: rint C ⊂ C ⊂ rcl C ⊂ fl C. The radial boundary ∂C of C is defined as ∂C = rcl C \ rint C. Proposition 1.4.2 Let C ⊂ E be a convex set. If x ∈ rint C, y ∈ rcl C, and 0  t < 1, then (1 − t)x + ty ∈ rint C, i.e., x, y \ {y} ⊂ rint C. 3.2.3 Proposition 1.4.3 For every two convex sets C, D ⊂ E, rint(C ∩ D) = rint C ∩ rint D if rint C ∩ rint D = ∅; (C ∩ D)x = Cx ∩ Dx for any x ∈ C ∩ D.

3.2.7(2) 3.2.7(1)

When a convex set C is contained in a flat F ⊂ E, the following set is called the core of C in F :17 coreF C = {x ∈ C | ∀y ∈ F, ∃δ > 0 such that |t| < δ ⇒ (1 − t)x + ty ∈ C} = {x ∈ C | ∀y ∈ F, ∃s > 0 such that (1 − s)x + sy ∈ C}, where the last equality comes from the convexity of C. Then, we have the following fact (cf. Fact on p. 87 in [GAGT]): • coreF C = ∅ ⇔ fl C = F . When F = E, we can omit the phrase “in E” and simply write core C. Proposition 1.4.4 For every convex set C ⊂ E, corefl C C = rint C, which is also convex. Then, core C = ∅ implies core C = rint C, and core core C = core C. 3.3.2 For a convex set C in a linear space E with 0 ∈ core C, the Minkowski functional pC : E → R+ for C can be defined as follows: pC (x) = inf{s > 0 | x ∈ sC} = inf{s > 0 | s −1 x ∈ C}. 17 The

core is defined for any subset of F , but we assume that C is convex.

1.4 Linear Spaces and Convex Sets

23

Then, pC is sublinear, that is, the following hold: pC (x + y)  pC (x) + pC (y) for each x, y ∈ E, pC (tx) = tpC (x) for each x ∈ E and t > 0. Using the Minkowski functional, the radial interior, the radial closure, and the radial boundary can be written as follows: −1 −1 −1 rint C = core C = pC ([0, 1)), rcl C = pC (I) and ∂C = pC (1),

whose proof can be read in [GAGT, Proposition 3.3.4]. Let X and Y be flats (resp. convex sets) in linear spaces E and E  , respectively. A function f : X → Y is said to be affine (or linear in the affine sense) provided that f ((1 − t)x + ty) = (1 − t)f (x) + tf (y) for every x, y ∈ X and t ∈ R (resp. t ∈ I). For convex sets C ⊂ E and D ⊂ E  , every affine function f : C → D can be uniquely extended to an affine function f˜ : fl C → fl D between their flat hull. A topological linear space E is a linear space equipped with a topology such that the algebraic operations, the addition (x, y) → x + y and the scalar multiplication (t, x) → tx, are continuous.18 A neighborhood basis at 0 in a topological linear space can be characterized as follows: Proposition 1.4.5 Let E be a topological linear space. Then, a neighborhood basis U at 0 in E has the following properties: (i) (ii) (iii) (iv) (v)

For each U, V ∈ U, U ∩ V contains some W ∈ U; For each U ∈ U, there is some V ∈ U such that V + V ⊂ U ; For each U ∈ U, there is some V ∈ U such that [−1, 1]V ⊂ U ; For each x ∈ E and U ∈ U, there is some a > 0 such that x ∈ aU ;  U = {0}.

Conversely, let U be a collection of subsets of a linear space E satisfying the above properties (i)–(v). Then, E has a topology which makes E a topological linear space such that U is a neighborhood basis at 0. 3.4.1 It is said that A ⊂ E is circled if [−1, 1]A ⊂ A (i.e., tA ⊂ A for every t ∈ [−1, 1]). In Proposition 1.4.5 above, W = [−1, 1]V is a circled neighborhood of 0 in E.

18 Recall that topological spaces are assumed to be Hausdorff. For topological linear spaces (more generally for topological groups), it suffices to assume the axiom T0 , which implies the regularity (cf. Proposition 2.4.2 in [GAGT]).

24

1 Preliminaries and Background Results

Concerning convex sets in a topological linear space E, we have the following: Proposition 1.4.6 For each convex set C in a topological linear space E with −1 ([0, 1)) = 0 ∈ core C, the Minkowski functional pC : E → R is continuous, pC −1 −1 int C = rint C, pC (I) = cl C = rcl C, and pC (1) = bd C = ∂C. 3.4.12 Theorem 1.4.7 Every open convex set V in a topological linear space E is homeomorphic to E itself. 3.4.15 Concerning the finite-dimensionality, we have the following: Proposition 1.4.8 Every finite-dimensional flat F in any linear space E has the unique topology such that the following operation is continuous: F × F × R  (x, y, t) → (1 − t)x + ty ∈ F. With respect to this topology, every affine bijection f : Rn → F is a homeomorphism, where n = dim F . 3.5.1 As a corollary, we have the following: Corollary 1.4.9 Every finite-dimensional linear space E has the unique topology compatible with the algebraic operations (addition and scalar multiplication), and then it is linearly homeomorphic to Rn , where n = dim E. 3.5.2 Concerning the radial interior of a finite-dimensional convex set, we have the following: Proposition 1.4.10 For every finite-dimensional convex set C ⊂ E, rint C = intfl C C with respect to the unique topology for fl C as in Proposition 1.4.8. 3.5.5 For the radial closure, we have the following similar result, which was an overlooked result in [GAGT]: Proposition 1.4.11 For every finite-dimensional convex set C ⊂ E, rcl C = clfl C C with respect to the unique topology for fl C as in Proposition 1.4.8, hence ∂C = bdfl C C. Proof Let n = dim C. Since fl C is affinely homeomorphic to Rn , we may assume that C ⊂ Rn . Since int C = ∅ as in the proof of [GAGT, Proposition 3.5.5], there is v ∈ int C. Then, choose δ > 0 so that v − u < δ ⇒ u ∈ C, where  ·  is the norm for Rn . As is easily observed, rcl C ⊂ cl C [GAGT, Proposition 3.4.7(1)]. To see the converse inclusion, for each x ∈ cl C and 0 < t < 1, let z = (1 − t)x + tv. Take y ∈ C so that x − y < tδ/(1 − t) and let u=

1−t 1−t 1−t 1 z− y= x+v− y. t t t t

1.4 Linear Spaces and Convex Sets

25

Then, (1 − t)y + tu = z and   1 − t 1−t   = 1 − t x − y < δ, x − y u − v =    t t t which implies u ∈ C, so z ∈ C. Thus, rintx, v ⊂ C, hence x ∈ rcl C.

" !

The finite-dimensionality of a topological linear space can be topologically characterized as follows: Theorem 1.4.12 Let E be a topological linear space. The following are equivalent: (a) E is finite-dimensional; (b) E is locally compact; (c) 0 ∈ E has a totally bounded neighborhood in E.

3.5.9

In the above, a subset A of a topological linear space E is totally bounded provided, for each neighborhood U of 0 ∈ E, there exists a finite set M ⊂ E such that  A ⊂ U + M = x∈M (U + x). In this definition, M can be taken as a subset of A. Indeed, there is a neighborhood V  of 0 in E such that V − V ⊂ U . Then, we can take x1 , . . . , xn ⊂ E so that A ⊂ ni=1 (V + xi ) and (V + xi ) ∩ A = ∅ for every i = 1, . . . , n. For each i = 1, . . . , n, choose yi ∈ (V +xi )∩A. Since xi ∈ −Vi +yi , it follows that V + xi ⊂ V − V + yi = (V − V ) + yi ⊂ U + yi .  Hence, we have A ⊂ ni=1 (U + yi ). For the metrizability of a topological linear space, we have the following very simple characterization: Theorem 1.4.13 A topological linear space E is metrizable if and only if 0 ∈ E 3.6.1 has a countable neighborhood basis. A metric d on a linear space E is said to be invariant if d(x + z, y + z) = d(x, y) for every x, y, z ∈ E. It is equivalent to this condition that d(x, y) = d(x − y, 0) for every x, y ∈ E. With respect to an invariant metric on E, addition on E is clearly continuous. Moreover, scalar multiplication on E is continuous if and only if the following three conditions are satisfied: (i) d(xn , 0) → 0 ⇒ ∀t ∈ R, d(txn , 0) → 0; (ii) tn → 0 ⇒ ∀x ∈ E, d(tn x, 0) → 0; (iii) d(xn , 0) → 0, tn → 0 ⇒ d(tn xn , 0) → 0. An invariant metric d on E satisfying these conditions is called a linear metric. A metric linear space E = (E, d) is a linear space E with a linear metric d.

26

1 Preliminaries and Background Results

Evidently, every linear metric on a linear space E induces a topology of E that makes E a topological linear space. The following is a combination of the Fact on p. 112 of [GAGT] and Theorem 3.6.6 of [GAGT]: Theorem 1.4.14 Every admissible invariant metric for a (completely) metrizable topological linear space is a (complete) linear metric. A functional  ·  : E → R on a linear space E is called an F -norm19 if it satisfies the following conditions: (F1 ) (F2 ) (F3 ) (F4 ) (F5 ) (F6 )

x  0; x = 0 ⇒ x = 0; |t|  1 ⇒ tx  x; x + y  x + y; xn  → 0 ⇒ txn  → 0;20 tn → 0 ⇒ tn x → 0.

It should be observed that the converse of (F2 ) is true because 0 = 0 by the last condition (F6 ). Then, x = 0 if and only if x = 0. A linear space E given an F -norm  ·  is called an F -normed linear space. Every norm is an F -norm, hence every normed linear space is an F -normed space. An F -norm  ·  induces the linear metric d(x, y) = x − y. Then, every F -normed linear space is a metric linear space. An F -norm on a topological linear space E is said to be admissible if it induces the topology for E. Theorem 1.4.15 A topological linear space has an admissible F -norm if and only if it is metrizable. 3.6.3 Every metrizable topological linear space has an admissible F -norm with the following stronger condition than (F3 ): (F3∗ )

x = 0, |t| < 1 ⇒ tx < x,

which implies that sx < tx for each x = 0 and 0 < s < t. Due to Theorem 1.4.15, every metrizable topological linear space has an admissible invariant metric. When a topological linear space E is metrizable, the total boundedness of a subset A ⊂ E is equivalent to the total boundedness with respect to an invariant metric d ∈ Metr(E), that is: Fact A is totally bounded in E if and only if for each ε > 0, A is covered by finite subsets A1 , . . . , An with diam  ε. In fact, since d is invariant, it follows that Bd (x, ε) = Bd (0, ε) + x for each x ∈ E and ε > 0.

19 In

the Köthe book [(21)], it is called an (F)-norm. mentioned on p. 112 of [GAGT], (F5 ) is unnecessary because it comes from (F3 ) and (F4 ). But we follow the definition in Köthe’s book [(21)].

20 As

1.5 Cell Complexes and Simplicial Complexes

27

If nA is totally bounded in E, then for each ε > 0, there is a1 , . . . , an ∈ A such that A ⊂ i=1 B(ai , ε/2), where diam B(ai , ε/2)  ε. Hence, A is totally bounded with respect to d. Conversely, assume A is totally bounded with respect to d. For each open neighborhood U of 0 in E, choose ε > 0 so that B(0, ε) ⊂ U . Then, A is covered by finite subsets A1 , . . . , A n with diam  ε. Take ai ∈ Ai , i = 1, . . . , n. Since Ai ⊂ B(ai , ε) ⊂ U + ai , we have A ⊂ ni=1 (U + ai ). Hence A is totally bounded in E.

A topological linear space E is locally convex if 0 ∈ E has a neighborhood basis consisting of (open) convex sets; equivalently, open convex sets make up an open basis for E. A completely metrizable locally convex topological linear space is called a Fréchet space. Theorem 1.4.16 (BARTLE–GRAVES–MICHAEL) Let E be a locally convex metric linear space and F be a linear subspace of E that is complete (so a Fréchet space). Then, E ≈ F × E/F . In particular, E ≈ R × G for some metric linear space G. 3.8.9 Theorem 1.4.17 (BANACH–MAZUR, KLEE) For every Banach space E, there exists a continuous linear surjection q : 1 () → E, where card  = dens E. 3.8.11

As a corollary of the above two theorems, we have the following: Corollary 1.4.18 For every Banach space E, there exists a Banach space F such that E × F ≈ 1 (), where card  = dens E. 3.8.12

1.5 Cell Complexes and Simplicial Complexes Let E be a linear space. The convex hull C = A of a non-empty finite set A ⊂ E is called a (convex) linear cell (or just a cell). An n-dimensional cell is called an n-cell. According to Proposition 1.4.8, the flat hull fl C (= fl A) has the unique topology such that the following operation is continuous: fl C × fl C × R  (x, y, t) → (1 − t)x + ty ∈ fl C. Then, C is a compact subspace of fl C with this topology because C = A is the image of A × A × I by this operation. Hence, rcl C = C by Proposition 1.4.11. Recall rint C = intfl C C by Propositions 1.4.10. Hence, ∂C = C \ rint C. Note Due to [GAGT, Proposition 3.5.8], a cell C = A has the unique topology such that the following operation is continuous: C × C × I  (x, y, t) → (1 − t)x + ty ∈ C.

Using affine functionals, we can characterize cells as follows: Proposition 1.5.1 A non-degenerate C ⊂ E is a cell if and only if fl C is finitedimensional, x + R+ (y − x) ⊂ C for each distinct points x, y ∈ C, and there

28

1 Preliminaries and Background Results

are finitely many 1 , . . . , fk : fl C → R such that  non-constant affine functionals f C = fl C ∩ ki=1 fi−1 (R+ ), where rint C = fl C ∩ ki=1 fi−1 ((0, ∞)).21 4.1.8 Every cell C has the smallest finite subset C (0) such that C (0)  = C. Then, ⊂ ∂C. Each point of C (0) is called a vertex of C. When C (0) = {v1 , . . . , vn } is affinely independent, C is called a simplex and dim C = n − 1 (n = card C (0) ). An n-dimensional simplex is called an n-simplex. A cell D is said to be a face of C if D = Cx for some x ∈ C; this is denoted by D  C or C  D. If D  C, then D (0) = C (0) ∩ D (cf. [GAGT, Proposition 4.1.4]). A face D  C is called a proper face of C if D = C; this is denoted by D < C or C > D. Then, ∂C = D · · · > Cn ∈ K, vC1 > · · · > vCn , and each Ci is the smallest cell of K containing vC1 , . . . , vCn . In the above proof (Uniqueness), let x = σˆ . Then, C1 = cK (σˆ ) and rint σ ⊂ rint C1 , hence C > C1 and vC >vC1 . Thus, vC σ = , vCn  ∈ K  . Therefore, Ln ⊂ K  . Since {rint σ | σ ∈ n∈N Ln } = |K|, it vC , vC1 , . . . follows that n∈N Ln = K  .

1.6 Simplicial Subdivisions

35

A cell complex K with such an order as in Theorem 1.6.1 is called an ordered cell complex. In the case where K is simplicial complex, such an order gives a total order on the vertices of each simplex in K. Then, an ordered simplicial complex is a simplicial complex K with an order on K (0) such that σ (0) is totally ordered for each σ ∈ K. For cell complexes K and L, we define the product cell complex K ×c L as follows:    K ×c L = C × D  C ∈ K, D ∈ L . Then, |K ×c L| = |K| × |L| as sets but |K ×c L| = |K| × |L| as spaces in general (cf. [GAGT, Proposition 4.3.2]). Theorem 1.6.2 For each pair of cell complexes K and L, |K ×c L| = |K| × |L| as spaces if (1) both K and L are locally countable or (2) one of K and L is locally finite. 4.3.1 For simplicial complexes K and L, K ×c L is not simplicial but does have a simplicial subdivision. In fact, giving an order on (K ×c L)(0) = K (0) × L(0) so that K ×c L is an ordered cell complex, we can obtain a simplicial subdivision of K ×c L with the same vertices as K ×c L (Theorem 1.6.1). When K and L are ordered simplicial complexes, such an order on K (0) × L(0) can be defined as follows: (u, v)  (u , v  ) ⇔ u  u , v  v  . def

Let K ×s L denote the simplicial subdivision of K ×c L defined by using this order, which can be written as follows:   K ×s L = (u1 , v1 ), . . . , (uk , vk )  ∃σ ∈ K, ∃τ ∈ L

 such that u1  · · ·  uk ∈ σ (0), v1  · · ·  vk ∈ τ (0) .

This simplicial complex K ×s L is called the product simplicial complex of K and L. Overlapping cell complexes, we can obtain a cell complex as in the following proposition: Proposition 1.6.3 For each pair of cell complexes K1 and K2 , the following K is a cell complex with |K| = |K1 | ∩ |K2 | as sets:    K = C ∩ D  C ∈ K1 , D ∈ K2 , C ∩ D = ∅ , where if K0 is a subcomplex of both K1 and K2 , then K0 is a subcomplex of K. Moreover, if each cell of Ki is covered by only finitely many cells of K3−i for i = 1, 2, then |K| is a closed subspace of both |K1 | and |K2 |. 4.2.12

36

1 Preliminaries and Background Results

As a special case, the following can be obtained: Theorem 1.6.4 Let K1 and K2 be cell complexes such that |K1 | = |K2 | as spaces: Then, K1 and K2 have a common subdivision K. In addition, if K0 is a subcomplex of both K1 and K2 , then K0 is also a subcomplex of K. 4.2.11 The following is a combination of Theorems 1.6.1 and 1.6.4: Theorem 1.6.5 Each pair of cell complexes K1 and K2 with |K1 | = |K2 | as spaces have a common simplicial subdivision K. In addition, if they have a common (simplicial) subcomplex K0 , then K can be taken so as to contain K0 as a subcomplex. 4.2.13 The following is obtained from the definition of a subpolyhedron and Theorem 1.6.5 (which was overlooked in [GAGT]): Proposition 1.6.6 Let K be a cell complex. For each subpolyhedron P of |K|, there exists a simplicial subdivision K   K such that P is triangulated by a subcomplex L of K  , that is, P = |L|. " ! The following lemma is useful to construct simplicial subdivision of cell complexes: Lemma 1.6.7 Let C be an n-cell and v0 ∈ rint C. Given a triangulation L of ∂C (i.e., L  F (∂C)), let K = v0 ∗ L, that is,    K = L ∪ {v0 } ∪ v0 σ  σ ∈ L . Then, K is a triangulation of C (i.e., K  F (C)) such that L is a full subcomplex of K. 4.6.1 Using Lemma 1.6.7 in the skeletonwise construction, we can obtain the following: Proposition 1.6.8 Let K be a cell complex and L be a simplicial subdivision of a subcomplex L ⊂ K. Given vC ∈ rint C for each C ∈ K \ L, there is a simplicial subdivision K  of K such that L is a full subcomplex of K  and    K (0) = L(0) ∪ K (0) ∪ vC  C ∈ K \ L .

4.6.2

In the case L = ∅ in Proposition 1.6.8 above, the obtained subdivision is written as follows:    K  = vC1 , . . . , vCn   C1 < · · · < Cn ∈ K , which is called a derived subdivision of K. For a simplicial K, as n complex −1 v , where vσ ∈ rint σ , σ ∈ K, we can take the barycenter σˆ = n i i=1 σ = v1 , . . . , vn  (n = dim σ + 1). Then, the obtained derived subdivision K  of K is called the barycentric subdivision of K and denoted by Sd K.

1.7 The Metric Topology of Polyhedra

37

When L = L in Proposition 1.6.8, the obtained subdivision K  is called a derived subdivision of K relative to L, where    K  = L ∪ vC1 , . . . , vCn   C1 < · · · < Cn ∈ K \ L   ∪ v1 , . . . , vm , vC1 , . . . , vCn   v1 , . . . , vm  ∈ L,  C1 < · · · < Cn ∈ K \ L with v1 , . . . , vm  < C1 . When K and L are simplicial and vσ = σˆ for each σ ∈ K \ L, the obtained derived subdivision of K relative to L is called the barycentric subdivision of K relative to L and denoted by SdL K. Note that L is a full subcomplex of SdL K (Proposition 1.6.8). For a simplicial complex K and x ∈ |K| \ K (0) , we can define the following simplicial subdivision of K:    Kx = (K \ K[x]) ∪ xσ  σ ∈ St(cK (x)) \ K[x] , where xσ is the join of x and σ , i.e., xσ = {x} ∪ σ (0) . The operation K → Kx (or Kx itself) is called the starring of K at x. Then, as is easily observed, | St(x, Kx )| = | St(cK (x), K)| and Lk(x, Kx ) = F (∂cK (x)) ∗ Lk(cK (x), K). A subdivision obtained by finite starrings is known as a stellar subdivision. In general, (Kx )y = (Ky )x for distinct two points x, y ∈ |K| \ K (0). When K is finite, every derived subdivision of K is a stellar subdivision. The following theorem is well known and very important (e.g., the proof of the paracompactness of polyhedra is dependent on this theorem). Almost all textbooks mentioning or citing the result do not treat the proof. For the proof, we used to refer to the original article by J.H.C. Whitehead [154]. But now, the complete proof is found in [GAGT, Section 4.7]. Theorem 1.6.9 (J.H.C. WHITEHEAD) Let K be an arbitrary simplicial complex. For any open cover U of |K|, K has a simplicial subdivision K  such that SK  ≺ U. 4.7.11

1.7 The Metric Topology of Polyhedra Let K be a simplicial complex. There exist maps βvK : |K| → I, v ∈ K (0) , such that every βvK is affine on each simplex of K,   x= βvK (x)v, βvK (x) = 1, and v∈K (0)

v∈K (0)

  cK (x)(0) = v ∈ K (0)  βvK (x) > 0 , 

38

1 Preliminaries and Background Results

where βvK (x) is called the barycentric coordinate of x at v with respect to K. Note that (βvK )−1 ((0, 1]) = OK (v) for every v ∈ K (0). The injection β K : |K| → 1 (K (0) ) defined by β K (x) = (βvK (x))v∈K (0) is called the canonical representation of K. Observe that β K (v) = ev is the unit vector of 1 (K (0) ) for each v ∈ K (0) . We define the metric ρK on |K| as follows: ρK (x, y) = β K (x) − β K (y) =

   β K (x) − β K (y). v

v

v∈K (0)

The topology on |K| induced by the metric ρK is called the metric topology. The space |K| with this topology is denoted by |K|m . The following fact is easily seen: Fact Each map f : X → |K|m of an arbitrary space X is continuous if and only if βvK f is continuous for every v ∈ K (0). The identity id|K| : |K| → |K|m is continuous but not a homeomorphism unless K is locally finite. Proposition 1.7.1 For a simplicial complex K, the metric topology on |K| coincides with the Whitehead topology (i.e., |K|m = |K| as spaces) if and only if K is locally finite. 4.5.6 The complete metrizability of |K|m can be characterized as follows: Theorem 1.7.2 For a simplicial complex K, the following are equivalent: (a) |K|m is completely metrizable; (b) K contains no infinite full complexes as subcomplexes; (c) ρK is complete.

4.5.9

There exists the following connection between |K| and |K|m : Theorem 1.7.3 For every simplicial complex K, the identity ϕ = id : |K| → |K|m is a homotopy equivalence with a homotopy inverse ψ : |K|m → |K| such that ψϕ K id and ϕψ K id, where ψϕ OK id and ϕψ OK id are also valid. These homotopies are realized by the straight-line homotopy. 4.9.6 Let K be a simplicial complex and K  be a simplicial subdivision of K. In general, the metric ρK  is not admissible for |K|m , that is, |K  |m = |K|m . We call K  an admissible subdivision of K if the metric ρK  is admissible for |K|m ; equivalently, |K  |m = |K|m . For example, the barycentric subdivision is admissible. Of course, for a locally finite simplicial complex, every subdivision is admissible. We have the following characterization of admissible subdivisions: Theorem 1.7.4 Let K be a simplicial complex and K  be a simplicial subdivision of K. Then, the following are equivalent: (a) K  is an admissible subdivision of K; (b) the open star OK  (v) of every vertex v ∈ K (0) is open in |K|m ; (c) the set K (0) of vertices of K  is discrete in |K|m .

4.8.1 4.8.4

1.8 PL Maps and Simplicial Maps

39

The following is the metric topology version of Theorem 1.6.9: Theorem 1.7.5 (HENDERSON–SAKAI) Let K be an arbitrary simplicial complex. For any open cover U of |K|m , there is an admissible subdivision K  of K such that SK  ≺ U. 4.8.8 A map f : X → Y is called a fine homotopy equivalence provided that, for each open cover U ∈ cov(Y ), there is a map g : Y → X such that fg U id and gf f −1 (U) id. By virtue of Theorem 1.7.5, Theorem 1.7.3 can be strengthened as follows: Theorem 1.7.6 For every simplicial complex K, the identity ϕ = id : |K| → |K|m is a fine homotopy equivalence.25 Proof Let U ∈ cov(|K|m ). By Theorem 1.7.5, K has an admissible simplicial subdivision K  such that SK  ≺ U. Then, applying Theorem 1.7.3, we can obtain a map ψ : |K  |m → |K  | such that ψϕ K  id and ϕψ K  id. Since |K  | = |K| and |K  |m = |K|m as spaces, and K ≺ U, ψ : |K|m → |K| is a map such that ψϕ U id and ϕψ ϕ −1 (U) id. " !

1.8 PL Maps and Simplicial Maps Let K and L be cell complexes. A map f : |K| → |L| is said to be piecewise linear (abbrev., PL) if there is a subdivision K  of K such that f |C is affine for each C ∈ K  . In this definition, we can take K  so that f (K  ) ≺ L, that is, for each C ∈ K  , f (C) is contained in some cell in L [GAGT, Lemma 4.4.3]. Using this fact, we can easily prove that the composition of PL maps is also PL [GAGT, Proposition 4.4.4]. We have the following characterization of PL maps: Theorem 1.8.1 Let P and Q be polyhedra. A map f : P → Q is PL if and only if the graph G(f ) = {(x, f (x)) | x ∈ |K|} ⊂ P × Q is a polyhedron. 4.4.2 Remark 1.4 In the above, when P = |K| and Q = |L| for some cell complexes K and L, P × Q = |K ×c L| as spaces in general (cf. Theorem 1.6.2). However, we can prove that the topology of the graph G(f ) is equal to the one inherited from |K ×c L| [GAGT, Lemma 4.4.1]. It should also be remarked that the image of a PL map is, in general, not a polyhedron. Such an example can be seen in [GAGT, Remark 4 on p. 158]. For a homeomorphism f : |K| → |L|, if f is PL, then the inverse f −1 is also PL by Theorem 1.8.1 above. A homeomorphism f : |K| → |L| being PL is called a piecewise linear (PL) homeomorphism. Every bijective PL map between compact polyhedra is a PL homeomorphism but this is not true in general [GAGT, Remark 5

25 Cf.

Notes for Chapter 4 (p. 247) in [GAGT].

40

1 Preliminaries and Background Results

on p. 160]. Note that the composition of PL homeomorphisms is also PL. It is said that the polyhedra |K| and |L| are PL homeomorphic or |K| is PL homeomorphic to |L| if there exists a PL homeomorphism f : |K| → |L|. An embedding f : |K| → |L| is called a PL embedding if f (|K|) is a subpolyhedron of |L| and f : |K| → f (|K|) is a PL homeomorphism. Note that every PL embedding is a closed embedding. Let (K, K0 ) and (L, L0 ) be pairs of cell complexes and subcomplexes. For a map f : (|K|, |K0 |) → (|L|, |L0 |), if f : |K| → |L| is PL, then f ||K0 | : |K0 | → |L0 | is also PL. Such a map f : (|K|, |K0 |) → (|L|, |L0 |) is said to be piecewise linear (PL). For a homeomorphism f : (|K|, |K0 |) → (|L|, |L0 |), if f : |K| → |L| is a PL homeomorphism, then f ||K0 | : |K0 | → |L0 | is also a PL homeomorphism. Such a homeomorphism f : (|K|, |K0 |) → (|L|, |L0 |) is called a piecewise linear (PL) homeomorphism. It is said that the pairs of polyhedra (|K|, |K0 |) and (|L|, |L0 |) are PL homeomorphic or (|K|, |K0 |) is PL homeomorphic to (|L|, |L0 |) if there exists a PL homeomorphism f : (|K|, |K0 |) → (|L|, |L0 |). An embedding f : (|K|, |K0 |) → (|L|, |L0 |) is called a PL embedding if f and f ||K0 | : |K0 | → |L0 | are PL embeddings. Now, let K and L be simplicial complexes. A map f : |K| → |L| is said to be simplicial with respect to K and L (or simply a simplicial map from K to L) if f |σ is affine and f (σ ) ∈ L for each σ ∈ K, where dim f (σ )  dim σ . Evidently, f (K (0) ) ⊂ L(0) and f  (K) = {f (σ ) | σ ∈ K} is a subcomplex of L. When σ = v1 , . . . , vn  ∈ K, we have f (σ ) = f (v1 ), . . . , f (vn ) ∈ L and f

n n    t v t f (v ) for t , . . . , t ∈ I with ti = 1, = i i i i 1 n i=1

 n

i=1

i=1

where it is possible that f (vi ) = f (vj ) for i = j . Obviously, every simplicial map is PL. A simplicial map f : |K| → |L| from K to L is also written as f : K → L.26 Every simplicial map f : K → L is uniquely determined by the function f0 = f ||K (0) : K (0) → L(0) . Proposition 1.8.2 Let K and L be simplicial complexes. For a function f0 : K (0) → L(0) , the following are equivalent: (a) f0 extends to a simplicial map; (0) (b) f 0 (σ ) ∈ L for each σ ∈ K; (c) v∈σ (0) OL (f0 (v)) = ∅ for each σ ∈ K. In this case, the simplicial extension f of f0 is unique.

4.4.5

26 Note that K and L are collections of simplexes. This notation f : K → L means a simplicial map f : |K| → |L| with respect to K and L, but it does not mean that f itself is a function between collections. Of course, a simplicial map f : |K| → |L| with respect to K and L induces a function from K to L.

1.8 PL Maps and Simplicial Maps

41

It might be expected that every PL map f : |K| → |L| is simplicial with respect to some simplicial subdivisions of K and L but this is not true. In fact, there exists a PL map f : |K| → |L| that is not simplicial with respect to any simplicial subdivisions of K and L [GAGT, Remark 6 on p. 162]. However, we have the following: Theorem 1.8.3 Let K and L be cell complexes. A proper map f : |K| → |L| is PL if and only if f is simplicial with respect to some simplicial subdivisions of K and 4.6.5 L. For simplicial complexes K and L, if a bijective map f : |K| → |L| is simplicial with respect to K and L, then the inverse f −1 is a simplicial map from L to K. A bijective simplicial map f : K → L is called a simplicial isomorphism or a simplicial homeomorphism. It is said that K and L are simplicially isomorphic or K is simplicially isomorphic to L if there is a simplicial isomorphism f : K → L. An embedding f : |K| → |L| is called a simplicial embedding if f (K) = {f (σ ) | σ ∈ K} is a subcomplex of L and f : K → f (K) is a simplicial isomorphism. Obviously, every simplicial homeomorphism is a PL homeomorphism and every simplicial embedding is a PL embedding. The proof of Theorem 4.4.8 in [GAGT] shows the following: Theorem 1.8.4 Let P and Q be polyhedra. In order that f : P → Q is a PL homeomorphism, it is necessary and sufficient that there exist triangulations K and L of P and Q respectively such that f : K → L is a simplicial isomorphism. " ! Let (K, K0 ) and (L, L0 ) be pairs of simplicial complexes and subcomplexes. If f : |K| → |L| is simplicial with respect to K and L, then f ||K0 | : |K0 | → |L0 | is also simplicial with respect to K0 and L0 . Such a map f : (|K|, |K0 |) → (|L|, |L0 |) is said to be simplicial with respect to (K, K0 ) and (L, L0 ) (or from (K, K0 ) to (L, L0 )) and is also written as f : (K, K0 ) → (L, L0 ). A homeomorphism (a bijective simplicial map) f : (K, K0 ) → (L, L0 ) is called a simplicial homeomorphism or a simplicial isomorphism. Then, it is said that (K, K0 ) and (L, L0 ) are simplicially isomorphic or (K, K0 ) is simplicially isomorphic to (L, L0 ). Note Let K and L be cell complexes. In a similar way to a simplicial map, we can define a cellular map from K to L as a map f : |K| → |L| such that f |C is affine and f (C) ∈ L for each C ∈ K.27 However, such a map is not practical because of the requirement for a cellular map to be affine on each cell. For example, there are no affine homeomorphisms from the unit square I2 to the rectangle 0, e1 , e2 , v ⊂ R2 , where v = (1/2, 1) ∈ R2 . Here, it should be noted that there is a cellular homeomorphism from the boundary ∂I2 onto ∂0, e1 , e2 , v. As can be seen from this example, even if there exist a bijection η : K → L (between collections of cells) such that for σ, τ ∈ K, σ < τ if and only if η(σ ) < η(τ ), there are no cellular homeomorphisms from |K| to |L|. For CW-complexes X and Y , a cellular map f : X → Y is defined as a map such that f (X n ) ⊂ Y n for each n ∈ N. Any cell complex K defines a CW-complex X = |K| with

27 In fact,

this definition is employed in the textbook of C.P. Rourke and B.J. Sanderson [(17)], p. 16.

42

1 Preliminaries and Background Results X n = |K (n) |, n ∈ ω. Evidently, a cellular map between cell complexes is cellular as a map between CW-complexes but the converse is not true.

For two cell complexes K and L, it is said that K is congruent to L, or K and L are congruent, written as K ≡ L, if there exists a bijection η : K → L (between collections of cells) such that C, D ∈ K, D < C ⇔ η(D) < η(C).28 Then, η|K (0) : K (0) → L(0) is a bijection, and for each C ∈ K, η(C)(0) = η(C (0) ) and dim η(C) = dim C. For pairs (K, K0 ) and (L, L0 ) of cell complexes and subcomplexes, when the above bijection η : K → L exists and satisfies η(K0 ) = L0 , it is said that (K, K0 ) is congruent to (L, L0 ), or (K, K0 ) and (L, L0 ) are congruent, written as (K, K0 ) ≡ (L, L0 ). Obviously, ≡ is an equivalence relation among cell complexes (or pairs of cell complexes). In [GAGT], the notation ≡ was used only for simplicial complexes, which means “is simplicially isomorphic to.” This usage is consistent by the following proposition: Proposition 1.8.5 Two simplicial complexes K and L are simplicially isomorphic if and only if they are congruent, that is, K ≡ L. For pairs (K, K0 ) and (L, L0 ) of simplicial complexes and subcomplexes, (K, K0 ) and (L, L0 ) are simplicially isomorphic if and only if they are congruent, that is, (K, K0 ) ≡ (L, L0 ). Proof Evidently, a simplicial isomorphism f : K → L induces the bijection η in the definition of congruence. Conversely, assume K ≡ L, that is, there exists a bijection η : K → L such that σ, τ ∈ K, σ < τ if and only if η(σ ) < η(τ ). For each σ ∈ K, the bijection η|σ (0) : σ (0) → η(σ )(0) induces an affine homeomorphism fσ : σ → η(σ ). Then, a simplicial isomorphism f : K → L is defined by f |σ = fσ for each σ ∈ K. Thus, we have the first statement of the proposition. It is easy to translate the first statement to the second, namely the pair version. " ! Even when K and L are not simplicial complexes, K ≡ L implies that |K| is PL homeomorphic to |L|. To see this fact, let η : K → L be a bijection defining K ≡ L. Then, we can give orders on K (0) and L(0) so that K and L are ordered cell complexes and η|K (0) : K (0) → L(0) is an order isomorphism. Then, applying Theorem 1.6.1, we can obtain simplicially isomorphic simplicial subdivisions K   K and L  L such that K (0) = K (0) and L(0) = L(0) . In fact,    K  = vC1 , . . . , vCn   C1 > · · · > Cn ∈ K , where vC = max C (0) for each C ∈ K. Since η|K (0) is an order isomorphism, η(vC ) = max η(C (0) ) = max η(C)(0) = vη(C) for each C ∈ K.

28 Here,

η is a function between collections K and L but not a map from |K| to |L|.

1.8 PL Maps and Simplicial Maps

43

Therefore, we have    L = vη(C1 ) , . . . , vη(Cn )   η(C1 ) > · · · > η(Cn ) ∈ L    = η(vC1 ), . . . , η(vCn )  C1 > · · · > Cn ∈ K . Then, it follows that η induces a bijection η : K  → L defined as follows: η (vC1 , . . . , vCn ) = η(vC1 ), . . . , η(vCn ) for C1 > · · · > Cn ∈ K. Evidently, for σ, τ ∈ K  , σ  τ if and only if η (σ )  η (τ ). Hence, we have K  ≡ L , so K  is simplicially isomorphic to L by Proposition 1.8.5 above. Namely, |K| is PL homeomorphic to |L|. The above argument is valid for pairs of cell complexes and subcomplexes. Thus, we have the following: Proposition 1.8.6 If cell complexes K and L are congruent, then |K| and |L| are PL homeomorphic. If pairs (K, K0 ) and (L, L0 ) of cell complexes and subcomplexes are congruent, then (|K|, |K0 |) and (|L|, |L0 |) are PL homeomorphic. " ! For cell complexes K and L, it is said that K is combinatorially equivalent to L or K and L are combinatorially equivalent and denoted by K ∼ = L if they have subdivisions which are congruent. As we just saw above, subdivisions can be simplicial, that is, K∼ = L ⇔ ∃K   K, ∃L  L such that K  ≡ L . For pairs (K, K0 ) and (L, L0 ) of cell complexes and subcomplexes, it is said that (K, K0 ) is combinatorially equivalent to (L, L0 ) or (K, K0 ) and (L, L0 ) are combinatorially equivalent and denoted by (K, K0 ) ∼ = (L, L0 ) if they have (simplicial) subdivisions which are congruent. Namely, we have the following equivalence: (K, K0 ) ∼ = (L, L0 ) ⇔ ∃(K  , K0 )  (K, K0 ), ∃(L , L0 )  (L, L0 ) such that (K  , K0 ) ≡ (L , L0 ). As is easily observed, Theorem 4.4.8 in [GAGT] is valid for cell complexes, that is, we have the following theorem, from which it follows that ∼ = is an equivalence relation. Theorem 1.8.7 Two cell complexes K and L are combinatorially equivalent if and 4.4.8 only if their polyhedra |K| and |L| are PL homeomorphic. Remark 1.5 The above can be easily translated to pairs (K, K0 ) and (L, L0 ) of cell complexes and subcomplexes, that is, (K, K0 ) ∼ = (L, L0 ) ⇔ (|K|, |K0 |) and (|L|, |L0 |) are PL homeomorphic.

44

1 Preliminaries and Background Results

For cell (or simplicial) complexes K and L, the following implications are trivial: K≡L ⇒ K∼ = L ⇒ |K| ≈ |L|. It goes without saying that the converse of the first implication does not hold. But, the converse of the second implication is non-trivial. It was well known as a longstanding problem and is called the Hauptvermutung (the main conjecture).29 It had taken a long time to show that this conjecture is false. First, the Hauptvermutung for polyhedra was disproved in 1961 by Milnor [107], and then Kirby and Siebenmann [94] demonstrated in 1969 that this conjecture is false for triangulations of nmanifolds for n  5. However, it should be noted that |K| = |L| implies K ∼ = L by Theorems 1.6.1 and 1.6.4. Let K and L be simplicial complexes. A simplicial map g : K → L is called a simplicial approximation of a map f : |K| → |L| if each g(x) is contained in the carrier cL (f (x)) of f (x) in L, where f L g by the straight line homotopy. When K and L are locally finite, if f is proper, then g is also proper. Indeed, since | St(v, L)| is compact for each v ∈ L(0) and Ocl L = SL , we can apply Proposition 1.3.8. Thus, we can assert as follows: • Every simplicial approximation of a proper map is proper. As easy applications of Theorem 1.6.9, the following approximation theorems can be obtained: Theorem 1.8.8 (SIMPLICIAL APPROXIMATION) Let K and L be simplicial complexes. Then, each map f : |K| → |L| has a simplicial approximation g : K  → L for some simplicial subdivision K   K. 4.7.14 Theorem 1.8.9 (PL APPROXIMATION) Let P and Q be polyhedra. For each map f : P → Q and any open cover U of Q, there exists a PL map g : P → Q that is U-homotopic to f . 4.7.15 The following GENERAL POSITION LEMMA is an important tool in both PL Topology and Dimension Theory, and we will also use it in Chap. 6: Lemma 1.8.10 (GENERAL POSITION) Let {Ui | i ∈ N} be a countable open collection in Rn and A ⊂ Rn with card A  ℵ0 such that all n + 1 many points of A are affinely independent. Then, there exists B = {vi | i ∈ N} such that vi ∈ Ui \ A for each i ∈ N and all n + 1 many points of A ∪ B are affinely independent. 5.8.4 In the same way as the General Position Lemma 1.8.10 above, we can prove the following PL EMBEDDING APPROXIMATION THEOREM (that is not treated in [GAGT]):

29 This German terminology was introduced in 1921 by Kneser, but Poincaré claimed this toward the end of nineteenth century and it was formulated as a conjecture in 1908 by Steiniz and Tieze. Refer to Introduction of [113].

1.8 PL Maps and Simplicial Maps

45

Theorem 1.8.11 (PL EMBEDDING APPROXIMATION FOR PRODUCTS) Let P be a compact polyhedron with P0 ⊂ P a subpolyhedron, and let Q be another polyhedron. Suppose that f : P → Q × In is a map such that f |P0 is a PL embedding and n  2 dim P + 1. For each U ∈ cov(Q × I), there exists a PL embedding h : P → Q × In such that h|P0 = f |P0 and h U f rel. P0 . Proof Because prQ f (P ) is compact, it is contained in some compact subpolyhedron of Q, which is PL homeomorphic to a subpolyhedron of Rm for some m ∈ N. Then, Q itself can be regarded as a subpolyhedron of Rm . As an admissible metric for Q × In , we employ the metric induced from the Euclidean metric for Rm × I ⊂ Rm+n . Let ε > 0 be a Lebesgue number for U. By Proposition 1.6.8 and Theorem 1.6.9, we have a simplicial complex K and its subcomplex K0 such that |K| = P , |K0 | = P0 , f |σ is affine on each σ ∈ K0 , (0) and diam f (σ ) < ε/4 for every σ ∈ K. Let K (0) \ K0 = {vi | i = 1, . . . , k}. By induction, using the Baire Category Theorem 1.3.15 at each step, we can obtain a map g0 : K (0) → In such that g0 |K0(0) = prIn f |K0(0), g0 (vi ) − prIn f (vi ) < ε/8, i = 1, . . . , k, and    fl{x1 , . . . , xn }  xi ∈ prIn f (K0(0) ) ∪ {g0 (v1 ), . . . , g0 (vi−1 )} , g0 (vi ) ∈ where the above set is nowhere dense in In . Then, it follows that, for each σ ∈ K\K0 (0) and τ ∈ K, g0 (σ (0) ) ∪ g0 (τ (0) ) is affinely independent. Note that g0 |K (0) \ K0 is an injection and g0 (K (0) \ K0(0) ) ∩ g0 (K0(0)) = ∅, but g0 |K0(0) is not injective in general. We define a map h0 : K (0) → Q × In by h0 (v) = (prQ f (v), g0 (v)) for each v ∈ K (0). Then, h0 is an injection, h0 |K0(0) = f |K0(0), and h0 (v) − f (v) < ε/8 for each (0) v ∈ K (0) \ K0 . For each σ ∈ K \ K0 and τ ∈ K, h0 (σ (0) ) ∪ h0 (τ (0)) is affinely independent, where h0 (σ (0) )∪h0 (τ (0)) might be affinely dependent for some distinct σ, τ ∈ K0 . This injection h0 extends to a PL embedding h : P = |K| → Q × In such that h|σ is affine on each σ ∈ K. For each σ ∈ K, diam h(σ ) = diam h0 (σ (0) ) < diam f (σ (0) ) + ε/4  diam f (σ ) + ε/4 < ε/2. If σ ∈ K0 , then h|σ = f |σ because they are affine on σ = σ (0)  and h|σ (0) = h0 |σ (0) = f |σ (0) . Since P0 = |K0 |, it follows that g|P0 = f |P0 . For each x ∈ P = |K|, let σ = cK (x) ∈ K be the carrier of x and v ∈ σ (0) . Then, it follows that h(x) − f (x)  diam h(σ ) + h(v) − f (v) + diam f (σ ) < ε/2 + ε/8 + ε/4 < ε. Consequently, h U f rel. P0 by the straight line homotopy.

" !

46

1 Preliminaries and Background Results

In Theorem 1.8.11 above, the product Q × In can be replaced with a PL nmanifold. which is proved in the Appendix (Theorem A.4.1).

1.9 Derived and Regular Neighborhoods The natural triangulation of the unit interval I = [0, 1] is the simplicial complex I = {0, 1, I}. Let K be a simplicial complex. For a subcomplex L of K, we have the simplicial map ϕL : K → I defined by ϕL (v) =

 0 1

if v ∈ L(0) , if v ∈ K (0) \ L(0) ,

that is, ϕL : |K| → I is defined by 

ϕL (x) =

βvK (x) for each x ∈ |K|.

v∈K (0) \L(0)

Then, L(0) = K (0) ∩ ϕL−1 (0) = (ϕL |K (0) )−1 (0) and ϕL (|L|) = 0, so |L| ⊂ ϕL−1 (0). However, |L| = ϕL−1 (0) if L is not full in K. Using this map, we can characterize a full subcomplex of K as follows: Proposition 1.9.1 Let K be a simplicial complex. A subcomplex L of K is full in K (i.e., L is a full subcomplex of K) if and only if |L| = ϕL−1 (0). Proof To see part, let x ∈ ϕL−1 (0). Since ϕL is simplicial, we have  the “only if” K ϕL (x) = v∈cK (x)(0) βv (x)ϕL (v), where cK (x) ∈ K is the carrier of x and (0) cK (x) = {v ∈ K (0) | βvK (x) > 0}. Hence, cK (x)(0) ⊂ ϕL−1 (0) = L(0) . Since L is full in K, it follows that cK (x) ∈ L, so x ∈ |L|. Thus, ϕL−1 (0) ⊂ |L|, which implies that |L| = ϕL−1 (0). To see the “if” part, let σ ∈ K with σ (0) ⊂ L(0) . Since ϕL (σ (0) ) = 0 and ϕL is simplicial, we have ϕL (σ ) = 0. Hence, σ ⊂ ϕL−1 (0) = |L|, which means that σ ∈ L. Thus, L is full in K. " ! Remark 1.6 Observe that the complement C(L, K) of L in K is a full subcomplex of K. Since K (0) \ C(L, K)(0) = L(0) , it follows that ϕC(L,K)(x) =

 v∈L(0)



βvK (x) = 1 −

v∈K (0) \L(0)

−1 Hence, |C(L, K)| = ϕC(L,K) (0) = ϕL−1 (1).

βvK (x) = 1 − ϕL (x).

1.9 Derived and Regular Neighborhoods

47

For a simplicial complex K and its full subcomplex L, let K  be a derived subdivision relative to L ∪ C(L, K). Then, N(L, K  ) is called a derived neighborhood of L in K. The following uniqueness of derived neighborhoods easily follows from the definition: Lemma 1.9.2 Let K be a simplicial complex and L a full subcomplex of K. For any two derived subdivisions K  and K  of K relative to L ∪ C(L, K), there exists a simplicial isomorphism h : K  → K  such that h|L ∪ C(L, K) = id and h(N(L, K  )) = N(L, K  ). In particular, N(L, K  ) ≡ N(L, K  ), that is, these derived neighborhoods are simplicial isomorphic. " ! For 0 < ε < 1, the following is called the ε-neighborhood of L in K:    Nε (L, K) = σ ∩ ϕL−1 (τ )  σ ∈ K[|L|], τ  [0, ε] , which is a cell complex with L ⊂ Nε (L, K) and |Nε (L, K)| = ϕL−1 ([0, ε]) (Fig. 1.1). We have also the following cell complex:    Cε (L, K) = σ ∩ ϕL−1 (τ )  σ ∈ K \ L, τ  [ε, 1] , where C(L, K) ⊂ Cε (L, K) and |Cε (L, K)| = ϕL−1 ([ε, 1]). Then, Nε (L, K) and Cε (L, K) are subcomplexes of the following cell complex subdividing K: KLε = Nε (L, K) ∪ Cε (L, K).

Fig. 1.1 N(L, K  )  Nε (L, K)

48

1 Preliminaries and Background Results

As is easily observed, |Nε (L, K)| \ |L| = ϕL−1 ((0, ε]) and bd |Nε (L, K)| = ϕL−1 (ε) = |Nε (L, K)| ∩ |Cε (L, K)|. It should be noted that K \ (L ∪ C(L, K)) = K[|L|] \ L. Due to Proposition 1.6.8, choosing vσ ∈ rint σ ∩ ϕL−1 (ε) for each σ ∈ K[|L|] \ L, we can define a derived subdivision K  of K relative to L ∪ C(L, K). Then, as is easily observed, K   KLε and N(L, K  )  Nε (L, K), where N(L, K  ) is a derived neighborhood of L in K. Lemma 1.9.3 Let K be a locally finite simplicial complex and let L be a finite full subcomplex of K. Suppose that K1 is a simplicial subdivision of K and L1 is the simplicial subdivision of L induced from K1 , where it follows that L1 is full in K1 . Then, there exist derived subdivisions K  and K1 of K and K1 relative to L ∪ C(L, K) and L1 ∪ C(L1 , K1 ), respectively, such that |N(L, K  )| = |N(L1 , K1 )|. Proof Since K1 is locally finite and L1 is finite, we can find ε > 0 such that ϕL−1 ([0, ε]) ∩ |C(L1 , K1 )| = ∅, i.e., ϕL−1 ([0, ε]) ⊂ int |N(L1 , K1 )|. As mentioned above, by choosing all the new vertices on ϕL−1 (ε), we can define derived subdivisions K   K and K1  K1 relative to L ∪ C(L, K) and L1 ∪ C(L1 , K1 ), respectively. Both N(L, K  ) and N(L1 , K1 ) are simplicial subdivisions of Nε (L, K). Hence, |N(L, K  )| = |N(L1 , K1 )| = |Nε (L, K)|. " ! Let P be a polyhedron with a subpolyhedron Q ⊂ P . A subpolyhedron N ⊂ P is called a regular neighborhood of Q in P if there exist a triangulation K of P , a full subcomplex L ⊂ K with |L| = Q, and a derived subdivision K  of K relative to L ∪ C(L, K) such that P = |K|, Q = |L|, and N = |N(L, K  )|, where Propositions 1.6.6 and 1.6.8 guarantee the existence of such a pair (K, L). Roughly speaking, a regular neighborhood is a polyhedral neighborhood triangulated by a derived neighborhood. As mentioned before Lemma 1.9.3, for any simplicial complex K and any full subcomplex L of K, and 0 < ε < 1, we can take a derived subdivision K  of K relative to L ∪ C(L, K) such that N(L, K  )  Nε (L, K). Thus, the polyhedron of an ε-neighborhood is a regular neighborhood, that is: Lemma 1.9.4 For any simplicial complex K, any full subcomplex L of K, and 0 < ε < 1, |Nε (L, K)| is a regular neighborhood of |L| in |K|. " ! Concerning regular neighborhoods, we have the following uniqueness: Theorem 1.9.5 (UNIQUENESS OF REGULAR NEIGHBORHOODS) Let P be a locally compact polyhedron with a compact subpolyhedron Q ⊂ P . If N1 and N2 are regular neighborhoods of a subpolyhedron Q in P , then there is a PL homeomorphism h : P → P such that h(N1 ) = N2 , h|Q = id, and h|P \ U = id for some open neighborhood U of Q in P .

1.9 Derived and Regular Neighborhoods

49

Proof Let (Ki , Li ), i = 1, 2, be pairs of simplicial complexes and their full subcomplexes and let Ki be a derived subdivision of Ki relative to Li ∪ C(Li , Ki ) such that P = |Ki |, Q = |Li |, and Ni = |N(Li , Ki )|. By Theorem 1.6.5, P has a triangulation K that is a common subdivision of K1 and K2 , where K is locally finite because P is locally compact. Let L be a subcomplex of K with |L| = Q. Then, L is finite because Q is compact. As is easily observed, L is full in K. By Lemma 1.9.3, we can obtain derived subdivisions K  and K1 of K and K1 relative to L ∪ C(L, K) and L1 ∪ C(L1 , K1 ), respectively, such that |N(L, K  )| = |N(L1 , K1 )|. Similarly, we have derived subdivisions K  and K2 of K and K2 relative to L ∪ C(L, K) and L2 ∪ C(L2 , K2 ), respectively, such that |N(L, K  )| = |N(L2 , K2 )|. By Lemma 1.9.2, there exist simplicial isomorphisms fi : Ki → Ki , i = 1, 2, and g : K  → K  such that fi |Li ∪ C(Li , Ki ) = id, fi (N(Li , Ki )) = N(Li , Ki ), g|L ∪ C(L, K) = id, and g(N(L, K  )) = N(L, K  ). Then, h = f2−1 gf1 : P → P is the desired PL homeomorphism. Indeed, h(N1 ) = f2−1 gf1 (|N(L1 , K1 )|) = f2−1 g(|N(L1 , K1 )|) = f2−1 g(|N(L, K  )|) = f2−1 (|N(L, K  )|) = f2−1 (|N(L2 , K2 )|) = |N(L2 , K2 )| = N2 . Because Q = |L1 | = |L2 | = |L|, we have h|Q = f2−1 gf1 |Q = f2−1 g|Q = f2−1 |Q = id. Moreover, the following is an open neighborhood of Q in P : U = P \ (|C(L1 , K1 )| ∩ |C(L2 , K2 )|). Then, P \ U ⊂ |C(Li , K)| ⊂ |C(L, K)|, i = 1, 2. Hence, h|P \ U = f2−1 gf1 |P \ U = f2−1 g|P \ U = f2−1 |P \ U = id. This completes the proof.

" !

A subpolyhedron A of P is said to be PL bi-collared (resp. PL collared) in P if there is a PL embedding k : A × [−1, 1] → P (resp. k : A × I → P ) such that k(x, 0) = x for every x ∈ A and k(A × (−1, 1)) (resp. k(A × [0, 1))) is open in P (i.e., k|A × (−1, 1) (resp. k|A × [0, 1)) is an open embedding in P ), where k is

50

1 Preliminaries and Background Results

called a PL bi-collar (resp. a PL collar) in P .30 A subset A ⊂ X is called a strong deformation retract of X if there is a homotopy h : X × I → X such that h0 = id, ht |A = id for each t ∈ I, and h1 (X) = A.31 Theorem 1.9.6 Let P be a locally compact polyhedron and let N be a regular neighborhood of a compact subpolyhedron Q in P . The boundary bd N is PL bicollared in P (and PL collared in N). Furthermore, we have a homeomorphism h : bd N × R+ → N \ Q such that h(x, 0) = x for every x ∈ bd N, so bd N is a strong deformation retract of N \ Q. Proof We have a triangulation K of P and a full subcomplex L with |L| = Q. Let 0 < ε < 1. Take a derived subdivision K  of K relative to L ∪ C(L, K) so that K (0) \ K (0) ⊂ ϕL−1 (ε). As observed before, |N(L, K  )| = |Nε (L, K)|. By the Uniqueness of Regular Neighborhoods (Theorem 1.9.5), we may assume that N = |Nε (L, K)| = ϕL−1 ([0, ε]). Then, bd N = ϕL−1 (ε). We have the following subcomplex of Nε (L, K):    ∂Nε (L, K) = σ ∩ ϕL−1 (ε)  σ ∈ K[|L|] \ L . Let J = {±1, [−1, 1]} be the natural triangulation of [−1, 1] and let M = ∂Nε (L, K) ×c J. Then, |∂Nε (L, K)| = bd N and |M| = bd N × [−1, 1]. For each C ∈ M, we define vC ∈ rint C as follows: vC =

 (vσ , 0)

if C = (σ ∩ ϕL−1 (ε)) × [−1, 1],

(vσ , ±1) if C = (σ ∩ ϕL−1 (ε)) × {±1},

where σ ∈ K[|L|] \ L. Let M  be the derived subdivision of M defined by using these points. Then, M0 = {τ ∈ M  | τ ⊂ bd N × {0}} is a subcomplex of M  with |M0 | = bd N × {0}. Now, take 0 < a < ε < b < 1 and define the cell complex as follows:    B = σ ∩ ϕL−1 (τ )  σ ∈ K[|L|] \ L, τ  [a, b] . Then, |B| = ϕL−1 ([a, b]), which is a subpolyhedron of P . Let B  be a derived subdivision of B, where we choose vD = vσ ∈ rint σ ∩ ϕL−1 (ε) for D =

30 The definitions of a collar and a collared set are given on p. 129. A PL collar is a closed collar, where a closed collar is defined on p. 135. It is known that every collared closed set in a paracompact space has a closed collar (Proposition 2.5.5). For a bi-collar and a bi-collared set, refer to Remark 2.7. 31 A deformation retract will be defined on p. 68. These concepts will be discussed in Sect. 1.13.

1.9 Derived and Regular Neighborhoods

51

σ ∩ ϕL−1 ([a, b]), σ ∈ K[|L|] \ L. Then, B0 = {τ ∈ B  | τ ⊂ ϕL−1 (ε)} is a subcomplex of B  with |B0 | = bd N = ϕL−1 (ε). We have a bijection η : M → B defined by η((σ ∩ ϕL−1 (ε)) × [−1, 1]) = σ ∩ ϕL−1 ([a, b]), η((σ ∩ ϕL−1 (ε)) × {1}) = σ ∩ ϕL−1 (a), and η((σ ∩ ϕL−1 (ε)) × {−1}) = σ ∩ ϕL−1 (b), where σ ∈ K[|L|] \ L. Observe that for C, D ∈ M, C  D ⇔ η(C)  η(D). Hence, we can define a simplicial isomorphism k : M  → B  as follows: k(vC ) = vη(C) for each C ∈ M. When C = (σ ∩ ϕL−1 (ε)) × [−1, 1], σ ∈ K[|L|] \ L, since vC = (vσ , 0) and η(C) = σ ∩ ϕL−1 ([a, b]), we have vη(C) = vσ . Because k is simplicial, it follows that k(x, 0) = x for every x ∈ bd N. Consequently, k : bd N × [−1, 1] → P is a PL bi-collar, so bd N is PL bi-collared in P . Moreover, since k(bd N × [0, 1]) = ϕL−1 ([a, ε]) ⊂ ϕL−1 ([0, ε]) = N, the restriction k| bd N ×[0, 1] : bd N ×[0, 1] → N is a PL collar in N. To see the additional statement, take a = a1 > a2 > · · · > 0 so that infn∈N an = 0, and let a0 = ε. Then, N \ Q = ϕL−1 ((0, ε]) =



ϕL−1 ([an , an−1 ]).

n∈N

As shown above, we have homeomorphisms kn : ϕL−1 (an−1 ) × I → ϕL−1 ([an , an−1 ]), n ∈ N, such that kn (x, 0) = x for every x ∈ ϕL−1 (an−1 ) and kn (ϕL−1 (an−1 ) × {1}) = ϕL−1 (an ). By induction, homeomorphisms     hn : bd N × [n − 1, n], bd N × {n} → ϕL−1 ([an , an−1 ]), ϕL−1 (an ) , n ∈ N, can be defined as follows: for each (x, t) ∈ bd N × [n − 1, n], hn (x, t) = kn (hn−1 (x, n − 1), t − n + 1).

52

1 Preliminaries and Background Results

where h0 (x, 0) = x, so h1 (x, t) = k1 (x, t). Then, for every x ∈ bd N and n ∈ N, hn (x, n − 1) = kn (hn−1 (x, n − 1), 0) = hn−1 (x, n − 1). So, we can define a homeomorphism h : bd N × R+ → N \ Q by h| bd N × [n − 1, n] = hn , n ∈ N. Then, h(x, 0) = k1 (x, 0) = x for every x ∈ bd N. Remark 1.7 In the above, observe that |N(L, K)| \ |L ∪ C(L, K)| = Since ϕC(L,K)(x) = 1 − ϕL (x) (cf. Remark 1.6), it follows that

" ! ϕL−1 ((0, 1)).

−1 |N1−ε (C(L, K), K)| = ϕC(L,K) ([0, 1 − ε]) = ϕL−1 ([ε, 1]), −1 bd |N1−ε (C(L, K), K)| = ϕC(L,K) (1 − ε) = ϕL−1 (ε) = bd N, and −1 ((0, 1 − ε]) = ϕL−1 ([ε, 1)). |N1−ε (C(L, K), K)| \ |C(L, K)| = ϕC(L,K)

By the above result, we have a homeomorphism h : bd N × R+ → ϕL−1 ([ε, 1)) such that h (x, 0) = x for every x ∈ bd N. Combining h above with this h , we can easily define a homeomorphism h˜ : bd N × R → ϕL−1 ((0, 1)) = |N(L, K)| \ |L ∪ C(L, K)| ˜ such that h(x, 0) = x for every x ∈ bd N = ϕL−1 (ε). Thus, we have the following: |N(L, K)| \ |L ∪ C(L, K)| ≈ bd N × R. So, it follows that bd N is a strong deformation retract of |N(L, K)|\|L∪C(L, K)|. For regular neighborhoods, we can also prove the following: Theorem 1.9.7 Let P be a locally compact polyhedron. Then, each compact subpolyhedron Q of P is a strong deformation retract of any regular neighborhood N of Q in P . Proof As in the above proof, we may assume that N = |Nε (L, K)| = ϕL−1 ([0, ε]), where 0 < ε < 1, K is a triangulation of P , and L isits full subcomplex with |L| = Q. Recall that ϕL : |K| → I is defined by ϕ( x) = v∈K (0)\L(0) βvK (x). Then,  1 −ϕL (x) = v∈L(0) βvK (x). Since Q = ϕL−1 (0) and 0 < ϕL (x)  ε for x ∈ N \Q,

1.10 The Homotopy Type of Simplicial Complexes

53

we can define a homotopy h : N × I → N as follows: for each t ∈ I, ht |Q = id and h(x, t) = (1 − t)x +

 t βvK (x)v 1 − ϕL (x) (0) v∈L

= (1 − t)



βvK (x)v

 + 1+

v∈K (0) \L(0)

tϕL (x) 1 − ϕL (x)

 

βvK (x)v

v∈L(0)

if x ∈ N \ Q. Then, h0 = id and h1 : N → Q is a retraction. Therefore, Q is a strong deformation retract of N. " !

1.10 The Homotopy Type of Simplicial Complexes Given a map f : X → Y , the mapping cylinder Mf of f is defined as the following adjunction space: Mf = Y ∪f prX |X×{0} X × I. The collapsing map cf : Mf → Y is defined by cf |Y = idY and cf |X × (0, 1] = f prX . Let if : X → Mf be the natural embedding defined by if (x) = (x, 1). Then, cf if = f and cf  id rel. Y in Mf by the homotopy hf : Mf × I → f f f Mf defined by h0 = cf , ht |Y = id for every t ∈ I, and ht (x, s) = (x, st) for (x, s) × (0, 1] and t ∈ (0, 1]. When Y is a singleton, the mapping cylinder Mf is naturally homeomorphic to the cone CX = (X × I)/(X × {0}) over X. As is easily observed, if f : X → Y is a proper map, then the collapsing map cf : Mf → Y is proper, and the homotopy hf : Mf × I → Mf is also proper, hence cf p id rel. Y in Mf . For a proper (perfect) map f : X → Y between metrizable spaces, the mapping cylinder Mf is metrizable because it is the perfect image of a metrizable space Y ⊕ X × I (Theorem 1.3.6(1)). For spaces X and Y with A ⊂ X ∩ Y , we write XY

rel. A

if there exist homotopy equivalences f : X → Y and g : Y → X such that f |A = g|A = id, gf  idX rel. A, and fg  idY rel. A. In the above, if f and g are proper, gf p idX rel. A, and fg p idY rel. A, we write X p Y

rel. A.

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1 Preliminaries and Background Results

Proposition 1.10.1 Concerning the mapping cylinders, the following hold: (1) For (proper) maps f, g : X → Y , 

f  g ⇔ Mf  Mg rel. Y ∪ (X × {1})

 f p g ⇔ Mf p Mg rel. Y ∪ (X × {1}) .

4.11.1

(2) For (proper) maps f : X → Y and g : Y → Z, 

Mgf  Mg ∪ig Mf rel. Z ∪ (X × {1})

 Mgf p Mg ∪ig Mf rel. Z ∪ (X × {1}) .

4.11.2

In Theorems 4.11.1 and 4.11.2 of [GAGT], the proper cases are not mentioned but the same proofs are valid, where all homotopies are proper. Let A be a closed set in a space X with a closed collar k : A × I → X (i.e., k is a closed embedding such that k(A × [0, 1)) is open in X and k(x, 0) = x for every x ∈ A).32 Then, it follows that     A × I) ∪ (X × {1}), A × {0} ≈ X, A . For a map f : A → Y from A to another space Y , the adjunction space Y ∪f X can be regarded as the space Mf ∪ (X × {1}). As a corollary of Proposition 1.10.1(1), we have the following: Corollary 1.10.2 Let A be a closed set in a space X with a closed collar k : A × I → X and let f, g : A → Y be maps to another space Y . If f  g, then Y ∪f X  Y ∪g X rel. Y ∪ (X \ k(A × [0, 1)). " ! Let K and L be simplicial complexes, where K (0) is given an order so that K is an ordered simplicial complex. For a simplicial map f : K → L, the simplicial mapping cylinder Zf of f is defined as the following simplicial complex:   Zf = K ∪ L ∪ f (v1 ), . . . , f (vm ), vm , . . . , vn  

 v1 , . . . , vn , v1 < · · · < vn , 1  m  m  n , 33

The simplicial collapsing map c¯f : Zf → L is defined by c¯f (v) = f (v) for v ∈ K (0) and c¯f (v) = v for L(0) . When L = {v0 } (hence f is a constant map), the simplicial mapping cylinder Zf is equal to the simplicial cone v0 ∗ K over K. If K and L are locally finite and f is a proper simplicial map, then the simplicial mapping cylinder Zf is locally finite. 32 See

Footnote 30 (p. 50). simplicial mapping cylinder is dependent on such an order of K (0) . See Fig. 4.15 (p. 217) of [GAGT].

33 The

1.10 The Homotopy Type of Simplicial Complexes

55

For an ordered simplicial complex K and a subdivision K   K, we define the simplicial subdivision I (K  , K) of the product simplicial complex K ×s I as follows: I (K  , K) = (K  × {0}) ∪ (K × {1})   ∪ σ × {0} ∪ vm , . . . , vn  × {1}  σ ∈ K 

 v1 , . . . , vn  ∈ K, v1 < · · · < vn , σ ⊂ v1 , . . . , vm  ,

where K  × {0} = {σ × {0} | σ ∈ K  } and K × {1} = {τ × {1} | τ ∈ K}. For a simplicial map f : K  → L, give an order on K (0) so that K  is an ordered simplicial complex and define the simplicial mapping cylinder Zf . Identifying K  ⊂ Zf with K  × {0} ⊂ I (K  , K), we can obtain the simplicial complex: ZfK = Zf ∪K  =K  ×{0} I (K  , K). Then, |ZfK | = |Zf | ∪pr|K| ||K|×{0} |K| × I. If K and L are locally finite and f is a proper simplicial map, then ZfK is also locally finite. Theorem 1.10.3 Let K and K  be ordered simplicial complexes such that K   K and let L be a simplicial complex. For any simplicial approximation f : K  → L of a map g : |K| → |L|, |ZfK |  Mg

rel. |L| ∪ (|K| × {1}).

When K and L are locally finite and g is proper, |ZfK | p Mg

rel. |L| ∪ (|K| × {1}).34

4.11.3

It is said that X is homotopy dominated by Y (or Y homotopy dominates X) if there are maps f : X → Y and g : Y → X such that gf  id, where g : Y → X is called a homotopy domination. Moreover, if gf U id for a given open cover U ∈ cov(X), it is said that X is U-homotopy dominated by Y (or Y U-homotopy dominates X) and g : Y → X is called a U-homotopy domination. The homotopy type of a simplicial complex K means the homotopy type of the polyhedron |K| (or |K|m ). We say that X is homotopy dominated by a simplicial complex K if X is homotopy dominated by the polyhedron |K| (or |K|m ). The following theorem on the homotopy type of simplicial complexes is called the WHITEHEAD–MILNOR THEOREM:

34 The proper case of Theorem 1.10.3 is not mentioned in Theorem 4.11.3 of [GAGT] but the same proof is valid.

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1 Preliminaries and Background Results

Theorem 1.10.4 (J.H.C. WHITEHEAD; MILNOR) A space X homotopy dominated by a simplicial complex has the homotopy type of a simplicial complex K with card K (0) = dens |K|  dens X. If X is separable, X has the homotopy type of a countable complex. 4.12.1 Moreover, we have the following: Theorem 1.10.5 Every simplicial complex has the homotopy type of a locally finitedimensional simplicial complex with the same density. In addition, every countable simplicial complex has the homotopy type of a countable locally finite simplicial 4.12.3 complex with the same density. The following is a combination of the above two theorems: Corollary 1.10.6 A space X homotopy dominated by a simplicial complex has the homotopy type of a locally finite-dimensional simplicial complex K with card K (0) = dens |K|  dens X. If X is separable, X has the homotopy type of a countable locally finite simplicial complex. 4.12.4 It is said that X is proper homotopy dominated by Y (or Y proper homotopy dominates X) if there are proper maps f : X → Y and g : Y → X such that gf p id, where g : Y → X is called a proper homotopy domination. For a locally compact space, we can show the following proper homotopy version of Theorem 1.10.4: Theorem 1.10.7 Let X be a locally compact space. If X is proper homotopy dominated by a locally finite simplicial complex K, then X × R+ has the proper homotopy type of a locally finite simplicial complex L. Proof To prove that X × R has the proper homotopy type of a locally finite simplicial complex L, we can perform the same proof as Theorem 4.12.1 of [GAGT]. For the reader’s convenience, we give its outline below: Let f : X → |K| and g : |K| → X be proper maps such that gf p id. By the Simplicial Approximation Theorem 1.8.8, f g has a simplicial approximation ϕ : K  → K for some K   K Since f g is proper, ϕ is also proper. Giving orders on K (0) and K (0) so that K  and K are ordered simplicial complexes, we can obtain the simplicial complex ZϕK . Since K is locally finite and ϕ is proper, ZϕK is locally finite. For each n ∈ Z, let Ln be a copy of ZϕK . Identifying K ⊂ L n+1 with K × {1} ⊂ Ln for each n ∈ Z, we have a locally finite simplicial complex L = n∈Z Ln . For each n ∈ Z, let M2n−1 and M2n be copies of Mf and Mg , respectively. Identifying X × {1} ⊂ M2n−1  and |K| × {1} ⊂ M2n ] with X ⊂ M2n and |K| ⊂ M2n+1 respectively, we define M = n∈Z Mn . Then, by using Proposition 1.10.1 and Theorem 1.10.3 we can prove that |L| p |M| p X × R. Refer to Fig. 4.19 (p. 223) of [GAGT].

As is easily observed (see Fig. 1.2), we have I × R+ ≈ I × [0, 1) ≈ (0, 1) × [0, 1) ≈ R × R+ .

1.11 The Nerves of Open Covers

I

0 1

57

0 1

0 1

Fig. 1.2 I × [0, 1) ≈ (0, 1) × [0, 1)

Since X × R+ p X × I × R+ , it follows that X × R+ p X × R × R+ p |L| × R+ = |L ×s R+ |, where R+ is the natural triangulation of R+ , i.e., R+ = {[n, n + 1], n | n ∈ ω}. Thus, X × R+ has the proper homotopy type of L ×s R+ , which is a locally finite simplicial complex. " ! Remark 1.8 In Theorem 1.10.7, if X is connected, then L is countable because L is also connected. Remark 1.9 Relating to Theorem 1.10.4 and Corollary 1.10.6, the question arises whether a compact space homotopy dominated by a simplicial complex has the homotopy type of a finite simplicial complex. This is equivalent to asking whether a compact space having the homotopy type of a simplicial complex has the homotopy type of a finite simplicial complex. As will be seen in Sect. 1.13, every ANR has the homotopy type of a simplicial complex and every simplicial complex has the homotopy type of an ANR. Thus, the question is equivalent to asking whether a compact ANR has the homotopy type of a finite simplicial complex. The positive answer has been expected and it is called the BORSUK CONJECTURE (cf. [33]). This conjecture was proved by J. West [152, 153]. We will discuss this conjecture in Chap. 4.

1.11 The Nerves of Open Covers Let V be an arbitrary set. Recall that Fin(V ) is the collection of all non-empty finite subsets of V . An abstract complex K over V is a subcollection K ⊂ Fin(V ) satisfying the following condition: (AC)

if A ∈ K and ∅ = B ⊂ A then B ∈ K.

A subcollection L ⊂ K satisfying (AC) is called a subcomplex of K. In particular, Fin(V ) is an abstract complex and every abstract complex K over V is a subcomplex

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1 Preliminaries and Background Results

of Fin(V ). For each n ∈ ω, the n-skeleton K(n) of K is defined by    K(n) = A ∈ K  card A  n + 1 , where we regard K(0) ⊂ V , and so Fin(V )(0) = V . Each K(n) is a subcomplex of K. If K = K(n) , we say that K is at most n-dimensional and write dim K  n. It is said that K is n-dimensional (written as dim K = n) if dim K  n and dim K  n − 1. Note that every abstract complex K over V with dim K  n is a subcomplex of Fin(V )(n) . For any simplicial complex K, K = {σ (0) | σ ∈ K} ⊂ Fin(K (0) ) is an abstract complex, which is called the abstract complex defined by K. Each K(n) is defined by K (n) . In particular, K(0) = K (0). Conversely, each abstract complex K over V is defined by some simplicial complex. In fact, consider the linear space RVf . By identifying each v ∈ V with ev ∈ RVf defined by ev (v) = 1 and ev (v  ) = 0 if v = v  , we can regard V as a Hamel basis for RVf . Then, K = {A | A ∈ K} is a simplicial complex that defines K. This K is called the simplicial complex defined by K. Then each K (n) is defined by K(n) and K (0) = K(0). Note that the full simplicial complex F (V ) is the simplicial complex defined by Fin(V ). Remark 1.10 When V is a subset of a linear space E, for an abstract complex K over V , K = {A | A ∈ K} is a simplicial complex that defines K if and only if each A ∈ K is affinely independent and A ∩ A  = A ∩ A  for each A, A ∈ K. In particular, K is a simplicial complex if K satisfies the condition: () A ∪ A is affinely independent for each A, A ∈ K. Due to the General Position Lemma 1.8.10, there exists a countable (discrete) set V in R2n+1 such that all 2n + 2 many points of V are affinely independent. Then, it follows that every countable complex K with dim K  n is simplicially isomorphic to a simplicial complex in R2n+1 . Now, consider two abstract complexes K and L over V and W , respectively. Let K and L be the simplicial complexes defined by K and L, respectively. Recall K(0) = K (0) and L(0) = L(0) . Suppose that a function ϕ : K (0) → L(0) satisfies the following condition: (∗) A ∈ K implies ϕ(A) ∈ L. Then, ϕ : K (0) → L(0) induces the simplicial map f : K → L with f |K (0) = ϕ. Conversely, for any simplicial map f : K → L, the restriction ϕ = f |K (0) : K (0) → L(0) satisfies condition (∗) and f itself is the simplicial map induced by this ϕ. Such a function ϕ : K (0) → L(0) is also called a simplicial map from K to L, and is written as ϕ : K → L. If a bijection ϕ : K (0) → L(0) satisfies the condition

1.11 The Nerves of Open Covers

59

that A ∈ K if and only if ϕ(A) ∈ L, then ϕ induces the simplicial isomorphism f : K → L with f |K (0) = ϕ, and any simplicial isomorphism f : K → L is induced by such a bijection ϕ : K (0) → L(0) . Such a bijection ϕ : K (0) → L(0) is also called a simplicial isomorphism from K to L. It is said that K is simplicially isomorphic to L (denoted by K ≡ L) if there is a simplicial isomorphism from K to L. For any open cover U of a space X, we define the abstract complex N(U) over U \ {∅} as follows:    N(U) = {U1 , · · · , Un } ∈ Fin(U)  U1 ∩ · · · ∩ Un = ∅ . The simplicial complex N(U) defined by N(U) is called the nerve of U. A map f : X → |N(U)| (or f : X → |N(U)|m ) is called a canonical map for U if f −1 (ON(U) (U )) ⊂ U for each U ∈ N(U)(0) = U. Observe that every compact set in |N(U)| meets ON(U) (U ) for only finitely many U ∈ U. Hence, if every U ∈ U has the compact closure in X, then each canonical map f : X → |N(U)| is proper. We can characterize canonical maps as follows: Proposition 1.11.1 For an open cover U of X, a map f : X → |N(U)| (or f : X → |N(U)|m ) is a canonical map if and only if cN(U) (f (x))(0) ⊂ U[x] for each x ∈ X, where cN(U) (f (x)) ∈ N(U) is the carrier of f (x). If U[x] is finite, this condition is equivalent to f (x) ∈ U[x] ∈ N(U). 4.9.1 Proposition 1.11.1 yields the following: Corollary 1.11.2 Let U be an open cover of X. Then, any two canonical maps f, g : X → |N(U)| (or f, g : X → |N(U)|m ) are contiguous. 4.9.2 For each open refinement V of U ∈ cov(X), we have a simplicial map ϕ : N(V) → N(U) such that V ⊂ ϕ(V ) for each V ∈ V = N(V)(0) . Such a simplicial map is called a refining simplicial map. Corollary 1.11.3 Let U and V be open covers of X with V ≺ U and ϕ : N(V) → N(U) be a refining simplicial map. If f : X → |N(V)| (or f : X → |N(V)|m ) is a canonical map for V, then ϕf : X → |N(U)| (or ϕf : X → |N(U)|m ) is also a canonical map for U. 4.9.3 Concerning the existence of canonical maps, we have the following: Theorem 1.11.4 For every open cover U of a paracompact space X, there exists a canonical map f : X → |N(U)| such that f : X → |N(U)|m is also a canonical map. 4.9.5 The following is trivial:

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Proposition 1.11.5 Let K be a simplicial complex and X an arbitrary space. If two maps f, g : X → |K|m are contiguous then f K g by the straight line homotopy. In Proposition 1.11.5, |K|m cannot be replaced by |K| in general. Such an example can be seen in Remark 2 for Proposition 4.3.4 in [GAGT]. When X is locally compact (more generally it is a k-space), |K|m can be replaced by |K|. If we abandon the straight line homotopy, then we can obtain the following by combining Proposition 1.11.5 with Theorem 1.11.4: Proposition 1.11.6 Let K be a simplicial complex and X an arbitrary space. If two 4.9.7 maps f, g : X → |K| are contiguous, then f K g. Combining Corollary 1.11.2 with Proposition 1.11.6 (or 1.11.5), we have the following corollary: Corollary 1.11.7 Let U be an open cover of a space X. Then, f N(U) g for any two canonical maps f, g : X → |N(U)| (or f, g : X → |N(U)|m ). 4.9.8 An open cover U of a space X is said to be star-finite if U[U ] is finite for each U ∈ U, which is equivalent to the condition that the nerve N(U) is locally finite. In fact, St(U, N(U))(0) = U[U ] for each U ∈ U = N(U)(0). Thus, the star-finiteness of an open cover characterizes the local finiteness of its nerve. On the other hand, the nerve N(U) is locally finite-dimensional (l.f.d.) if and only if supx∈U card U[x] < ∞ for each U ∈ U. In this case, we have sup card U[x] = dim St(U, N(U)) + 1. x∈U

Note that every star-finite open cover is locally finite and its nerve is locally finite-dimensional, and that if an open cover is locally finite or its nerve is locally finite-dimensional then it is point-finite, that is, we have the following implications: locally finite nerve ≡ star-finite

locally finite

locally finite-dimensional nerve

point-finite

In the above, the converse implications do not hold and there are no connections between the local finiteness of an open cover and the local finite-dimensionality of its nerve. For details, refer to [GAGT, p. 201]. Theorem 1.11.8 Every open cover of a paracompact space has a locally finite σ 4.9.9 discrete open refinement with the locally finite-dimensional nerve. When X is a locally compact paracompact space, each U ∈ cov(X) has a locally finite open refinement V such that cl V is compact for each V ∈ V. Then, as is easily observed, V is star-finite, so the nerve N(V) is locally finite. Thus, we have the following theorem:

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61

Theorem 1.11.9 Every open cover of a locally compact paracompact space has a star-finite open refinement whose nerve is locally finite. 4.9.10 In addition, we can also show that: Theorem 1.11.10 Every open cover of a regular Lindelöf space has a countable 4.9.11 star-finite open refinement whose nerve is countable and locally finite.

1.12 Dimensions For an open cover U of a space X, ord U = sup{card U[x] | x ∈ X} is called the order of U. Note that ord U = dim N(U) + 1, where N(U) is the nerve of U. The (covering) dimension of X (denoted by dim X) is defined as follows: dim X  n if each finite open cover U of X has a finite open refinement V with ord V  n + 1. It can be proved that the finiteness of V is not necessary in the definition (cf. [GAGT, Lemma 5.2.1]). When X is paracompact, not only V but also U is not required to be finite, that is, dim X  n ⇔ ∀ U ∈ cov(X), ∃V ∈ cov(X) such that V ≺ U, ord V  n + 1, where V can be locally finite σ -discrete in X (cf. [GAGT, Theorem 5.2.4]). For a metric space X, the following characterization is valid (cf. [GAGT, Theorem 5.3.1]): dim X  n ⇔ ∃ U1  U2  · · · ∈ cov(X) such that lim mesh Ui = 0, ord Ui  n + 1,

i→∞

where each Ui can be locally finite σ -discrete in X like the paracompact case. We define dim X = n if dim X  n and dim X < n. It is said that X is finitedimensional (f.d.) (dim X < ∞) if dim X  n for some n ∈ ω. Otherwise, X is said to be infinite-dimensional (i.d.) (dim X = ∞). The dimension of a normal space can be characterized by extendability of maps from closed sets to the sphere, that is: Theorem 1.12.1 For a normal space X and n ∈ ω, dim X  n if and only if each map f : A → Sn of any closed set A in X extends over X. 5.2.3 As a matter of course, the following hold:

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Fact (1) For each n ∈ N, dim Bn = dim Rn = n. (2) For any simplicial complex K, dim K = dim |K| = dim |K|m .

5.2.7 5.2.11 5.2.10

In particular, dim Bn  n − 1 is obtained by combining Theorem 1.12.1 with the following NO RETRACTION THEOREM: Theorem 1.12.2 (NO RETRACTION) For any n ∈ N, there does not exist any map r : Bn → Sn−1 such that r|Sn−1 = id.35 5.1.5 The No Retraction Theorem 1.12.2 is also applied to prove the following characterization of boundary points of a closed subset Rn : Theorem 1.12.3 Let X be a closed set in Rn and x ∈ X. In order that x ∈ bd X, it is necessary and sufficient that each neighborhood U of x in X contains a neighborhood V of x in X such that every map f : X \ V → Sn−1 extends to a map f˜ : X → Sn−1 . 5.1.7 We can use this theorem to prove the following: Theorem 1.12.4 For any open set U ⊂ Rn , every continuous injection f : U → Rn is an open embedding. Proof 36 Let x ∈ U and take any compact neighborhood C in Rn with C ⊂ U . Then, f |C : C → h(C) is a homeomorphism. We show that f (x) ∈ int f (C). On the contrary, assume that f (x) ∈ bd f (C). For each neighborhood V of x in C, since f (V ) is a neighborhood of f (x) in f (C), we can apply Theorem 1.12.3 to find a neighborhood W of f (x) in f (C) such that W ⊂ h(V ) and every map g : f (C) \ W → Sn−1 extends to a map g˜ : f (C) → Sn−1 . Then, f −1 (W ) is a neighborhood of x in C with f −1 (W ) ⊂ V . For every map h : C\f −1 (W ) → Sn−1 , hf −1 : f (C) \ W → Sn−1 extends to a map g˜ : f (C) → Sn−1 . Thus, we have a map gf ˜ : C → Sn−1 such that gf ˜ |C \ f −1 (W ) = (g|f ˜ (C) \ W )(f |C \ f −1 (W )) = hf −1 (f |C \ f −1 (W )) = h. Due to Theorem 1.12.3, this means that x ∈ bd C, which is a contradiction. Therefore, f (C) is a neighborhood of x in Rn and f (C) ⊂ f (U ). Consequently, f (U ) is open in Rn and f : U → f (U ) is a homeomorphism, hence f is an open embedding. " ! The INVARIANCE OF DOMAIN can be obtained as a corollary of the above theorem: Corollary 1.12.5 (INVARIANCE OF DOMAIN) For each pair of subsets X, Y ⊂ Rn , X ≈ Y implies int X ≈ int Y . 5.1.8 35 Such 36 The

a map r is called a retraction, which is discussed in Sect. 1.13. proof is the same as the Invariance of Domain in [GAGT, p. E2].

1.12 Dimensions

63

Indeed, let h : X → Y be a homeomorphism. By Theorem 1.12.4, h| int X : int X → Rn and h−1 | int Y : int Y → Rn are open embeddings. Then, it follows that h(int X) ⊂ int Y and h−1 (int Y ) ⊂ int X. Therefore, h(int X) = int Y , which means that int X ≈ int Y .

Let A and B be disjoint closed sets in a space X. A closed set C in X is called a partition between A and B in X if there exist disjoint open sets U and V in X such that A ⊂ U , B ⊂ V and X \ C = U ∪ V . A family (Aγ , Bγ )γ ∈ of pairs of disjoint closed sets in X is inessential in X if there are partitions Lγ between Aγ and Bγ with γ ∈ Lγ = ∅. Note that if one of Aγ or Bγ isempty then (Aγ , Bγ ) is inessential. If (Aγ , Bγ )γ ∈ is not inessential in X (i.e., γ ∈ Lγ = ∅ for any partitions Lγ between Aγ and Bγ ), it is said to be essential in X. For example, the −1 n n n-cube In has an essential family (pr−1 i (0), pri (1))i=1 , where pri : I → I is the projection onto the i-th factor (refer to [GAGT, Corollary 5.2.16]). Using essential families, we can also characterize dimension as follows: Theorem 1.12.6 (EILENBERG–OTTO CHARACTERIZATION) For a normal space X and n ∈ N, dim X  n if and only if X has an essential family of n pairs of disjoint closed sets. 5.2.17 By definition, it is obvious that dim Y  dim X if Y is a closed subspace of X. However, there is a zero-dimensional compact space X containing a subspace Y with dim Y > 0 (cf. [GAGT, Theorem 5.5.3]). For metrizable spaces, the following SUBSET THEOREM holds: Theorem 1.12.7 (SUBSET) For every subset Y of a metrizable space X, dim Y  dim X. 5.3.3 Additional fundamental theorems on Dimension Theory are listed below:  Theorem 1.12.8 (COUNTABLE SUM) Let X = i∈N Fi be normal and n ∈ ω, where each Fi is closed in X. If dim Fi  n for every i ∈ N, then dim X  n. 5.4.1 Theorem 1.12.9 (LOCALLY FINITE SUM) Let X be normal and n ∈ ω. If X has a locally finite closed cover {Fλ | λ ∈ } such that dim Fλ  n for every λ ∈ , then dim X  n. 5.4.2 Theorem 1.12.10 (DECOMPOSITION) Let X be metrizable and n ∈ ω. Then, dim X  n if and only if X is covered by n + 1 many subsets X1 , . . . , Xn+1 with dim Xi  0. 5.4.5 In the Decomposition Theorem 1.12.10 above, each Xi is not closed in X in general. In fact, a countable union of zero-dimensional closed subspaces is zerodimensional by Theorem 1.12.8. Theorem 1.12.11 (ADDITION) For any two subspaces X and Y of a metrizable space, dim X ∪ Y  dim X + dim Y + 1.

5.4.8

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Theorem 1.12.12 (PRODUCT) For any two metrizable spaces X and Y , dim X × Y  dim X + dim Y.

5.4.9

In the above, the equality dim X × Y = dim X + dim Y does not hold. However, if one of X or Y is a locally compact polyhedron (or a metric polyhedron), then the equality does hold. This fact can be proved by using the cohomological dimension (cf. Theorem 7.9.7 of [GAGT]). According to the Eilenberg–Otto Characterization Theorem 1.12.6, the infinitedimensionality can be characterized as follows: dim X = ∞ ⇔ ∀n ∈ N, X has an essential family of n pairs of disjoint closed sets. A space X is said to be strongly infinite-dimensional (s.i.d.) if X has an infinite essential family of pairs of disjoint closed sets. For example, the Hilbert cube Q = [−1, 1]N is strongly infinite-dimensional because it has an essential family −1 (pr−1 i (−1), pri (1))i∈N , where pri : Q → [−1, 1] is the projection onto the i-th factor (refer to [GAGT, Theorem 5.6.1]). It easily follows from the definition that a space is strongly infinite-dimensional if it contains an strongly infinite-dimensional closed subspace. Thus, RN and space 2 is also strongly infinite-dimensional  Hilbert−n N 2 because R ⊃ Q and  ⊃ n∈N [−2 , 2−n ] ≈ QN . Obviously, if X is s.i.d. then dim X = ∞. It is said that X is weakly infinite-dimensional (w.i.d.) if dim X = ∞ and X is not s.i.d., that is, for every family (Ai , Bi )i∈N of pairs  of disjoint closed sets in X, there are partitions Li between Ai and Bi such that i∈N Li = ∅.37 It is said that X is (strongly) countable-dimensional ((s.)c.d.) if X is a countable union of f.d. normal (closed) subspaces. Obviously, every strongly countabledimensional space is countable-dimensional. By virtue of Theorem 1.12.10, we have the following characterization of the countable-dimensionality: • A metrizable space is countable-dimensional if and only if it is a countable union of zero-dimensional subspaces. It should be remarked that a countable-dimensional space is never strong infinitedimensional, that is: Theorem 1.12.13 A countable-dimensional metrizable space X with dim X = ∞ is weakly infinite-dimensional. Equivalently, any strongly infinite-dimensional metrizable space is not countable-dimensional. 5.6.2 The following EMBEDDING THEOREM will be used in Chaps. 5 and 6:

37 In many articles, the infinite-dimensionality is not assumed, i.e., w.i.d. = not s.i.d., so f.d. implies w.i.d. However, here we assume the infinite-dimensionality because we discuss the difference among infinite-dimensional spaces.

1.13 Absolute Neighborhood Retracts

65

Theorem 1.12.14 (EMBEDDING) Every separable metrizable space with dim  n can be embedded in I2n+1 , hence it can be embedded in the Euclidean space R2n+1 . 5.8.1 This Embedding Theorem can be obtained as a corollary of the following strengthened result: Theorem 1.12.15 (EMBEDDING APPROXIMATION) For a separable metrizable space X with dim X  n, the following hold: (1) If X is compact, then every map f : X → I2n+1 can be approximated by embeddings, that is, for each ε > 0, there is an embedding h : X → I2n+1 that is ε-close to f . 5.8.5 (2) If X is locally compact, then every proper map f : X → R2n+1 can be approximated by closed embeddings, that is, for each U ∈ cov(R2n+1 ), there is a closed embedding h : X → R2n+1 that is U-close to f . 5.8.10

1.13 Absolute Neighborhood Retracts A subset A of a space X is called a retract of X if there is a map r : X → A such that r|A = id, which is called a retraction. As is easily observed, every retract of a space X is closed in X. A neighborhood retract of X is a closed set in X that is a retract of some neighborhood in X. A metrizable space X is called an absolute neighborhood retract (ANR) (resp. an absolute retract (AR)) if X is a neighborhood retract (resp. a retract) of an arbitrary metrizable space containing X as a closed subspace. A space Y is called an absolute neighborhood extensor for metrizable spaces (ANE) if each map f : A → Y from any closed set A in an arbitrary metrizable space X extends over some neighborhood U of A in X. When f can always be extended over X (i.e., U = X in the above), we call Y an absolute extensor for metrizable spaces (AE). As is easily observed, every metrizable ANE (resp. a metrizable AE) is an ANR (resp. an AR). Moreover, every neighborhood retract of an ANE is an ANE and every retract of an AE is an AE. The following theorems are fundamental to ANR theory. Theorem 1.13.1 (DUGUNDJI EXTENSION) Let E be a locally convex topological linear space and X be a metrizable space with a closed set A ⊂ X. Then, each map f :A→E can be extended to a map f˜ : X→E such that the image f˜(X) is contained in the convex hull f (A) of f (A), hence every convex set is an AE. 6.1.1 Remark 1.11 In the above, when E = (E,  · ) is a normed linear space and f is bounded, the extension f˜ can be taken so as to be bounded and have the same sup-norm as f , (i.e., f˜ = f ). Cf. Remark 2 in [GAGT, Chapter 6]. Theorem 1.13.2 (ARENS–EELLS EMBEDDING) Every (complete) metric space X can be isometrically embedded in a (complete) normed linear space E with dens E = ℵ0 dens X as a linearly independent closed set. 6.2.1

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By combining these two theorems, we can obtain the following: • a metrizable space X is an ANR (resp. an AR) if and only if X is an ANE (resp. an AE). Let X be a space and A ⊂ X. It is said that A is contractible in X if the inclusion A ⊂ X is null-homotopic, that is, there is a homotopy h : X × I → X such that h0 = id and h1 is constant. Such a homotopy h is called a contraction of A in X. When A = X, we say that X is contractible and h is called a contraction of X. As is easily observed, X is contractible if and only if X  0 (X is homotopic to a singleton). It is said that X is locally contractible if any neighborhood U of each point x ∈ X contains a neighborhood of x that is contractible in U . Proposition 1.13.3 (1) A contractible ANE is an AE. (2) A metrizable space is an AR if and only if it is a contractible ANR. (3) Every ANR is locally contractible.

6.1.5 6.2.9 6.2.8

The converse of the above (3) does not hold in general, that is, locally contractible metrizable space is not necessary to be an ANR. For such an example, refer to [GAGT, Theorem 6.3.8]. For simplicial complexes, we have the following: Theorem 1.13.4 For any simplicial complex K, the polyhedron |K| is an ANE and 6.1.4 6.2.6 |K|m is an ANR. Basic properties of ANEs are listed below: Proposition 1.13.5 (Basic Properties of ANEs) (1) An arbitrary product of AEs is an AE and a finite product of ANEs is an ANE. 6.1.9(1)

(2) A retract of an AE is an AE and a neighborhood retract of an ANE is an ANE. 6.1.9(2)

(3) Any open set in an ANE is also an ANE. 6.1.9(3) (4) (HANNER THEOREM) A paracompact space is an ANE if it is locally an ANE, that is, each point has an ANE neighborhood. 6.1.9(6) (5) Let X = X1 ∪ X2 , where X1 and X2 are closed in X. If X1 , X2 , and X1 ∩ X2 are ANEs (AEs), then so is X. If X and X1 ∩ X2 are ANEs (AEs), then so are X1 and X2 . 6.1.9(7) Since a metrizable space is an ANR (resp. an AR) if and only if it is an ANE (resp. an AE), the above list can be translated to ANRs as follows: Proposition 1.13.6 (Basic Properties of ANRs) (1) An countable product of ARs is an AR and a finite product of ANRs is an ANR. 6.2.10(1)

(2) A retract of an AR is an AR and a neighborhood retract of an ANR is an ANR. 6.2.10(2)

1.13 Absolute Neighborhood Retracts

67

(3) Any open set in an ANR is also an ANR. 6.2.10(3) (4) (HANNER THEOREM) A paracompact space is an ANR if it is locally an ANR, that is, each point has an ANR neighborhood. 6.2.10(4) (5) Let X = X1 ∪ X2 , where X1 and X2 are closed in X. If X1 , X2 , and X1 ∩ X2 are ANRs (ARs), then so is X. If X and X1 ∩ X2 are ANRs (ARs), then so are X1 and X2 . 6.2.10(5) The following is a very useful procedure to extend homeomorphisms: Theorem 1.13.7 (KLEE’S TRICK) Let E and F be metrizable topological linear spaces which are AEs and let A and B be homeomorphic closed sets in E and F , respectively. Then, each homeomorphism f : A × {0} → {0} × B can be extended to a homeomorphism f˜ : E × F → E × F . 6.2.2 Klee’s Trick can be applied to prove Hausdorff’s Metric Extension Theorem: Theorem 1.13.8 (HAUSDORFF’S METRIC EXTENSION) Let A be a closed set in a (completely) metrizable space X. Every admissible (complete) metric on A extends to an admissible (complete) metric on X. 6.2.3 In Theorem 1.13.2, if an embedding is not required to be an isometry, we have the following alteration: Theorem 1.13.9 Every completely metrizable space can be embedded in a Hilbert space with the same density as a closed set. And every metrizable space can be embedded in a pre-Hilbert space (that is, a linear subspace of a Hilbert space) with 6.2.4 the same density as a closed set. Let V be an open refinement of an open cover U of a space X. We call V an h-refinement (resp. h ¯ -refinement) of U if any two V-close maps f, g : Y → X defined on an arbitrary space Y are U-homotopic (resp. U-homotopic rel. {y ∈ Y | f (y) = g(y)}), where we write V ≺ U or U  V h

h

resp. V ≺ U h¯

or U  V . h¯

Theorem 1.13.10 Every open cover of an ANR has an h¯ -refinement (hence it has 6.3.5 an h-refinement). The following is frequently used in this book: Theorem 1.13.11 (HOMOTOPY EXTENSION) Let Y be an ANE, U be an open cover of Y , and h : A × I → Y be a U-homotopy of a closed set A in a metrizable space X. If h0 is extended to a map f : X → Y , then h can be extended to a U-homotopy h˜ : X × I → Y with h˜ 0 = f . 6.4.1 Let X be a subspace of Y . A homotopy h : X × I → Y with h0 = id is called deformation of X in Y . When X = Y , h is called a deformation of X. A subset A of a space X is said to be a deformation retract of X if there is a deformation h : X × I → X such that h1 is a retraction of X onto A, where h1 is called a

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1 Preliminaries and Background Results

deformation retraction of X onto A. When ht |A = id for all t ∈ I in the above, we call A a strong deformation retract of X and h1 a strong deformation retraction of X onto A.38 A retraction r : X → A is a deformation retraction (resp. strong deformation retraction) if and only if r  id (resp. r  id rel. A). A closed set A ⊂ X is a neighborhood deformation retract of X if A has a neighborhood U in X with a homotopy h : U × I → X such that h0 = id and h1 is a retraction of U onto A, where h1 is called a deformation retraction of U onto A in X. When ht |A = id for all t ∈ I in the above, we call A a strong neighborhood deformation retract of X and h1 a strong deformation retraction of U onto A in X. Proposition 1.13.12 A retract of an AR is a strong deformation retract and a neighborhood retract of an ANR is a strong neighborhood deformation 6.2.7 retract. Using the Homotopy Extension Theorem 1.13.11, we can also show the following: Proposition 1.13.13 A deformation retract of an ANR is a strong deformation retract. 6.4.4 It is said that X is deformable into A (⊂ X) if there is a deformation h : X × I → X with h1 (X) ⊂ A. Obviously, if A is a deformation retract of X, then X is deformable into A and A is a retract of X. The converse also holds, that is: Proposition 1.13.14 A subset A of a space X is a deformation retract of X if and p. E3 only if X is deformable into A and A is a retract of X.39 The following can be easily proved: Proposition 1.13.15 Let A be an ANR that is a closed subset of an ANR X. Then, the inclusion A ⊂ X is a homotopy equivalence if and only if A is a strong deformation retract of X. Proof The “if” part is trivial. In fact, a strong deformation retraction of X onto A is a homotopy inverse of the inclusion A ⊂ X. To see the “only if”, let f : X → A be a homotopy inverse of the inclusion A ⊂ X. Since f |A  id in A, we can apply the Homotopy Extension Theorem 1.13.11 to obtain a retraction r : X → A such that r  f in A. Moreover, because f  id in X, X is deformable into A. Hence, A is a deformation retract of X by Proposition 1.13.14. Then, it follows from Proposition 1.13.13 that A is a strong deformation retract of X. " ! The following is very useful: Theorem 1.13.16 (KRUSE–LIEBNITZ) Let X be metrizable and A be a strong neighborhood deformation retract of X. If A and X \ A are ANRs, then so is X. 6.5.2

38 A

strong deformation retract was defined on p. 50. to Erratum of [GAGT], Comment for p. 348.

39 Refer

1.13 Absolute Neighborhood Retracts

69

For a map f : X → Y , we define the mapping cylinder M(f ) as the space Y ∪ (X × (0, 1]) (the disjoint union) with the topology generated by open sets in X × (0, 1] and sets V ∪ f −1 (V ) × (0, ε), where V is open in Y and 0 < ε < 1. If X and Y are metrizable, then M(f ) is also metrizable, but Mf is not in general. Then, we call M(f ) the metrizable mapping cylinder to distinguish it from Mf . For the following facts, refer to Section 6.5 (pp. 362–363) in [GAGT]. Fact (1) For each map f : X → Y between completely metrizable spaces, M(f ) is also completely metrizable. (2) For a perfect map f : X → Y between metrizable spaces, Mf is metrizable and Mf ≈ M(f ). As an easy application of the Kruse–Liebnitz Theorem 1.13.16 above, we have the following: Corollary 1.13.17 For any map f : X → Y between ANRs, the mapping cylinder M(f ) is an ANR. 6.5.4 The (metrizable) cone C(X) (resp. the (metrizable) open cone C o (X)) over a (metrizable) space X is defined as the space {0} ∪ X × (0, 1] (resp. C o (X) = {0} ∪ X × (0, 1)) with the topology generated by open sets in X × (0, 1] (resp. X × (0, 1)) and sets {0} ∪ X × (0, ε), 0 < ε < 1. In other words, the cone C(X) is the mapping cylinder M(c0 ) of the constant map c0 : X → {0}. Recall that the usual cone CX is defined as the quotient space CX = (X × I)/(X × {0}), which is another mapping cylinder Mc0 of the constant map c0 : X → {0}. If X is metrizable, then C(X) is also metrizable but CX is not in general. Moreover, if X is completely metrizable, then so are C(X) and C o (X). When X is compact, CX ≈ C(X). The following is the special case of Corollary 1.13.17: Corollary 1.13.18 The cone C(X) over any ANR X is an AR, hence so is the open cone C o (X). 6.5.7 Let X be a space and U ∈ cov(X). Let K be a simplicial complex and L be a subcomplex of K with K (0) ⊂ L. Then, a map f : |L| → X is called a partial U-realization of K in X if {f (σ ∩ |L|) | σ ∈ K} ≺ U. A full U-realization of K in X is a map f : |K| → X such that {f (σ ) | σ ∈ K} ≺ U. We call V ∈ cov(X) a Lefschetz refinement of U and denote V ≺ U or U  V L

L

if V ≺ U and any partial V-realization of an arbitrary simplicial complex in X extends to a full U-realization in X. The following is called LEFSCHETZ’S CHARACTERIZATION of ANRs: Theorem 1.13.19 (LEFSCHETZ) A metrizable space X is an ANR if and only if any open cover of X admits a Lefschetz refinement. 6.6.1

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1 Preliminaries and Background Results

Recall that given U ∈ cov(X) (resp. ε > 0), X is U-homotopy dominated (resp. ε-homotopy dominated) by Y if there are maps f : X → Y and g : Y → X such that gf U idX (resp. gf ε idX ). The following is called HANNER’S CHARACTERIZATION of ANRs: Theorem 1.13.20 (HANNER) For a metric space X = (X, d), the following are equivalent: (a) X is an ANR; (b) For each open cover U of X, there is a simplicial complex K such that X is U-homotopy dominated by |K|; 6.6.2 (c) For any ε > 0, X is ε-homotopy dominated by an ANE. Remark 1.12 In condition (b) above, we can require K to be locally finitedimensional and card K (0)  w(X). If X is compact, then K can be replaced by a finite simplicial complex. Indeed, let f : X → |K| and g : |K| → X be maps such that gf U idX . Then, because of compactness, f (X) is contained in |L| for some finite subcomplex L of K. Hence, X is U-homotopy dominated by L. Moreover, when, X is separable, we can take a locally finite, countable simplicial complex as K. Indeed, in the proof of (a) ⇒ (b) of Theorem 6.6.2 in [GAGT], we can take a countable star-finite open cover W by Theorem 1.11.10. — Refer to Remark 9 for Theorem 6.6.2 in [GAGT]. Theorem 1.13.21 Let f : X → Y be a map from a paracompact space X to an ANR Y and U be an open cover of Y . Then, each open cover V of X has an open refinement W with a map ψ : |N(W)| → Y such that ψϕ U f for any canonical map ϕ : X → |N(W)|. 6.6.3 Remark 1.13 In the above theorem, we can take a locally finite σ -discrete open refinement W of V such that the nerve N(W) is locally finite-dimensional. When X is separable, we can take a star-finite countable open refinement W of V such that the nerve N(W) is locally finite. If X is compact, then a finite open refinement W can be taken. — Refer to Remark 10 for Corollary 6.6.3 in [GAGT]. By Corollary 1.10.6 and Theorem 1.13.20, we obtain the following: Theorem 1.13.22 Every ANR X has the homotopy type of a locally finitedimensional simplicial complex K with card K (0)  w(X). Moreover, every separable ANR has the homotopy type of a countable locally finite simplicial complex. 6.6.4 For locally compact ANRs, the following can be proved in the same way as the implication (a) ⇒ (b) of Theorem 1.13.20: Theorem 1.13.23 Let X be a locally compact ANR. For each open cover U of X, there is a locally finite simplicial complex K such that X is proper U-homotopy dominated by |K|.

1.13 Absolute Neighborhood Retracts

71

Sketch of Proof Take open refinements as follows: ∗

∗

U  U  V  V  W  W , h

L

where U is locally finite, every U ∈ U has the compact closure, and W is star-finite (cf. Theorem 1.11.9). Then, the nerve K of W is locally finite. Let f : X → |K| be a canonical map. Choosing g  (W ) ∈ W ∈ W , we have a partial W-realization g  : K (0) → X, which can be extended to a full V -realization g : |K| → X Then, observe that f and g are proper, p and gf W id (cf. Proposition 1.3.9).

Combining Theorems 1.10.7 and 1.13.23, we have the following: Theorem 1.13.24 For every locally compact ANR X, the product space X × R+ has the proper homotopy type of a locally finite simplicial complex. " ! A subset A of a space X said to be homotopy dense in X if there exists a homotopy h : X × I → X such that h0 = id and h(X × (0, 1]) ⊂ A. Proposition 1.13.25 A subset X of a metrizable space Y is homotopy dense in Y if and only if the inclusion map X ⊂ Y is a fine homotopy equivalence. 6.7.1 Theorem 1.13.26 Let X be a metrizable space and A be a homotopy dense subset of X. Then, X is an ANR if and only if A is an ANR. 6.6.7 Proposition 1.13.27 Let x0 ∈ A ⊂ X. If A is homotopy dense in X and X is N contractible, then the following set AN f is homotopy dense in X :    N  AN x(n) = x0 except for finitely many n ∈ N . f = x ∈A

6.6.8

A space X is said to be homotopically trivial provided, for each n ∈ N, every map f : Sn−1 → X is null-homotopic, that is, f extends over Bn . Proposition 1.13.3(2) can be improved as follows: Theorem 1.13.28 A metrizable space is an AR if and only if it is a homotopically trivial ANR. 6.6.10 A map f : X → Y is called a weak homotopy equivalence40 provided that, for each n ∈ N and each map α : Sn−1 → X, if f α extends to a map β : Bn → Y , then α extends to a map α˜ : Bn → X such that f α˜  β in Y rel. Sn−1 : X α

Sn−1

f

α˜

⊂

Y β

Bn

In some literature, the following is adopted as the definition: 40 It

is also called an ∞-equivalence because it is an n-equivalence for every n ∈ N.

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1 Preliminaries and Background Results

Theorem 1.13.29 A map f : X → Y is a weak homotopy equivalence if and only if f induces the bijection between the path-components and the isomorphisms between the homotopy groups. 4.14.12 Theorem 1.13.30 Let X and Y be ANRs. Then, every weak homotopy equivalence f : X → Y is a homotopy equivalence. 6.6.6 Let X = (X, dX ) and Y = (Y, dY ) be metric spaces and A be a closed set in X. A map f : X → Y is said to be uniformly continuous at A if, for each ε > 0, there is some δ > 0 such that for each a ∈ A and x ∈ X, dX (a, x) < δ implies dY (f (a), f (x)) < ε. We call A a uniform retract of X if there is a retraction r : X → A that is uniformly continuous at A. A subset U ⊂ X such that dist(A, X \ U ) > 0 is called a uniform neighborhood of A in X. When A has a uniform neighborhood U in X and A is a uniform retract of U , A is called a uniform neighborhood retract of X. As is easily observed, every compact (neighborhood) retract of X is a uniform (neighborhood) retract. A metric space X is called a uniform AR (resp. uniform ANR) if X is a uniform retract (resp. a uniform neighborhood retract) of an arbitrary metric space containing X as a closed set. It should be noted that every compact AR (ANR) is a uniform AR (ANR). Theorem 1.13.31 Every convex set in a locally convex metric linear space is a uniform AR. 6.8.5 By virtue of the following modification of Theorem 1.13.10, uniform ANRs are very useful: Theorem 1.13.32 Let X be a uniform ANR. For each ε > 0, there exists δ > 0 such that any two δ-close maps of an arbitrary space to X are ε-homotopic. 6.8.6 We need the following results concerning uniform ANRs. Theorem 1.13.33 A uniform ANR is a uniform AR if and only if it is homotopically trivial. 6.8.7 Proposition 1.13.34 A uniform ANR X is homotopy dense in every metric space Z that contains X isometrically as a dense subset. 6.8.8 Theorem 1.13.35 A metric space X is a uniform ANR (resp. a uniform AR) if and  of X is a uniform ANR (resp. a uniform AR) and X only if the metric completion X  is homotopy dense in X. 6.8.10 Proposition 1.13.36 Let X be a subset of a metric space M = (M, d). If X is an ANR, then X is homotopy dense in some Gδ -set Y in M, hence Y is also an ANR. 6.8.2

Theorem 1.13.37 For any admissible metric d on an AR (resp. ANR) X, X has an admissible metric ρ  d such that (X, ρ) is a uniform AR (resp. uniform ANR). If d is bounded, then so is ρ. 6.8.11

1.14 Locally Equi-Connected Spaces

73

Every ANR is locally contractible but the converse does not hold in general. However, we have the following: Theorem 1.13.38 (HAVER) metrizable space is an ANR.

Every countable-dimensional locally contractible 6.10.1

1.14 Locally Equi-Connected Spaces This section contains complementary results of Section 6.3 of the book [GAGT], which will be referred to in Sect. 6.13 of the present book.

For a space X, let X denote the diagonal of X2 , that is,    X = (x, x)  x ∈ X ⊂ X2 . For a each A ⊂ X2 and x ∈ X, we write A(x) = {y ∈ X | (x, y) ∈ A}. Each neighborhood U of X in X2 gives  every x ∈ X its neighborhood U (x) simultaneously. Given U ∈ cov(X), W = U ∈U U 2 is an open neighborhood of X in X2 , where W (x) = st(x, U) for each x ∈ X. A space X is said to be locally equi-connected (LEC) if the diagonal X of X2 has a neighborhood U with a map λ : U × I → X such that λ(x, y, 0) = x and λ(x, y, 0) = y for each (x, y) ∈ U and λ(x, x, t) = x for each x ∈ X and t ∈ I, that is, I  t → λ(x, y, t) ∈ X is a path from x to y in X, where a map λ is called an equi-connecting map for X. When U = X2 , X is said to be equi-connected (EC). Every topological linear space E is EC with the equi-connecting map defined by E 2 × I  (x, y, t) → (1 − t)x + ty ∈ E.41 Basic results are listed below: Proposition 1.14.1 (Basic Property of LEC Spaces) 6.3.1 (1) An AR is EC and an ANR is LEC. (2) A space X is EC (LEC) if and only if X is a strong deformation (neighborhood) retract of X2 . 6.3.2

41 More generally, it is known that every semi-locally contractible topological group is LEC (cf. [GAGT, p. 349]).

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1 Preliminaries and Background Results

(3) A metrizable space X is EC if and only if X is contractible and LEC.42 6.3.3 (4) A space X is LEC if and only if each open cover of X has an h¯ -refinement. 6.3.4

(5) An arbitrary product of EC spaces is EC and a finite product of LEC spaces is LEC. 6.3.9(1) (6) A retract of an EC space is EC and a neighborhood retract of an LEC space is LEC. 6.3.9(2) (7) Any open set in an LEC space is also LEC. 6.3.9(3) (8) A metrizable space is LEC if each point of X has an LEC neighborhood in X. 6.3.9(5)

For a metrizable LEC space X, Lefschetz’s Characterization of ANRs (Theorem 1.13.20) can be weakened as follows: Theorem 1.14.2 (DUGUNDJI) A metrizable LEC space X is an ANR if and only if each open cover V has an open refinement W such that for an arbitrarily simplicial complex K, every partial W-realization f : |K (0)| → X of K extends to a full Vrealization f˜ : |K| → X of K. Proof It is enough to show the “if” part. By Hanner’s Characterization of ANRs (Theorem 1.13.19), it suffices to show that for each V ∈ cov(X), there is a simplicial complex K such that X is V-homotopy dominated by |K|. Take V , V ∈ cov(X) so that st V ≺ V and V is an h¯ -refinement of V. By the condition, we have W ∈ cov(X) such that for an arbitrary simplicial complex K, every partial W -realization g : |K (0)| → X extends to a full V -realization g˜ : |K| → X. Let W ∈ cov(X) be a star-refinement of W . Now, let K be the nerve of W with f : X → |K| a canonical map. We will show that X is V-homotopy dominated by |K|. Choosing a point g0 (W ) ∈ W for each W ∈ W = K (0), we define a map g0  : |K (0)| → X. For each simplex n σ g0 (σ (0) ) ⊂ n= W0 , . . . , Wn  ∈ K, we can take x0 ∈ i=0 Wi ( = ∅). Then, (0) i=0 Wi ⊂ st(x0 , W). Since W is a star-refinement of W , g0 (σ ) is contained in some member of W . Thus, g0 is a partial W -realization of K. Hence, g0 extends to a full V -realization g : |K| → X of K. To see that gf V id, it suffices to show that gf is V -close to id. Due to Proposition 1.11.1, for each x ∈ X, cK (f (x))(0) = {W0 , . . . , Wn } ⊂ W[x]. As we saw above, g(cK (f (x))(0) ) ⊂ st(x, W). Since g is a full V realization of K, g(cK (f (x))) is contained in some V ∈ V . Hence, x, gf (x) ∈ st(V , W) ⊂ st(V , V ). Because st V ≺ V , it follows that gf is V -close to id. The proof is completed. " ! Remark 1.14 In the above proof, it suffices to take V as an h-refinement of V instead of h-refinement. Therefore, Theorem 1.14.2 is valid for a metrizable unified ¯ locally contractile (ULC) space X. Refer to Theorem 6.3.6 in [GAGT].43

42 More

generally, this is valid for a space X such that X 2 is normal. is said that X is unified locally contractile (ULC) if each neighborhood U of X in X 2 contains a neighborhood V of X with a homotopy h : V × I → U such that h0 = id, 43 It

1.14 Locally Equi-Connected Spaces

75

Let X be an LEC space with an equi-connecting map λ : U × I → X. A subset A ⊂ X is said to be λ-convex if A2 ⊂ U and λ(A2 × I) ⊂ A, that is, λ(x, y, t) ∈ A for every x, y ∈ A and t ∈ I. For A ⊂ X with A2 ⊂ U , the λ-extension of A ⊂ X is the smallest subset Aλ ⊂ X such that A × Aλ ⊂ U , and λ(A × Aλ × I) ⊂ Aλ , where A ⊂ λ(A × Aλ × I) = Aλ because Aλ ⊂ λ(A × Aλ × I) ⊂ Aλ . We can construct such an Aλ as follows: First, let A0 = A and inductively define An = λ(A × An−1 × I). Then, we have Aλ = n∈ω An . Indeed, λ(A × Aλ × I) =



λ(A × An × I) ⊂

n∈ω



An ⊂ Aλ .

n∈N

If λ(A × B × I) ⊂ B, then A0 = A ⊂ B and it inductively follows that An = λ(A × An−1 × I) ⊂ λ(A × B × I) ⊂ B. Hence, Aλ =

 n∈ω

An ⊂ B.

For A ⊂ B ⊂ X such that A2 ⊂ U , it is said that A is λ-stable in B if Aλ ⊂ B. Evidently, each λ-convex set A is λ-stable in itself, hence in any B containing A. We define a λ-stable refinement of an open cover V of X as an open refinement W such that W 2 ⊂ U for every W ∈ W and each W ∈ W is λ-stable in some V ∈ V. Theorem 1.14.3 (DUGUNDJI) Let X be a metrizable LEC space with an equiconnecting map λ : U × I → X. If every open cover V ∈ cov(X) has a λ-stable refinement W ∈ cov(X), then X is an ANR. Proof By virtue of Theorem 1.14.2, it suffices to show that if W is a λ-stable refinement of V ∈ cov(X) and K is a simplicial complex, then every partial Wrealization f : |K (0)| → X extends to a full V-realization f˜ : |K| → X. Let f : |K (0)| → X be a partial W-realization of K, where we may assume that K is an ordered simplicial complex. Each σ ∈ K with dim σ > 0 can be written σ = vσ σ∗ , where vσ = min σ (0) and σ∗ < σ is the opposite face of σ to vσ . If τ is an (n − 1)-face of σ and τ = σ∗ , then vτ = vσ and τ∗ < σ∗ . We can inductively define maps fn : σ → X, n ∈ ω, as follows: fn ((1 − t)vσ + tx) = λ(f (vσ ), x, t) for each n-simplex σ ∈ K, x ∈ σ∗ , and t ∈ I, where f0 = f . It can be inductively shown that fn is well-defined and fn ||K (n−1) | = fn−1 . Because fn (σ ) = λ({vσ } × σ∗ × I) for every n-simplex σ ∈ K, we can see by induction that f (σ (0) ) ⊂ W implies fn (σ ) ⊂ W λ for every n-simplex σ ∈ K. Since f is a partial W-realization of K and W is a λ-stable

h1 (V ) = X , and pr1 ht = pr1 |V for every t ∈ I. In Theorem 6.3.6 in [GAGT], it is proved that a paracompact space X is ULC if and only if each open cover of X has an h-refinement.

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1 Preliminaries and Background Results

refinement of V, the desired full V-realization f˜ : |K| → X can be defined by f˜||K (n) | = fn , n ∈ ω. ! " Let X be a metrizable LEC space with an equi-connecting map λ : U × I → X. Suppose that each point x ∈ X has arbitrarily small λ-convex neighborhoods, that is, for each open neighborhood V of x, there is a λ-convex neighborhood W of x such that W ⊂ V . Then, (int W )λ ⊂ W λ = W . Hence, every open cover V has a λ-stable open refinement W. Thus, we have the following: Corollary 1.14.4 A metrizable LEC (resp. EC) space X is an ANR (resp. AR) if X has an equi-connecting map λ such that each point of X has arbitrarily small λ-convex neighborhoods. " !

1.15 Cell-Like Maps and Fine Homotopy Equivalences A compact set A in a space X is said to be cell-like in X if A is contractible in every neighborhood of A in X. A compactum X is cell-like if X is cell-like in some metrizable space that contains X as a subspace. A cell-like (CE) map is a perfect map f : X → Y such that every fiber f −1 (y) is cell-like. There is a close relationship between cell-like maps and fine homotopy equivalences. A fine homotopy equivalence is defined on p. 39, for which the perfectness is not required. As a related concept, a soft map f : X → Y is defined as a map such that for any metrizable space Z (with dim Z  n), and any map g : C → X of a closed set C ⊂ Z, if fg extends to a map h : Z → Y , then g extends to a map g˜ : Z → X with f g˜ = h: X g

C

f g˜

⊂

Y h

Z

In the above, evidently, if Y is an AE or an ANE, then so is X. We have several varieties of soft maps. In the definition of a soft map, replacing (Z, C) with the pair of a polyhedron and its subpolyhderon C, a polyhedrally soft map f : X → Y can be defined. A homotopically soft map (resp. an approximately soft map) f : X → Y is defined as a map such that for any metrizable space Z, any map g : C → X of a closed set C ⊂ Z, and each open cover U of Y , if fg extends to a map h : Z → Y , then g extends to a map g˜ : Z → X with f g˜ U h (resp. f g˜ is U-close to h). In this definition, replacing (Z, C) with the pair of a polyhedron and its subpolyhderon C, a polyhedrally homotopically soft map (resp. a polyhedrally approximately soft map) f : X → Y can be defined:

1.15 Cell-Like Maps and Fine Homotopy Equivalences

X g

f

g˜

C

⊂

77

Y U

h

Z

We call a map f : X → Y a hereditary weak homotopy equivalence if f |f −1 (U ) : f −1 (U ) → U is a weak homotopy equivalence for every open set U ⊂ Y . A map f : X → Y is called a local n-connection if f (X) is dense in Y and every neighborhood of each y ∈ Y contains a neighborhood V of y such that, for each 0  i < n, every map g : Si → f −1 (V ) is null-homotopic in f −1 (U ), and, for each map g : Sn → f −1 (V ), fg is null-homotopic in U . A local ∞-connection is a map f : X → Y that is a local n-connection for every n ∈ ω. A map f : X → Y is called a local ∗-connection if f (X) is dense in Y and every neighborhood of each y ∈ Y contains a neighborhood V of y such that f −1 (V ) is contractible in f −1 (U ). By using concepts introduced above, fine homotopy equivalences between ANRs can be characterized as follows: Theorem 1.15.1 For a map f : X → Y between ANRs, the following are equivalent: (a) (b) (c) (d) (e) (f)

f f f f f f

is a fine homotopy equivalence; is approximately (= homotopically) soft; is polyhedrally approximately (= homotopically) soft; is a hereditary weak homotopy equivalence; is a local ∞-connection; is a local ∗-connection.

7.4.3

Remark 1.15 In the above theorem, instead of assuming that X is an ANR, it suffices to assume that X is an ANE with the following property: (∗) For each U ∈ cov(X), X is U-homotopy dominated by a polyhedron. Indeed, Proposition 7.4.2 in [GAGT] is valid even if “ANR” in (1) is replaced by “ANE” and “both X and Y are ANRs” in (2) is replaced by “X is an ANE and Y is an ANR” (by the same proof). Hence, the implications of the diagram in the proof of Theorem 7.4.3 in [GAGT] (p. 442) hold except (c) ⇒ (a). By Hanner’s Characterization 1.13.20, every ANR has property (∗), which is the property of X required in the proof of (c) ⇒ (a). In particular, Theorem 1.15.1 is valid for a (non-metrizable) polyhedron X and an ANR Y .44

The above theorem has the following corollaries: Corollary 1.15.2 Let f : X → Y be a fine homotopy equivalence between ANRs. Then, for every open set U in Y , f |f −1 (U ) : f −1 (U ) → U is also a fine homotopy equivalence. 7.4.4

44 More generally, Theorem 1.15.1 is valid if X and Y are ANRs for stratifiable spaces [23, Proposition 2].

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1 Preliminaries and Background Results

Corollary 1.15.3 Let f : X → Y be a fine homotopy equivalence between ANRs. If A ⊂ Y is contractible in an open neighborhood U in Y , then f −1 (A) is contractible in f −1 (U ). 7.4.5 If fi : Xi → Yi , i = 1, 2 are local ∗-connections, then f1 × f2 : X1 × X2 → Y1 × Y2 is clearly a local ∗-connection. Then, we have the following: Corollary 1.15.4 Let fi : Xi → Yi , i = 1, 2 be fine homotopy equivalences between ANRs. Then, f1 ×f2 : X1 ×X2 → Y1 ×Y2 is a fine homotopy equivalence.45 Remark 1.16 The above corollary is valid even if X1 and X2 are ANEs with the property (*) in Remark 1.15. Due to Proposition 1.13.25, a subset X of a metrizable space Y is homotopy dense in Y if and only if the inclusion map X ⊂ Y is a fine homotopy equivalent. As another corollary of Theorem 1.15.1, we have the following characterization of homotopy denseness: Corollary 1.15.5 Let X and Y be ANRs such that X is a dense subset of Y . Then, the following are equivalent: (a) X is homotopy dense in Y ; (b) For each open set U in Y , the inclusion U ∩ X ⊂ U is a weak homotopy equivalence, i.e., for each n ∈ N, if a map α : Sn−1 → U ∩ X extends to a map β : Bn → U then α extends to a map α˜ : Bn → U ∩ X such that α˜  β in U rel. Sn−1 ; (c) Every neighborhood U of each y in Y contains a neighborhood V of y in Y such that every map α : Sn−1 → V ∩ X is null-homotopic in U ∩ X for each n ∈ N. 7.4.6 From the equivalence of (a) and (b) in the above corollary, the following is easily derived: Corollary 1.15.6 Let X be a homotopy dense set in an ANR Y . Then, for every open set U in Y , X ∩ U is homotopy dense in U . " ! Applying Corollary 1.15.5, we can show that every locally homotopy dense subset of an ANR is homotopy dense, that is: Proposition 1.15.7 Let X be a subset of an ANR Y . If Y has an open cover U such that for each U ∈ U, X ∩ U is homotopy dense in U , then X is homotopy dense in Y. Proof First of all, note that each U ∈ U is an ANR as an open set in an ANR. It follows from Theorem 1.13.26 that X ∩ U is an ANR. By Proposition 1.13.6(4) (Hannar’s Theorem), X is an ANR.

45 This

corollary does not appear in [GAGT].

1.15 Cell-Like Maps and Fine Homotopy Equivalences

79

Let U be an open neighborhood of y ∈ Y . We have U  ∈ U such that y ∈ U  . As we saw above, U  and X ∩ U  are ANRs. By Corollary 1.15.5, U ∩ U  contains a neighborhood V of y in U  such that every map α : Sn−1 → V ∩ X ∩ U  (= V ∩ X) is null-homotopic in U ∩ U  for each n ∈ N. Then, every map α : Sn−1 → V ∩ X is null-homotopic in U for each n ∈ N. Due to Corollary 1.15.5, this means that X is homotopy dense in Y . " ! To Corollary 1.15.5, we can add one more condition, that is: Proposition 1.15.8 Let X and Y be ANRs with X ⊂ Y and d ∈ Metr(Y ). Then, X is homotopy dense in Y if and only if, for each compact set A in Y and ε > 0, there is a map f : A → X that is ε-close to id. Proof The “only if” part is trivial. To see the “if” part, take U , α, and β as in condition (b) of Corollary 1.15.5. Since U ∩ X is an ANR and Sn−1 is compact, we have δ > 0 such that a map α  : Sn−1 → U ∩ X, α   α in U ∩ X if α  is δ-close to α. Choose ε > 0 so that   ε < min δ, dist(β(Bn ), Y \ U ) . Applying the condition to β(Bn ) and ε > 0, we can find a map f : β(Bn ) → X that is ε-close to id. Thus, we can obtain a map fβ : Bn → X ∩ U such that fβ|Sn−1 is δ-close to β|Sn−1 = α, which implies that fβ|Sn−1  α in U ∩ X. We can apply the Homotopy Extension Theorem 1.13.11 to extend α to a map α˜ : Bn → U ∩ X. Therefore, X is homotopy dense in Y by Corollary 1.15.5. " ! It is said that a closed set A ⊂ X has Property U V n in X (or simply A is U V n in X) if each neighborhood U of A in X contains a neighborhood V of A such that, for each 0  i  n, every map α : Si → V is null-homotopic in U . We say that A has Property U V ∞ in X (or simply A is U V ∞ in X) if it is U V n in X for every n ∈ ω. It is said that A has Property U V ∗ in X (or simply A is U V ∗ in X) if each neighborhood U of A in X contains a neighborhood V of A that is contractible in U . A surjective map f : X → Y is called a U V ∗ map, a U V ∞ map, or a U V n map depending on if each fiber f −1 (y) is U V ∗ , U V ∞ , or U V n . For a map f : X → Y and a space Z, let f ∗ : [Y, Z] → [X, Z] be the function defined by f ∗ ([g]) = [gf ] for each map g : Y → Z: f

X

Y g

gf

Z A map f : X → Y is called a shape equivalence if the function f ∗ : [Y, P ] → [X, P ] is a bijection for every ANR P (equiv. for every polyhedron P ). We call a map f : X → Y a hereditary shape equivalence if f |f −1 (A) : f −1 (A) → A is a shape equivalence for any closed set A ⊂ Y .

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1 Preliminaries and Background Results

By using the above concepts, fine homotopy equivalence (or cell-like maps) between ANRs are characterized as follows: Theorem 1.15.9 For a closed surjective map f : X → Y between ANRs X and Y , the following are equivalent: (a) (b) (c) (d)

f f f f

is a fine homotopy equivalence; is a U V ∞ map (= a local ∞-connection); is a U V ∗ map (= a local ∗-connection); is a hereditary shape equivalence.

In the case where f is a perfect map, the following is also equivalent to the above: (e) f is a cell-like map.

7.5.4

As a corollary of the above characterization, the following can be obtained: Corollary 1.15.10 Let X, Y and Z be ANRs. For each two cell-like maps f : X → Y and g : Y → Z, the composition gf : X → Y is also a cell-like map. 7.5.5 Concerning the uniform limit of fine homotopy equivalences, we have the following: Theorem 1.15.11 Let X and Y be ANRs and f = limi→∞ fi : X → Y the uniform limit of fine homotopy equivalences with respect to some admissible metric d ∈ Metr(Y ). Then, f is also a fine homotopy equivalence. 7.4.7

Notes for Chapter 1 As mentioned at the beginning of this chapter, almost all results are treated in the book [GAGT] but we need more results in PL Topology (Combinatorial Topology) which are not covered by [GAGT]. For notes concerning results of this chapter, refer to Notes for corresponding chapters of [GAGT]. In Sect. 1.9, we proved basic results on regular neighborhoods, which are very useful and basic tools in PL Topology but missing in [GAGT]. The concept of regular neighborhoods is introduced by J.H.C. Whitehead [154]. For more information for them, refer to the books [(17)], [(18)], and [(19)]. For PL manifolds and combinatorial manifolds, refer to Appendix at the end of the book. Section 1.14 is a supplement to Section 6.3 of [GAGT]. The local equiconnectedness (LECness) was introduced by R.H. Fox [67]. Theorems 1.14.2 and 1.14.3 were proved by J. Dugundji [57]. For related results, refer to papers [50, 58, 86, 106, 118], etc.

Chapter 2

Fundamental Results on Infinite-Dimensional Manifolds

Given a paracompact (homogenuous) space E (called the model space), an Emanifold is a (topological) manifold modeled on E, that is, a paracompact space covered by open sets that are homeomorphic to open sets in E. Additionally, we postulate the following condition: •

An E -manifold has the same density as E.

When E is separable, every E-manifold is separable. An Rn -manifold (an In manifold) is a usual n-manifold (an n-manifold with boundary). An 2 ()-manifold and a Q-manifold are respectively called a Hilbert manifold and a Hilbert cube manifold, where  is an infinite set. Throughout this chapter, as a model space E, we consider the following space: • Let E = (E,  · ) be a normed linear space 1 such that E ≈ E N or E ≈ EfN . We are going to see that the Hilbert space 2 () satisfies this condition (the first homeomorphy), that is, 2 () ≈ 2 ()N (Corollary 2.2.13). In Theorem 5.5.10 (p. 407), it will be shown that normed linear spaces 2f (), 2f () × 2Q , 2f () × 2 , and 2 ()×2f satisfy this condition (the second homeomorphy).2 It should be noted that every E-manifold is metrizable because every locally metrizable paracompact space is metrizable (Proposition 1.3.12(5)).

1 In this chapter, many results are valid for locally convex metrizable topological linear spaces. In the latter half, we will use the fact that E is homeomorphic to the open cone. To prove this fact, we assume that E is a normed linear space. 2 For 2 () and 2 , see the definition on p. 353. f Q

© Springer Nature Singapore Pte Ltd. 2020 K. Sakai, Topology of Infinite-Dimensional Manifolds, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-15-7575-4_2

81

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2 Fundamental Results on Infinite-Dimensional Manifolds

In this chapter, we give fundamental results on E-manifolds (e.g., 2 ()manifolds) and Q-manifolds. In particular, it will be seen that every E-manifold is homeomorphic to some open set in E, and that two E-manifolds are homeomorphic if they have the same homotopy type. For Q-manifolds M such that M × R+ ≈ M, similar results will be proved. We prove the Toru´nczyk Factor Theorem 2.2.12 (resp. 2.2.14), that is, every completely metrizable AR (resp. ANR) X with dens(X)  card  is a topological factor of 2 () (resp. an 2 ()-manifold), i.e., X × 2 () ≈ 2 () (resp. X × 2 () is an 2 ()-manifold).

2.1 Remarks on the Model Spaces and Isotopies First, we remark that the model spaces E and Q are ARs by the Dugundji Extension Theorem 1.13.1. By Hanner’s Theorem 1.13.6(4), E-manifolds and Q-manifolds are ANRs. Recall that the weight, density and cellularity are equal in metrizable spaces (see Sect. 1.1). The density postulate on E-manifolds is equivalent to the following: •

The cardinality of components of an E-manifold is not greater than dens E.

This equivalence follows from the following lemma: Lemma 2.1.1 Let X be a paracompact space covered by open sets that are homeomorphic to open sets in E. If X is connected, then X has the same density as E. In general, every component of X has the same density as E. Proof Let U be a locally finite open cover of X such that each U ∈ U is homeomorphic to a non-empty open set in E. Note that each non-empty open set in E has the same density  as E. Each U ∈ U has a dense subset DU with card DU = dens E. Then, D = U ∈U DU is dense inX. Let U0 ∈ U and D0  = DU0 . For each n ∈ N, we define Un = stn (U0 , U) (= U[Un−1 ]) and Dn = U ∈U[Un−1 ] DU .   Since X is connected, it follows that X = n∈N Un = n∈N stn (U0 , U), hence D = n∈N Dn . Because Dn is dense in Un , we have U[Un ] = U[Dn ]. Since U is locally finite in X, it follows that card Dn−1  card Dn  card U[Dn−1 ] dens E  ℵ0 card Dn−1 dens E = card Dn−1 card D0  card Dn−1 card Dn−1 = card Dn−1 . By induction, we have card Dn = card D0 = dens E for every n ∈ N. Therefore, dens E = dens U0  dens X  card D = sup card Dn = dens E. n∈N

Thus, we have dens X = dens E.

" !

2.1 Remarks on the Model Spaces and Isotopies

83

When E is a Banach space (= a complete normed linear space), every E-manifold is completely metrizable (Proposition 1.3.12(5)). On the other hand, every Qmanifold is locally compact. Let M0 and M1 denote the classes of all compacta (= compact metrizable spaces) and of all complete metrizable spaces, respectively. For infinite cardinal τ , let M1 (τ ) be the subclass of M1 consisting of spaces with density  τ . In particular, M1 (ℵ0 ) is the class of separable completely metrizable spaces. For a space X, let FX denote the class of spaces that can be embedded in X as closed sets. Then, FQ = M0 and F2 () = M1 (τ ) (F2 = M1 (ℵ0 )), where τ = card  (Theorem 1.13.9). A space X is said to be (topologically) homogeneous if for every pair of points x0 , x1 ∈ X, there is a homeomorphism h : X → X such that h(x0 ) = x1 . Every topological group is homogeneous. In particular, E is homogeneous. Although finite-dimensional cubes In are not homogeneous, the Hilbert cube Q is homogeneous. To prove this, we use the metric d ∈ Metr(Q) defined by d(x, y) =



2−n |x(n) − y(n)|.

n∈N

Let Homeo(Q) be the space of all homeomorphisms of Q onto itself with the supmetric d(f, g) = sup d(f (x), g(x)). x∈Q

Then, Homeo(Q) is completely metrizable because it has the following admissible complete metric: d ∗ (f, g) = d(f, g) + d(f −1 , g −1 ). The admissibility of d ∗ can be shown as follows: Given f ∈ Homeo(Q) and ε > 0, we have 0 < δ < ε/2 such that d(f −1 (x), f −1 (y)) < ε/2 for every x, y ∈ Q with d(x, y) < δ. For g ∈ Homeo(Q) with d(f, g) < δ, d(f −1 , g −1 ) = d(f −1 g, id) = d(f −1 g, f −1 f )  ε/2, hence d ∗ (f, g) < ε. The completeness of d ∗ can be seen as follows: Let (fn )n∈N be a d ∗ -Cauchy sequence of Homeo(Q). Then, (fn )n∈N and (fn−1 )n∈N are d-Cauchy, hence they converge to f and g in C(Q, Q) because C(Q, Q) is d-complete. It easy to see that f ∈ Homeo(Q), g = f −1 , and d ∗ (fn , f ) → 0 as n → ∞.

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2 Fundamental Results on Infinite-Dimensional Manifolds

The subset (−1, 1)N ⊂ Q is called the pseudo-interior of Q and denoted by I (Q). Evidently, I (Q) ≈ RN = s.3 The complement Q \ I (Q) is called the pseudo-boundary of Q and denoted by B(Q). Theorem 2.1.2 The Hilbert cube Q is homogeneous. Proof For every pair of points x0 , x1 ∈ Q, we have to construct a homeomorphism h : Q → Q such that h(x0 ) = x1 . In the case that x0 , x1 ∈ I (Q), for each n ∈ N, there is a PL-homeomorphism hn : [−1, 1] → [−1, 1] such that hn (±1) = ±1 and hn (x0 (n)) = x1 (n). The desired homeomorphism h : Q → Q can be defined by h(x) = (hn (x(n)))n∈N for each x ∈ Q. In order to reduce the general case into the case above, it suffices to show that, for each x0 ∈ B(Q), there is a homeomorphism h : Q → Q such that h(x0 ) ∈ I (Q). For each n ∈ N, let    Gn = h ∈ Homeo(Q)  prn h(x0 ) ∈ (−1, 1) , where prn : Q → [−1, 1] is the projection onto the n-th factor (i.e., prn (x) = x(n)). Evidently, each Gn isopen in Homeo(Q). We shall show that each Gn is dense in Homeo(Q). Then, n∈N Gn is dense in Homeo(Q) by the Baire Category Theorem 1.3.15, hence h ∈ n∈N Gn is the desired homeomorphism. To show that each Gn is dense in Homeo(Q), let f ∈ Homeo(Q) and 0 < ε < 1. It suffices to find h ∈ Homeo(Q) such that d(h, id) < ε and prn hf (x0 ) ∈ (−1, 1), where it may be assumed that |prn f (x0 )| = 1. Take a homeomorphism ϕ : [−1, 1]2 → [−1, 1]2 (as illustrated in Fig. 2.1) such that ϕ|[−1 + ε/2, 1 − ε/2] × [−1, 1] = id, ϕ(−x) = −ϕ(x) and ϕ({1} × [−1, 1]) = [1 − 3ε/8, 1 − ε/8] × {1}. Choosing m > n so that 2−m+1 < ε/2, we define h ∈ Homeo(Q) as follows: pri h = pri for every i ∈ N \ {n, m} and (prn h(x), prm h(x)) = ϕ(x(n), x(m)) for each x ∈ Q. Then, d(h, id) < ε and prn hf (x0 ) ∈ (−1, 1). This completes the proof.

" !

By the homogeneity of Q (Theorem 2.1.2), we can show the following: Proposition 2.1.3 FQ\{0} = FQ×[0,1) is the class of locally compact separable metrizable spaces.4

3 In the book [(1)], it is denoted by P . In much of the literature, the pseudo-interior is denoted by s. In fact, we can naturally identify I (Q) = s. However, to avoid confusion, we distinguish them (i.e., I (Q) = (−1, 1)N = RN = s). 4 It will be shown in Sect. 2.7 that Q × [0, 1) ≈ Q \ {0}.

2.1 Remarks on the Model Spaces and Isotopies

85

1 1

2

2 Fig. 2.1 The homeomorphism ϕ

Proof A space homeomorphic to a closed subspace of Q\{0} or Q×[0, 1) is locally compact separable metrizable. Conversely, let X be a locally compact separable metrizable space. Then, we have an embedding f : αX → Q of the one-point compactification αX = X ∪ {∞} of X. By the homogeneity of Q, we may assume that f (∞) = 0. Then, f |X : X → Q \ {0} is a closed embedding. Moreover, using a map k : αM → I with k −1 (0) = {∞}, we can define a closed embedding h : M → Q × [0, 1) by h(x) = (f (x), 1 − k(x)) for each x ∈ M. " ! From Fig. 2.2, we can grasp the homeomorphisms in the following lemma: Lemma 2.1.4 [0, 1) × (0, 1) ≈ [0, 1) × [0, 1) ≈ [0, 1) × I.

" !

By using the above lemma, we can easily show that the Hilbert cube Q has an open basis consisting of open sets homeomorphic to Q × [0, 1). Hence, we have the following: Proposition 2.1.5 Every Q-manifold can be covered by arbitrarily small open sets homeomorphic to Q × [0, 1). " ! For each x ∈ E and ε > 0, the ε-neighborhood B(x, ε) (≈ BE \ SE ) is homeomorphic to E. Thus, we have the following: Proposition 2.1.6 Every E-manifold is covered by arbitrarily small open sets homeomorphic to E. " ! We show Riesz’s Lemma as follows: Lemma 2.1.7 (RIESZ) Let L = (L,  · ) be a normed linear space, F be a closed linear subspace of L with F  L, and 0 < δ < 1. Then, there exists v ∈ L such that v = 1 and dist(v, F )  δ.

0 1

0 1

0 1

Fig. 2.2 The homeomorphisms in Lemma 2.1.4

0 1

0 1

I

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2 Fundamental Results on Infinite-Dimensional Manifolds

Proof Take x ∈ L \ F and let r = dist(x, F ) > 0. Since 0 < δ < 1, we have y ∈ F such that x − y < δ −1 r. Then, v = x − y−1 (x − y) ∈ L is the desired one. Indeed, for each z ∈ F ,   v − z = x − y−1 (x − y) − z   = x − y−1 x − y − x − yz > r −1 δ · dist(x, F ) = δ.

" !

Let X = (X, d) be a metric space, A ⊂ X, and δ > 0. It is said that A is δdiscrete if d(x, y)  δ for each pair of distinct points x, y ∈ A. We say that A is uniformly discrete if it is δ-discrete for some δ > 0. As an application of Riesz’s Lemma, we can see that the unit sphere SL of a normed linear space L contains a uniformly discrete infinite set. Proposition 2.1.8 Let L = (L,  · ) be an infinite-dimensional normed linear space. For each 0 < δ < 1, the unit sphere SL of L contains a δ-discrete linearly independent set D with card D = dens L (= w(L)). Proof Using Zorn’s Lemma, we can find a maximal δ-discrete linearly independent subset D ⊂ SL . Then, card D = dens L. Indeed, if card D < dens L, then the linear span F of D is not dense in L, because D is finite or dens F = card D. Applying Riesz’s Lemma 2.1.7 to the closed linear subspace cl F  L, we have x ∈ SL such that dist(x, cl F ) = dist(x, F )  δ. Then, D ∪ {x} is a δ-discrete linearly independent subset of SL , which contradicts the maximality of D. " ! According to Theorem 1.4.12, no infinite-dimensional normed linear spaces are locally compact. For normed linear spaces, this fact also follows from Proposition 2.1.8 above. The following lemma is simple but important. Lemma 2.1.9 In the case E ≈ E N , E × RN ≈ E; in the case E ≈ EfN , E × RN f ≈ E. Under our assumption (E ≈ E N or E ≈ EfN ), it follows that E × R ≈ E. Proof Because of their similarity, we show only the case E ≈ E N . By the Bartle– Graves–Michael Theorem 1.4.16, E ≈ F × R for some space F . Then, E × RN ≈ E N × RN ≈ (F × R)N × RN ≈ F N × RN ≈ (F × R)N ≈ E N ≈ E.

" !

Lemma 2.1.10 There is a discrete open collection U in E such that card U = dens E and every U ∈ U is homeomorphic to E.

2.1 Remarks on the Model Spaces and Isotopies

87

Proof For each n ∈ N, we can use Zorn’s Lemma to find a maximal 2−n -discrete subset Dn of E. Then, B(x,2−n /3) ≈ E for each x ∈ Dn and {B(x, 2−n /3) | x ∈ Dn } is discrete in X. Since n∈N Dn is dense in E, it follows that dens E  card



Dn = sup card Dn  c(E) = dens E,

n∈N

n∈N

hence supn∈N card Dn = w(E). Let h : E × R → E be a homeomorphism. Then, we have the following discrete open collection in E:    U = h(B(x, 2−n /3) × (2n − 1, 2n))  n ∈ N, x ∈ Dn , where card U = supn∈N card Dn = w(E) and h(B(x, 2−n /3) × (2n − 1, 2n)) ≈ E × R ≈ E.

" !

Next, we show the following: Theorem 2.1.11 Every E-manifold M can be embedded in E as a closed set, namely M ∈ FE .5 Proof We apply Michael’s Theorem on local properties (Corollary 1.3.21). Let P be the property for closed sets A in M such that A ∈ FE . Then, M has P locally, that is, each point x ∈ M has a closed neighborhood having P. To see that M has P (i.e., M ∈ FE ), it suffices to show that P is F -hereditary, where (F-1) is obvious. (F-2): Let A1 , A2 be closed sets in M with A1 , A2 ∈ FE , that is, there are closed embeddings hi : Ai → E, i = 1, 2. Since E 2 × R ≈ E, it suffices to construct a closed embedding h : A1 ∪ A2 → E 2 × R. Using Klee’s Trick (1.13.7), we can obtain a homeomorphism f : E 2 → E 2 such that f (h1 (x), 0) = (0, h2 (x)) for every x ∈ A1 ∩ A2 . Let g : A1 → I be a map with g −1 (0) = A1 ∩ A2 . Then, a closed embedding h : A1 ∪ A2 → E 2 × R can be defined as follows:  h(x) =

(f (h1 (x), 0), g(x))

if x ∈ A1 ,

((0, h2 (x)), 0)

if x ∈ A2 .

(F-3): Let {Aλ | λ ∈ } be a discrete collection of closed sets in M such that Aλ ∈ FE for every λ ∈ . Then, card   w(E) because c(M) = w(M) = w(E) by Lemma 2.1.1. On the other hand, by Lemma 2.1.10, E has a discrete open collection U such that card U = w(E) and every U ∈ U is homeomorphic to E

5 Recall

our postulate that every E-manifold has the same density as E.

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2 Fundamental Results on Infinite-Dimensional Manifolds

itself. Since E × R ≈ E, it suffices to show that, for each U ∈ U, every A ∈ FE can be embedded in U × R as a closed set in E × R. Since U ≈ E, we have a closed embedding f : A → U . However, it should be noted that f (A) is not closed in E. We define an embedding h : A → U × R as follows:   h(x) = f (x), dist(f (x), E \ U )−1 . Then, it is easy to see that h is a closed embedding into E × R.

" !

It should be remarked that a Q-manifold cannot be embedded in Q unless it is compact. However, every Q-manifold M can be embedded in Q × [0, 1) as a closed set, because it is locally compact, separable, and metrizable (Proposition 2.1.3). A homotopy h : X × I → Y is called an isotopy if each ht : X → Y is a homeomorphism. For homeomorphisms f, g : X → Y , it is said that f is isotopic to g if there is an isotopy h : X × I → Y with h0 = f and h1 = g, where h is also called an isotopy from f to g. Like “homotopic,” if f is isotopic to g, then g is isotopic to f . For an open collection U in Y , an isotopy h is called a U-isotopy if the collection of its tracks refine U. Then, it is said that h0 is U-isotopic to h1 . An isotopy h : X × I → Y is said to be invertible if h∗ : Y × I → X defined ∗ by h∗ (y, t) = h−1 t (y) is continuous (hence h is also an isotopy). Equivalently, ¯ h¯ : X × I → Y × I defined by h(x, t) = (ht (x), t) is a homeomorphism. A map ϕ : X × I → Y × I is I-preserving if prI ϕ = prI . For such a map ϕ, we define maps ϕt : X → Y , t ∈ I, by ϕt (x) = prY ϕ(x, t) for each x ∈ X, so ϕ(x, t) = (ϕt (x), t) for each (x, t) ∈ X × I. Here, the notation ϕt is used differently than before. When ϕ : X × I → Y × I is considered as a homotopy or an isotopy, ϕt is defined by ϕt (x) = ϕ(x, t), hence that is a map from X to Y × I. However, there would be no confusion by the context. The homeomorphism h¯ defined above is I-preserving and h¯ t = ht for t ∈ I. For an I-preserving homeomorphism ϕ : X × I → Y × I, prY ϕ : X × I → Y is an invertible isotopy. When an isotopy h from f to g is invertible, f is said to be invertibly isotopic to g, where f −1 is isotopic to g −1 by the isotopy h∗ defined above. It should be remarked that, even if h is not invertible, an isotopy from g −1 to f −1 can also be defined by g −1 ht f −1 (or f −1 ht g −1 ). Then, observe that    −1 g ht f −1 (x)  t ∈ I = g −1 (h({f −1 (x)} × I))     −1 −1  or f ht g (x)  t ∈ I = f −1 (h({g −1 (x)} × I)) . Thus, if f is U-isotopic to g, then f is g −1 (U)-isotopic (or f −1 (U)-isotopic) to g. Proposition 2.1.12 There exists a non-invertible isotopy of 1 . Proof For each i ∈ N, let fi : I → [1/i, 1] be a surjective map such that fi ([0, 1/(i + 1)]) = fi ([1/i, 1]) = {1}. We define a non-invertible isotopy of h : 1 × I → 1 by h(x, t) = (x(i)fi (t))i∈N . Then, each ht : 1 → 1 is a

2.1 Remarks on the Model Spaces and Isotopies

89

¯ homeomorphism with the inverse h−1 t (y) = (y(i)/fi (t))i∈N . However, the map h defined above is not a homeomorphism. Indeed, let xi ∈ 1 such that xi (i) = 1/i and xi (j ) = 0 for j = i, and let ti ∈ fi−1 (1/i). Then, xi  = 1/i → 0 and 0 < ti < 1/i → 0 (i → ∞), but h¯ −1 (xi , ti ) = (ei , ti ), where ei (i) = 1 and ei (j ) = 0 for j = i. Hence, h¯ −1 is not continuous at (0, 0) ∈ 1 × I. " ! The following is another example: Proposition 2.1.13 There exists an AR X in Euclidean plane R2 which has a noninvertible isotopy. Proof Let X be the subspace of R2 defined as follows (see the left-hand side of Fig. 2.3): X = I × {0} ∪



{1/n} × I.

n∈N

For each n ∈ N, let pn = (1/n, 1/3n), qn = (1/n, 2/3) ∈ X. We can define a non-invertible isotopy h : X × I → X as follows: Let h0 = h2−2n+2 = id for each n ∈ N. If 2−2n  t  2−2n+2 , then ht |X \ ({1/n} × I) = id and ht |{1/n} × I is a homeomorphism of {1/n} × I onto itself such that ht is linear on each {1/n} × [0, 2/3] and {1/n} × [2/3, 1], and the point ht (qn ) is sliding on {1/n} × I from qn down to pn and back to qn as t moves from 2−2n+2 to 2−2n (see the right-hand side of Fig. 2.3). The precise definition is as follows: ⎧  2(22n−1 t − 1) 1 2 − 22n−1 t ⎪ ⎪ + ⎨ , 3n 3 ht (qn ) =  n 2n 2n t − 1  1 2(2 − 2 2 t) ⎪ ⎪ ⎩ , + n 3 3n

if 2−2n+1  t  2−2n+2 , if 2−2n  t  2−2n+1 .

The continuity of h at each point except (0, 0) = ((0, 0), 0) is trivial. Note that h({1/n} × [0, s] × I) ⊂ {1/n} × [0, s] and h([0, s] × {0} × I) = [0, s] × {0}. Clearly, h(B(0, ε) × I) ⊂ B(0, ε) for any ε > 0, hence h is also continuous at (0, 0). Thus, h is an isotopy of X. On the other hand, (pn , 2−2n+1 ) → (0, 0) in X × I as n → ∞,

q2

q1

qn

··· p1

p2 1 1 4 3

1 2

h 2−2n +2 h 2−2n ht

pn

1

Fig. 2.3 The space X and the graph of ht (2−2n  t  2−2n+2 )

h 2−2n +1 q n {1/ n} × I

90

2 Fundamental Results on Infinite-Dimensional Manifolds

but h (pn , 2−2n+1 ) = h−1 (p ) = qn is not convergent in X, that is, h is not 2−2n+1 n invertible. " ! When X and Y are compact, every isotopy h : X × I → Y is clearly invertible. This fact holds for locally compact spaces. Proposition 2.1.14 If X and Y are locally compact, then every isotopy h : X ×I → Y is invertible. ¯ Proof We define h¯ : X × I → Y × I by h(x, t) = (ht (x), t). It suffices to show ¯ that h is open. Let G be an open set in X × I. For each (x, t) ∈ G, choose open sets U ⊂ X and S ⊂ I so that cl U is compact and (x, t) ∈ U × S ⊂ G. Since ht is a homeomorphism, we have an open neighborhood V of ht (x) in Y such that cl V ⊂ ht (U ). Then, h(bd U × {t}) ⊂ Y \ cl V . Since bd U is compact, we have a neighborhood W of x in X and a connected neighborhood J of t in I such that J ⊂ S, h(W × J ) ⊂ V and h(bd U × J ) ⊂ Y \ cl V . ¯ t) = (ht (x), t) in Y × I. For each z ∈ W Then, ht (W ) × J is a neighborhood of h(x, −1 and s ∈ J , hs (h({z} × J )) is connected, −1 h−1 s (h({z} × J )) ⊂ hs (cl V ) ⊂ X \ bd U = U ∪ (X \ cl U ) and −1 −1 z ∈ h−1 s (h({z} × J )) ∩ ht (V ) ⊂ hs (h({z} × J )) ∩ U, −1 hence h−1 s (h({z} × J )) ⊂ U . In particular, hs ht (z) ∈ U , that is, ht (z) ∈ hs (U ). Thus, ht (W ) ⊂ hs (U ) for each s ∈ J . Then, we have

ht (W ) × J ⊂



 hs (U ) × J ⊂

s∈J



 ¯ × S) ⊂ h(G), ¯ hs (U ) × {s} = h(U

s∈S

¯ hence h(G) is open in Y × I.

" !

For an invertible isotopy h : X × I → X with h0 = idX , let h∗ : X × I → X be the isotopy defined by h−1 t , t ∈ I. For their tracks, it should be remarked that: • h∗ has tracks different from h in general. Moreover, for an open collection U in X, • even if h is a U-isotopy, h∗ need not be a U-isotopy. For example, let h : R2 × I → R2 be the isotopy defined as follows: ht (x, y) =

 (x, y + 2t) (x + 2t − 1, y + 1)

if 0  t  1/2, if 1/2  t  1.

2.1 Remarks on the Model Spaces and Isotopies

91

Then, the set ({0} × I) ∪ (I × {1}) is the track of the origin 0 = (0, 0) ∈ R2 by h. Observe that  (x, y − 2t) if 0  t  1/2, ∗ −1 ht (x, y) = ht (x, y) = (x − 2t + 1, y − 1) if 1/2  t  1. The set ({1} × I) ∪ (I × {0}) is the track of (1, 1) = h1 (0) by the isotopy defined by h∗ . Moreover, let U = {U (x, y) | (x, y) ∈ R2 } ∈ cov(R2 ), where U (x, y) = ((x − 1/2, x + 1/2) × (y − 1/2, y + 3/2)) ∪ ((x − 1/2, x + 3/2) × (y + 1/2, y + 3/2)). The above h is a U-isotopy but h∗ is not a U-isotopy.

Nevertheless, the following holds: Proposition 2.1.15 Let U be an open collection in X. For an invertible U-isotopy h : X × I → X with h0 = idX , the isotopy h∗ : X × I → X defined by h−1 t , t ∈ I, is a st U-isotopy. Proof Let x ∈ X such that h∗ ({x} × I) is non-degenerate. It suffices to show that h∗ ({x} × I) ⊂ st(x, U). For each t ∈ I, there is Ut ∈ U such that h({h−1 t (x)} × I) ⊂ −1 Ut . Since x = ht (h−1 (x)) ∈ h({h (x)} × I), it follows that U ∈ U[x], that is, t t t Ut ⊂ st(x, U). Observe that −1 −1 h−1 t (x) = h0 (ht (x)) ∈ h({ht (x)} × I) ⊂ Ut ⊂ st(x, U).

Hence, we have h∗ ({x} × I) = {h−1 t (x) | t ∈ I} ⊂ st(x, U).

" !

For embeddings f, g : X → Y (resp. subsets A, B ⊂ Y ), it is said that f is ambiently isotopic to g in Y (resp. A is ambiently isotopic to B in Y ) if there is an isotopy ϕ : Y × I → Y with ϕ0 = id and ϕ1 f = g (resp. ϕ1 (A) = B). This is called an ambient isotopy from f to g (resp. from A to B). For an ambient isotopy ϕ from f to g (resp. from A to B), an ambient isotopy from g to f (resp. from B to A) is defined by ϕ1−t ϕ1−1 ,6 so g is ambiently isotopic to f (resp. B is ambiently isotopic to A in Y ). When ϕ is an invertible isotopy, it is said that f is ambiently invertibly isotopic to g in Y (resp. A is ambiently invertibly isotopic to B in Y ), where an ambient isotopy from g to f (resp. from B to A) can also be defined by ϕt−1 because ϕ is invertible. Indeed, ϕ0−1 = id and ϕ1−1 g = f When f, g : X → Y are homeomorphisms, f is isotopic to g if and only if f is ambiently isotopic to g. Indeed, for an ambient isotopy ϕ from f to g, an isotopy from f to g is defined by ϕt f . Conversely, for an isotopy h from f to g, an ambient isotopy from f to g is defined by ht f −1 .

an isotopy can also be defined by ϕ1−1 ϕ1−t . However, when ϕ is a U-isotopy, this definition does not give a U-isotopy but the above does.

6 Such

92

2 Fundamental Results on Infinite-Dimensional Manifolds

We shall show the existence of very useful open covers of an open set in a metrizable space: Lemma 2.1.16 Every open set W in a metrizable space Y has an open cover U with the following properties: (i) Given a map f : X → Y , if a map g : f −1 (W ) → W is U-close to f |f −1 (W ) then g can be extended to the map g˜ : X → Y by g|X ˜ \ f −1 (W ) = f |X \ −1 f (W ); (ii) If a homeomorphism h : W → W is U-close to id, then h can be extended to ˜ \ W = id; the homeomorphism h˜ : Y → Y by h|Y (iii) Given a subset X ⊂ Y , if an embedding h : X ∩ W → W is U-close to id, then ˜ \ W = id. h can be extended to the embedding h˜ : X ∪ (Y \ W ) → Y by h|Y Moreover, if h is a closed embedding, i.e., h(X ∩ W ) is closed in W , then h˜ is also a closed embedding. Proof Take d ∈ Metr(Y ) and define    U = B(y, 2−1 d(y, Y \ W ))  y ∈ W ∈ cov(W ). To see that U satisfies (i), it suffices to show the continuity of g˜ in (i) at each x ∈ bd f −1 (W ). For each ε > 0, x has a neighborhood V in X such that f (V ) ⊂ B(f (x), ε/3). For each x  ∈ V ∩ f −1 (W ), there is some y ∈ W such that g(x  ), f (x  ) ∈ B(y, 2−1 d(y, Y \ W )). Since d(f (x  ), f (x)) < ε/3 and f (x) ∈ Y \ W , it follows that d(y, Y \ W )  d(y, f (x))  d(y, f (x  )) + d(f (x  ), f (x)) < 2−1 d(y, Y \ W ) + ε/3, which implies 2−1 d(y, Y \ W ) < ε/3. Then, we have d(g(x ˜  ), g(x)) ˜  d(g(x  ), f (x  )) + d(f (x  ), f (x)) < d(y, Y \ W ) + ε/3 < ε. Thus, g˜ is continuous at x. Now, we show that (ii) and (iii) follow from (i). First, applying (i) to f = idY , we can extend h in (ii) to the map h˜ : Y → Y . The inverse h−1 in (ii) is also U-close to id, so h˜ −1 is continuous by (i). Thus, h˜ is a homeomorphism. ˜ \ W = id. For (iii), h can be extended to the map h˜ : X ∪ (Y \ W ) → Y by h|Y ˜ Then, h is injective, ˜ ∪ (Y \ W )) = h(X ∩ W ) ∪ (Y \ W ) and h(X ˜ ∪ (Y \ W )) ∩ W = h(X ∩ W ). h(X

2.2 The Toru´nczyk Factor Theorem

93

Since h−1 : h(X ∩ W ) → W is U-close to id, it follows from (1) that h˜ −1 : ˜ ∪ (Y \ W )) → Y is continuous, hence h˜ is an embedding. If h(X ∩ W ) is closed h(X in W , then W ∩ clY h(X ∩ W ) = h(X ∩ W ), hence ˜ ∪ (Y \ W )) = clY h(X ∩ W ) ∪ (Y \ W ). h(X ˜ ∪ (Y \ W )) is closed in Y . Therefore, h(X

" !

For an open set W in a space Y , an open cover U ∈ cov(W ) is said to be fitting in Y if it satisfies the conditions of Lemma 2.1.16.7 As we saw in the above proof, if U ∈ cov(W ) satisfies condition (i), then it satisfies conditions (ii) and (iii), too. Obviously, every refinement V ∈ cov(W ) of U is also fitting in Y . Then, we have the following: Proposition 2.1.17 Let Y be a metrizable space with W an open set. Every open cover U of W has an open refinement V that is fitting in Y . " ! In the theory of Q-manifolds, it is important whether maps and homotopies are proper or not. When only compact Q-manifolds are considered, there is no problem because every map and homotopy are proper. However, for non-compact Q-manifolds, we must check to see if maps and homotopies are proper. It should be recalled that a proper map between metrizable spaces is identical to a perfect map (Proposition 1.3.7). Moreover, it really helps to keep in mind that, for an open over U of a Q-manifold M such that every U ∈ U has the compact closure in M, any map to M is proper if it U-close to a proper map (Proposition 1.3.8).

´ 2.2 The Torunczyk Factor Theorem In this section, let L = (L,  · ) be a normed linear space with dim L > 0. Recall that LN denotes the countable infinite product of L with the product topology, which is a topological linear space and contains the following dense linear subspace:    N  x(i) = 0 except for finitely many i ∈ N . LN f = x ∈L N Then, RN f can be embedded into Lf as a topological linear subspace because L contains a one-dimensional linear subspace (that is linearly isomorphic to R). Since N RN f is not normable (Proposition 1.2.1), it follows that any linear subspace F of L with LN f ⊂ F has no admissible norm.

7 In

[17], such an open cover is said to be normal to Y \ W .

94

2 Fundamental Results on Infinite-Dimensional Manifolds

We define the 1 -product 1 L of L as the following normed linear space:       1 L = x ∈ LN  (x(i))i∈N ∈ 1 = x ∈ LN  i∈N x(i) < ∞  N with the norm x1 = i∈N x(i). Then, LN f ⊂ 1 L ⊂ L , but the topology of 1 L is different from the product topology, as mentioned above. The normed linear subspace LN f of 1 L by 1 L, that is, f

   1 L = x ∈ LN  (x(i))i∈N ∈ 1f f

with the norm x1 =



i∈N x(i).

We call 1 L the 1f -product.8 f

The following can be proved in the same way as the completeness of 1 . Proposition 2.2.1 If L is a Banach space, then the 1 -product 1 L is also a Banach space. Sketch of Proof For each Cauchy sequence (xn )n∈N in 1 L, we have x ∈ LN such that x(i) = limn→∞ xn (i) for each i ∈ N because (xn (i))n∈N is Cauchy in L. Then, show that x ∈ 1 L and limn→∞ xn − x1 = 0.

Evidently, the 1 -product and the 1f -product of R = (R, | · |) are 1 and 1f , respectively. It is easy to construct a linear isometry from the 1 -product 1 1 () onto 1 (N × ), which sends the 1f -product 1 1f () onto 1f (N × ). Thus, we f have the following:   Proposition 2.2.2 The pair 1 1 (), 1 1f () is linearly isometric to the pair f

" !

(1 (), 1f ()) if  is infinite.

Remark 2.1 More generally, for a sequence (Li )i∈N of normed linear spaces Li = (Li ,  · i ), i ∈ N, we can similarly define the 1 -product and the 1f -product as the following normed linear spaces:  1 Li = x ∈ LN  1 Li = x ∈ LN f

   (x(i)i )i∈N ∈ 1 ,    (x(i)i )i∈N ∈ 1 , f

 with the norm |x|1 = i∈N x(i)i . This definition is available even if Li , i ∈ N, are F -normed spaces (cf. Sect. 1.4), where | · |1 is an F -norm. Using p instead of p 1 , the p -product and the f -product can also be defined.

 1 L is written as 1 L in Toru´nczyk’s papers [139, 140]. In the book [(2)] of f Bessaga and Pełczy´nski (p. 208, p. 255), the notations 1 L and 1 L are used instead of 1 L

8 The 1 -product f

f

and 1 L, respectively. f

2.2 The Toru´nczyk Factor Theorem

95

Recall that X is called a uniform retract of L if there is a retraction r : L → X that is uniformly continuous at X,9 that is, for each ε > 0, there is δ > 0 such that for each x ∈ X and y ∈ L, x − y < δ implies x − r(y) < ε (cf. Sect. 6.8 in [GAGT]). The following is the main theorem of this section: ´ Theorem 2.2.3 (TORU NCZYK ) Let X be a retract of a normed linear space L = (L,  · ) with a retraction r : L → X that is uniformly continuous at X. Then, X × 1 L ≈ 1 L. If X is  · -complete, then f

f



   X × 1 L, X × 1 L ≈ 1 L, 1 L . f

f

To prove Theorem 2.2.3, we may assume that 0 ∈ X. Since the case X = {0} is obvious, we may also assume that X = {0}. We will introduce a modification to the 1 -product. A function v : R+ → R+ is said to be sub-additive if v(s + t)  v(s) + v(t) for each s, t ∈ R+ . If v : R+ → R+ is a homeomorphism, then v(0) = 0 and v is increasing, i.e., s < t ⇒ v(s) < v(t). The continuity at 0 means that tn → 0 ⇒ v(tn ) → 0. Therefore, given a sub-additive homeomorphism v : R+ → R+ , for a normed linear space L = (L,  · ), an F -norm can be defined by v(x), x ∈ L. For a sub-additive homeomorphism v : R+ → R+ , we define  (1 ,v) L = x ∈ LN  (1 ,v)L = x ∈ LN f

 i   (2 v(x(i)))i∈N ∈ 1 ,  i   (2 v(x(i)))i∈N ∈ 1 , f



and xv = i∈N 2i v(x(i)) for each x ∈ (1 ,v) L (refer to Remark 2.1). For x, y ∈ (1 ,v)L, we have −x, x + y ∈ (1 ,v) L and −xv = xv and x + yv  xv + yv . By induction, nx ∈ (1 ,v) L for every n ∈ N. Moreover, if 0 < t < 1, then tx ∈ (1 ,v) L and txv < xv . For every a > 0, choose n ∈ N so that a < n. Then, ax = (a/n)(nx) ∈ (1 ,v)L and (−a)x = −(ax) ∈ (1 ,v) L. Thus, (1 ,v) L is a linear space. It should be remarked that  · v is not a norm, but an F -norm (cf. Sect. 1.4), which induces the linear metric d(x, y) = x − yv . We regard (1 ,v) L as a metric linear space with this metric. Then, each projection prn : (1 ,v) L → L is continuous.

9 In

[138] and [139], r is called a regular retraction.

96

2 Fundamental Results on Infinite-Dimensional Manifolds

Lemma 2.2.4 For any sub-additive homeomorphisms v : R+ → R+ , i ∈ N,     (1 ,v)L, (1 ,v) L ≈ 1 L, 1 L .10 f

f

Proof We define     ϕ : (1 ,v) L, (1 ,v) L → 1 L, 1 L and f f     ψ : 1 L, 1 L → (1 ,v) L, (1 ,v) L f

f

as follows: ⎧ i ⎪ ⎨ 2 v(x(i)) x(i) if ϕ(x)(i) = x(i) ⎪ ⎩0 if ⎧ −1 −i ⎪ ⎨ v (2 x(i)) x(i) ψ(x)(i) = x(i) ⎪ ⎩0

x(i) = 0, x(i) = 0; if x(i) = 0, if x(i) = 0.

Since ϕ(x)(i) = 2i v(x(i)) and 2i v(ψ(x)(i)) = x(i), it follows that ϕ and ψ are well-defined, ϕψ = id, and ψϕ = id. It remains to verify the continuity of ϕ and ψ. 

(Continuity of ϕ) For each x ∈ (1 ,v)L and ε > 0, choose m ∈ N so that i im 2 v(x(i)) < ε/4. Since each pri ϕ is continuous, we have δ > 0 such that y ∈ (1 ,v) L, x − yv < δ ⇒



ϕ(x)(i) − ϕ(y)(i) < ε/4.

i 0 such that  y ∈ 1 L, x − y1 < δ ⇒ 2i v(ψ(x)(i) − ψ(y)(i)) < ε/4. i m.

100

2 Fundamental Results on Infinite-Dimensional Manifolds

Then, it follows that 2v(y(1) + x) +

∞ 

2i v(y(i) + x − r(y(i − 1) + x))

i=2

 2v(y(1) + x) +

m 

2i v(y(i) + x − r(y(i − 1) + x))

i=2

+

∞ 

∞ 

2 v(y(i)) + 4 i

i=m+1

2i−1 v(y(i − 1)) < ∞,

i=m+1

that is, f (x, y) ∈ (1 ,v) L. (Continuity of f ) Let (x, y) ∈ X × (1 ,v) L and ε > 0. Since y ∈ (1 ,v) L, we can choose m ∈ N so that v(y(i)) < γ /2 for each i  m

and

∞ 

2i v(y(i)) < ε/30.

i=m

Since each pri f is continuous, we can find δ > 0 such that x − x   < δ and y − y  v < δ ⇒ 2v((y(1) + x) − (y  (1) + x  )) +

m 

2i v((y(i) + x − r(y(i − 1) + x))

i=2

− (y  (i) + x  − r(y  (i − 1) + x  ))) < ε/3. Note that if y − y  v < γ /2 and i  m, then v(y  (i))  v(y(i)) + v(y(i) − y  (i)) < γ . Now, assume that x − x   < δ and y − y  v < min{δ, γ /2, ε/15}. Then, using condition (∗), we can show that f (x, y) − f (x  , y  )v < ε as follows: f (x, y) − f (x  , y  )v < ε/3 +

∞ 

2i v(y(i) + x − r(y(i − 1) + x))

i=m+1

+

∞  i=m+1

2i v(y  (i) + x  − r(y  (i − 1) + x  ))

2.2 The Toru´nczyk Factor Theorem

101 ∞ 

 ε/3 +

i=m+1

+

∞  i=m+1

 ε/3 + 5

∞ 

2i−1 v(y  (i − 1))

i=m+1

2 v(y(i)) + 5 i

i=m

 ε/3 + 10

2i−1 v(y(i − 1))

i=m+1

2i v(y  (i)) + 4

∞ 

∞ 

2i v(y(i)) + 4

∞ 

∞ 

2i v(y  (i))

i=m

2i v(y(i)) + 5

i=m

∞ 

2i v(y(i) − y  (i))

i=m

< ε/3 + ε/3 + ε/3 = ε. " !

Therefore, f is continuous. By induction, we define gn : (1 ,v) L → L, n ∈ N, as follows: g1 (z) = z(1), g2 (z) = z(2) + r(z(1)), . . . , gn (z) = z(n) + rgn−1 (z), . . . ,

that is, g1 = pr1 and gn = prn + r ◦ gn−1 for n > 1. Since each projection prn : (1 ,v) L → L is continuous, the lemma below can be proved by induction: Lemma 2.2.7 Each gn : (1 ,v) L → L is continuous.

" !

For each z ∈ (1 ,v) L, there is some m ∈ N such that z(i) = 0 for i  m, f hence gm (z) = rgm−1 (z) ∈ X and gn+1 (z) = rgn (z) = gn (z) for n  m. Thus,   :  gn |(1 ,v) L n∈N converges to g∞ (1 ,v) L → X. When X is  · -complete, f

f

 can be extended to g :  g∞ ∞ (1 ,v) L → X. In fact, by the next lemma, g∞ can be defined as follows:

g∞ (z) = lim rgi (z) for each z ∈ (1 ,v) L. i→∞

Lemma 2.2.8 For each z ∈ (1 ,v)L, (rgn (z))n∈N is  · -Cauchy. Proof For each ε > 0, choose m ∈ N so that ∞  i=m+1

2i v(z(i)) < min{ε, γ }.

102

2 Fundamental Results on Infinite-Dimensional Manifolds

Then, v(z(i)) < γ for i > m. For each n > n  m, rgn (z) − rgn (z)  v(rgn (z) − rgn (z))  v(rgn (z) − rgn+1 (z)) + · · · + v(rgn −1 (z) − rgn (z)) = v(rgn (z) − r(z(n + 1) + rgn (z))) + · · · + v(rgn −1 (z) − r(z(n ) + rgn −1 (z)))  2v(z(n + 1)) + · · · + 2v(z(n )) 

∞ 

2i v(z(i)) < min{ε, γ }  ε.

i=m+1

Therefore, (rgn (z))n∈N is  · -Cauchy.

" !

From the proof of Lemma 2.2.8, the following can also be obtained: Lemma 2.2.9 When X is  · -complete, if v(z(i)) < γ for i > m, then rgm (z) − g∞ (z)  v(rgm (z) − g∞ (z))   ∞ ∞   i 2 v(z(i))  2 v(z(i)) . i=m+1

" !

i=m+1

 and g . Now, we can prove the continuity of g∞ ∞  is continuous. When X is  · -complete, g Lemma 2.2.10 The function g∞ ∞ is also continuous.

Proof Because of their similarity, we only prove the continuity of g∞ . Let z ∈ (1 ,v) L and ε > 0. Choose m ∈ N so that v(g∞ (z) − rgm (z)),

∞ 

  2i v(z(i)) < min ε/4, γ .

i=m

Then, g∞ (z) − rgm (z) < ε/4 and v(z(i)) < γ /2 for i  m. Since the continuity of rgm follows from Lemma 2.2.7, we can find δ > 0 such that z − z v < δ ⇒ rgm (z) − rgm (z ) < ε/4. If z − z v < γ /2 and i  m, then v(z (i))  v(z(i)) + v(z(i) − z (i)) < γ .

2.2 The Toru´nczyk Factor Theorem

103

On the other hand, if z − z v < ε/4, then ∞ 

2i v(z (i)) 

i=m+1

∞ 

2i v(z(i)) + z − z v < ε/4 + ε/4 = ε/2.

i=m+1

Hence, by Lemma 2.2.9, if z − z v < min{δ, γ /2, ε/4}, then g∞ (z) − g∞ (z )  g∞ (z) − rgm (z) + rgm (z) − rgm (z ) + g∞ (z ) − rgm (z ) < ε/4 + ε/4 +

∞ 

2i v(z (i)) < ε/2 + ε/2 = ε.

i=m+1

" !

Therefore, g∞ is continuous. We define g  : (1 ,v)L → X × (1 ,v) L as follows: f

f

   g  (z) = (g∞ (z), g1 (z) − g∞ (z), . . . , gi (z) − g∞ (z), . . . ).

When X is  · -complete, g : (1 ,v) L → X × (1 ,v) L is defined by g(z) = (g∞ (z), g1 (z) − g∞ (z), . . . , gi (z) − g∞ (z), . . . ). Then, g is an extension of g  . Lemma 2.2.11 The function g  is continuous. When X is  · -complete, g is welldefined and continuous. Proof Since the continuity of g  can be proved in the same way as for g, we only prove that g is well-defined and continuous. Evidently, it suffices to show the well-definedness and continuity of pg : (1 ,v) L → (1 ,v) L, where p : X × (1 ,v)L → (1 ,v) L is the projection. (Well-Definedness of pg) For each z ∈ (1 ,v) L, choose m ∈ N so that ∞ i i=m 2 v(z(i)) < γ . Since v(z(i)) < γ for i  m, it follows from Lemma 2.2.9 that ∞  i=m

2i v(gi (z) − g∞ (z)) 

∞ 

2i v(gi (z) − rgi−1 (z))

i=m

+

∞  i=m

2i v(g∞ (z) − rgi−1 (z))

104

2 Fundamental Results on Infinite-Dimensional Manifolds



∞ 

2i v(z(i)) + 2

i=m

0, choose m ∈ N so that ∞ 

  2i v(z(i)) < min ε/20, γ .

i=m

Then, v(z(i)) < γ /2 for i  m. Replacing γ by min{ε/20, γ } in the proof of the well-definedness of pg above, we have ∞ 

2i v(gi (z) − g∞ (z)) < ε/4.

i=m

If z − z v < ε/20, then ∞ 

2i v(z (i)) 

i=m

∞ 

2i v(z(i)) +

i=m

∞ 

2i v(z(i) − z (i))

i=m 

< ε/20 + z − z v < ε/10. If z − z v < γ /2 and i  m, then v(z (i))  v(z(i)) + v(z(i) − z (i)) < γ /2 + z − z v < γ . Thus, z − z v < min{ε/20, γ /2} implies ∞ 

2i v(z (i)) < ε/10 and v(z (i)) < γ for i  m.

i=m

In the same way as above, we can obtain ∞  i=m

2i v(gi (z ) − g∞ (z )) < ε/2.

2.2 The Toru´nczyk Factor Theorem

105

By the continuity of v, g∞ and gi − g∞ , i = 1, . . . , m − 1, we can find δ > 0 such that z − z v < δ ⇒

m−1 

2i v((gi (z) − g∞ (z)) − (gi (z ) − g∞ (z ))) < ε/4.

i=1

Hence, if z − z v < min{δ, ε/20, γ /2}, then pg(z) − pg(z )v 

m−1 

  2i v (gi (z) − g∞ (z)) − (gi (z ) − g∞ (z ))

i=1

+

∞ 

2i v(gi (z) − g∞ (z)) +

i=m

∞ 

2i v(gi (z ) − g∞ (z ))

i=m

< ε/4 + ε/4 + ε/2 = ε. " !

Therefore, pg is continuous. Now, we can prove Theorem 2.2.3. Proof (Theorem 2.2.3) For each x ∈ (1 ,v) L, f

   fg  (x) = f (g∞ (x), g1 (x) − g∞ (x), g2 (x) − g∞ (x), . . . )

= (g1 (x), g2 (x) − rg1 (x), g3 (x) − rg2 (x), . . . ) = (x(1), x(2), x(3), . . . ) = x, that is, fg  = id. For each (x, y) ∈ X × (1 ,v) L, f

g1 (f (x, y)) = y(1) + x, g2 (f (x, y)) = y(2) + x, . . . ,  (f (x, y)) = g (f (x, y)) = y(n) + x for sufficiently large n ∈ N. Since so g∞ n  (f (x, y)) = x. Then, it follows y(n) = 0 for sufficiently large n ∈ N, we have g∞ that   (f (x, y)), g1 (f (x, y)) − g∞ (f (x, y)), g  f (x, y) = (g∞  g2 (f (x, y)) − g∞ (f (x, y)), . . . )

= (x, y(1), y(2), . . . ) = (x, y). Hence, g  f |(1 ,v) L = id. Thus, we have X × (1 ,v) L ≈ (1 ,v) L. f

f

f

106

2 Fundamental Results on Infinite-Dimensional Manifolds

If X is  · -complete, then fg = id and gf = id because (1 ,v) L is dense in f (1 ,v) L. In this case, we have     X × (1 ,v) L, X × (1 ,v) L ≈ (1 ,v) L, (1 ,v) L . f

f

Then, Theorem 2.2.3 follows from Lemma 2.2.4.

" !

As a corollary of Theorem 2.2.3, we have the following result on completely ´ FACTOR THEOREM: metrizable ARs, which is called the TORU NCZYK ´ Theorem 2.2.12 (TORU NCZYK ) For every non-degenerate completely metrizable AR X, X × 1 () ≈ 1 (), where dens X = card .12 In general, for an arbitrary non-degenerate AR X, there exists a normed linear space L such that dens X = dens L and X × L ≈ L.

Proof In the completely metrizable case, by virtue of Theorem 1.13.37, we have an admissible complete metric d of X such that (X, d) is a uniform AR. By the Arens– Eells Embedding Theorem 1.13.2, X = (X, d) can be embedded isometrically into some Banach space L with dens X = dens L as a closed set. Then, there is a retraction r : L → X that is uniformly continuous at X. By Theorem 2.2.3, X × 1 L ≈ 1 L. Since 1 L is a Banach space, we can apply Corollary 1.4.18 to obtain a Banach space F such that 1 L × F ≈ 1 (), where card  = dens 1 L = dens L = dens X. Hence, we have the first statement. Without assuming the completeness of X = (X, d), X can be embedded isometrically into some normed linear space L with dens X = dens L as a closed set. In this case, we have X × 1 L ≈ 1 L. Thus, the general case also holds. ! " f

f

Because 1 () ≈ 2 () (Theorem 1.2.3), 1 () in Theorem 2.2.12 can be replaced with 2 (). Since 2 ()N is a completely metrizable AR, we have 2 ()N ≈ 2 ()N × 2 () ≈ 2 (). Thus, we have the following: Corollary 2.2.13 For an arbitrary infinite set , 2 ()N ≈ 2 ().

" !

Recall that the metrizable cone C(X) over a metrizable space X is the space {0} ∪ (X × (0, 1]) with the topology generated by open sets in the product space X × (0, 1] and sets {0} ∪ (X × (0, ε)), ε ∈ (0, 1). Then, C(X) has the same density as X if X is not finite. If X is compact, then C(X) is homeomorphic to the usual cone CX that is defined as the quotient space (X × I)/(X × {0}). The metrizable

12 Note

that dens X is infinite because X is non-degenerate.

2.2 The Toru´nczyk Factor Theorem

107

open cone over X is the subspace {0}∪(X ×(0, 1)) of C(X) and denoted by C o (X). It should be noted that if X is completely metrizable, then so are C(X) and C o (X). Moreover, the cone C(X) and the open cone C o (X) over an ANR X are ARs by Corollary 1.13.18. By using the metrizable open cone, we can easily prove the following theorem, ´ which is also called the TORU NCZYK FACTOR THEOREM: ´ ) For every completely metrizable ANR X with Theorem 2.2.14 (TORU NCZYK dens X  card , X × 2 () is an 2 ()-manifold. In fact, X × 2 () is homeomorphic to an open set in 2 ().

Proof As mentioned above, the open cone C o (X) is a completely metrizable AR and clearly dens C o (X)  card . Then, C o (X) × 2 () ≈ 2 () by Theorem 2.2.12. Since 2 () × (0, 1) ≈ 2 (), X × 2 () is homeomorphic to the open set (0, 1) × X × 2 () in C o (X) × 2 (). Thus, X × 2 () can be embedded in 2 () as an open set, hence it is an 2 ()-manifold. " ! For a locally finite-dimensional simplicial complex K, |K|m is a completely metrizable ANR by Theorems 1.13.4 and 1.7.2. Recall that 1 () ≈ 2 () (Theorem 1.2.3). Hence, we have the following corollary: Corollary 2.2.15 For every locally finite-dimensional simplicial complex K with " ! card K (0)  card , |K|m × 2 () is an 2 ()-manifold. For the subspace 2f () of 2 (), we have the following result: Proposition 2.2.16 For every full simplicial complex F with card F (0)  card , |F |m × 2f () ≈ 2f (). Proof Identifying F (0) is a subset of the unit vectors of 1f (), we can regard |F |m ⊂ 1f (). Then, |F | is a convex set 1f () and the metric ρF is induced by the norm of 1f (). Because (|F |, ρF ) is a uniform AR by Theorem 1.13.31, there is a retraction r : 1f () → |F | that is uniformly continuous at |F |. Applying Theorem 2.2.3, we have |F |m × 1 1f () ≈ 1 1f (). Since 1 1f () ≈ f

f

f

1f () ≈ 2f () by Proposition 2.2.2 and Theorem 1.2.3, it follows that |F |m × 2f () ≈ 2f (). " ! Moreover, we have the following: Proposition 2.2.17 For every locally finite simplicial complex K with card K (0)  card , |K| × 2f () is an 2f ()-manifold. If K is a contractible finite simplicial complex, then |K| × 2f () ≈ 2f ().13 Proof First, we show the case where K is a contractible finite simplicial complex. As in the proof of Proposition 2.2.16, we can regard |K| ⊂ 1f () and the metric 13 |K| m

= |K| for a locally finite simplicial complex K (Proposition 1.7.1).

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2 Fundamental Results on Infinite-Dimensional Manifolds

ρK is induced by the norm of 1f (). Since (|K|, ρK ) is a compact AR (hence a uniform AR), there is a retraction r : 1f () → |K| that is uniformly continuous at |K|. By the same argument as the proof of Proposition 2.2.16, we can obtain |K| × 2f () ≈ 2f (). Now, we show the general case. For every v ∈ K (0), since St(v, K) is a contractible finite simplicial complex, | St(v, K)| × 2f () ≈ 2f () by the above case. Hence, OK (v) × 2f () is homeomorphic to an open set in 2f (). Thus, |K|×2f () is covered by open sets OK (v)×2f (), v ∈ K (0) . Namely, |K|×2f () " ! is an 2f ()-manifold. Remark 2.2 Proposition 2.2.17 holds without the local finite condition, which will be proved in Chapter 5 (combination of Theorems 5.10.18, 5.5.1, Proposition 5.10.3(2)).

2.3 Stability and Deficiency A map f : X → Y is called a near-homeomorphism provided, for each U ∈ cov(Y ), f is U-close to a homeomorphism h : X → Y . A space X is said to be E-stable if X × E ≈ X. A subset A of X said to be E-deficient if there is a homeomorphism h : X → X × E such that h(A) ⊂ X × {0}. In this section, it is proved that if X is E-stable, then the projection prX : X × E → X is a nearhomeomorphism and every open set in X is also E-stable. Moreover, we prove that every E-manifold is E-stable, and observe some properties of E-deficient sets. First, we introduce the reduced products, which will also be used in the following sections. Let X and Y be spaces and A be a closed set in X. The product of X and Y reduced over A is the space   (X × Y )A = (X \ A) × Y ∪ A, with the topology generated by open sets in the product space (X \ A) × Y and sets   (U \ A) × Y ∪ (U ∩ A), where U is open in X. Then, the projection prX : X × Y → X is factored through (X × Y )A into two natural maps qA : X × Y → (X × Y )A and pA : (X × Y )A → X defined by  qA |(X \ A) × Y = id, qA |A × Y = prA

and

 pA |(X \ A) × Y = prX\A , pA |A = id.

2.3 Stability and Deficiency

109

Thus, we have the following commutative diagram: prX

X×Y qA

X pA

(X × Y )A The topology of (X × Y )A is the coarsest topology such that the map pA is continuous and the product space (X \ A) × Y is its subspace. By the Bing Metrization Theorem 1.3.3, if X and Y are metrizable, then (X × Y )A is also metrizable. In the case where Y is compact, (X × Y )A is homeomorphic to the adjunction space A∪prA (X×Y ), where prA : A×Y → A is the projection. It should also be remarked that (I × X){0} is naturally homeomorphic to the (metrizable) cone C(X), so we sometimes regard C(X) = (I × X){0} . To prove the stability of E-manifolds, our main tool is the coordinate-switching pseudo-isotopy. For simplicity, we use the following notational convention: • Given a space X, let X∗ denote X × E N or X × EfN depending on E ≈ E N or E ≈ EfN , and let p0 : X∗ → X and pn : X∗ → X × E n , n ∈ N, be the projections. Additionally, let X0 = X × {0} ⊂ X∗ . X∗

We define a homotopy θ E : X∗ × E × I → X∗ as follows: For (x, y1 , y2 , . . . ) ∈ (x ∈ X, yi ∈ E, i ∈ N) and z ∈ E, θ1E (x, y1 , y2 , . . . , z) = (x, z, −y1 , −y2 , . . . ), θ2E−1 (x, y1 , y2 , . . . , z) = (x, y1 , z, −y2 , −y3 , . . . ), .. . θ2E−n (x, y1 , y2 , . . . , z) = (x, y1 , . . . , yn , z, −yn+1 , −yn+2 , . . . ), θ2E−(n+1) (x, y1 , y2 , . . . , z) = (x, y1 , . . . , yn , yn+1 , z, , −yn+2 , . . . ), .. . θ0E (x, y1 , y2 , . . . , z) = (x, y1 , y2 , . . . ),

and for 2−(n+1)  t  2−n , n ∈ ω, θtE (x, y1 , y2 , . . . , z) = (x, y1 , . . . , yn , z cos π(1 − 2n t) + yn+1 sin π(1 − 2n t), z sin π(1 − 2n t) − yn+1 cos π(1 − 2n t), − yn+2 , −yn+3 , . . . ).

110

2 Fundamental Results on Infinite-Dimensional Manifolds

Then, it should be remarked that: () ()

θtE (x, 0) = x for each x ∈ X0 and t ∈ I. t  2−n ⇒ pn θtE = pn prX∗ .

In addition, θ E induces a homeomorphism θ˜ E : X∗ × E × (0, 1] → X∗ × (0, 1] defined by θ˜ E (x, y, z, t) = (θ E (x, y, z, t), t) for each (x, y) ∈ X∗ , z ∈ E, and t ∈ (0, 1]. ∗ ∗ This means that (θ E )−1 t : X → X ×E is also continuous with respect to t ∈ (0, 1]. E We call θ the coordinate-switching pseudo-isotopy for X∗ . Given a map τ : X∗ → I, we define a map θτE : (X∗ × E)τ −1 (0) → X∗ by

θτE |τ −1 (0) = id and θτE (w, z) = θτE(w) (w, z) if τ (w) = 0. Then, by (), the following is satisfied: θτE (x, 0) = x for every x ∈ X0 \ τ −1 (0). The continuity of θτE at w ∈ τ −1 (0) follows from (). Indeed, each neighborhood U of θτE (w) = w in X∗ contains pn−1 (V ), where V is an open neighborhood of pn (w) in X × E n . Then, the following is an open neighborhood of w in (X∗ × E)τ −1 (0) :   W = (pn−1 (V ) ∩ τ −1 ((0, 2−n ))) × E ∪ (pn−1 (V ) ∩ τ −1 (0)). Since pn θτE |τ −1 ((0, 2−n )) = pn prX∗ |τ −1 ((0, 2−n )) by (), we have θτE ((pn−1 (V ) ∩ τ −1 ((0, 2−n ))) × E) ⊂ pn−1 (V ). Hence, it follows that θτE (W ) ⊂ pn−1 (V ) ⊂ U . Lemma 2.3.1 Let τ : X∗ → I be a map. Suppose X∗ \ τ −1 (0) = some ∅ = D0 ⊂ D1 ⊂ · · · such that:



n∈N Dn

(i) τ (w)  2−n−1 for each w ∈ X∗ \ Dn and (ii) w ∈ Dn , pn (w) = pn (w ) ⇒ τ (w) = τ (w ). Then, θτE is a homeomorphism. Proof For simplicity, let A = τ −1 (0). First, observe the following two facts: (1) τ (θτE (w, z)) = τ (w) for each (w, z) ∈ (X∗ \ A) × E. (2) τ (prX∗ (θτE(w))−1 (w)) = τ (w) for each w ∈ X∗ \ A.

for

2.3 Stability and Deficiency

111

(1): Let (w, z) ∈ (X∗ \ A) × E. Then, w ∈ Dn \ Dn−1 for some n ∈ N, whence τ (w)  2−n by (i). Thus, it follows from () that pn (θτE (w, z)) = pn θτE(w) (w, z) = pn prX∗ (w, z) = pn (w), which implies τ (θτE (w, z)) = τ (w) by (ii). (2): Each w ∈ X∗ \ A is contained in some Dn \ Dn−1 . In the same way as the above, we have pn θτE(w) = pn prX∗ . Then, pn (w) = pn (prX∗ (θτE(w) )−1 (w)), which implies τ (prX∗ (θτE(w) )−1 (w)) = τ (w) by (ii). Now, we define ητE : X∗ → (X∗ × E)A by ητE |A = id and ητE (w) = E (θτ (w) )−1 (w) for w ∈ X∗ \ A (= X∗ \ τ −1 (0)). Then, by (2), w ∈ X∗ \ A ⇒ ητE (w) = (θτE(w) )−1 (w) ∈ (X∗ \ A) × E = (X∗ × E)A \ A. Since τ (prX∗ ητE (w)) = τ (w) for w ∈ X∗ \ A, it is easy to see that ητE θτE = id and θτE ητE = id, namely that ητE is the inverse of θτE . The continuity of ητE remains to be proved. It follows from the continuity of (θ˜ E )−1 that ητE is continuous at each point of X∗ \ A. We shall show the continuity of ητE at w ∈ A. Each neighborhood of w = ητE (w) in (X∗ × E)A contains (W ∩ A) ∪ ((W \ A) × E) for some open neighborhood W of w in X∗ . We can find n ∈ N and an open neighborhood W0 of pn (w) in X × E n such that pn−1 (W0 ) ⊂ W . Then, V = pn−1 (W0 ) ∩ τ −1 ([0, 2−n )) is an open neighborhood of w in X∗ . Moreover, ητE (V ) ⊂ (W ∩ A) ∪ ((W \ A) × E). Indeed, ητE (V ∩ A) = V ∩ A ⊂ pn−1 (W0 ) ∩ A ⊂ W ∩ A. For each v ∈ V \ A, we write ητE (v) = (θτE(v))−1 (v) = (v  , z) ∈ (X∗ \ A) × E. Since τ (v  ) = τ (v) < 2−n by (2), it follows from () that pn (v  ) = pn θτE(v  ) (v  , z) = pn θτE(v)(v  , z) = pn (v) ∈ W0 , hence v  ∈ W \ A. Thus, we have ητE (v) = (v  , z) ∈ (W \ A) × E.

" !

Lemma 2.3.2 Let X be an E-stable perfectly normal space, A be an E-deficient set, A0 ⊂ A1 be closed sets, and W be an open set in X such that cl(A1 \ A0 ) ⊂ W . Then, there is a homeomorphism h : (X × E)A0 → (X × E)A1 such that (i) h(x, 0) = (x, 0) for each x ∈ A \ A1 , (ii) h(x, 0) = x for each x ∈ A ∩ (A1 \ A0 ), and (iii) h|(X \ (A0 ∪ W )) × E ∪ A0 = id.

112

2 Fundamental Results on Infinite-Dimensional Manifolds (X × E ) A 0

(X × E ) A1 h

A0

A1 W

W E

E

0

1

A1 A0

X W

Fig. 2.5 Homeomorphism h

Proof Since X ×E ≈ X and E ≈ E N or E ≈ EfN , we may replace X by X∗ , where A can be regarded as a subset of X0 = X × {0} ⊂ X∗ because A is E-deficient. For j j j = 0, 1, we shall construct maps τj : X∗ → I and ∅ = D0 ⊂ D1 ⊂ · · · so that  j X∗ \ τj−1 (0) = n∈N Dn and those satisfy the conditions of Lemma 2.3.1 and the following additional conditions: (0) τ0 |X∗ \ W = τ1 |X∗ \ W and τ0−1 (0) = A0 ⊂ A1 = τ1−1 (0). Then, we would have homeomorphisms θτEj : (X∗ × E)Ai → X∗ , j = 0, 1, that satisfy the following: (1) θτEj (x, 0) = x for every x ∈ X0 \ Aj , (2) θτEj |Aj = idAj , and (3) θτE0 |(X∗ \ (A0 ∪ W )) × E = θτE1 |(X∗ \ (A0 ∪ W )) × E. Then, h = (θτE1 )−1 θτE0 is the desired homeomorphism (Fig. 2.5). We write A2 = A0 ∪ (X∗ \ W ). Then, X∗ \ A2 = W \ A0 . Now, we take Bi ∈ cov(X∗ \ Ai ), i = 1, 2, which consist of basic open sets in X∗ .14 Here, B ⊂ X∗ is said to be n-basic if pn−1 (pn (B)) = B. Note that each n-basic subset of X∗ is m-basic for every m  n. For each n ∈ N, let Bi[n] =

   B ∈ Bi  B is n-basic , i = 1, 2.

Then, Bi[n] is n-basic and pn (Bi[n] ) is open in X×E n . Since X×E n ≈ X is perfectly normal, we have a map ki,n : X × E n → I such that −1 ki,n (0) = (X × E n ) \ pn (Bi[n] ) = pn (X∗ \ Bi[n] ).

14 I.e.,

open sets pn−1 (U ), n ∈ N, where U is open in X × E n .

2.3 Stability and Deficiency

113

n = k −1 ([1/m, 1]). Then, p (B [n] ) =  n For each m ∈ N, let Di,m n i m∈N Di,m , each i,n n n n n Di,m is closed in X × E , and Di,m ⊂ int Di,m+1 . For each n ∈ N, we define

Di[n] =



  −1 m  pm (Di,n ) m  n , i = 1, 2.

Observe that each Di[n] is n-basic, Di[n] ⊂ int Di[n+1] , and For each n ∈ N, let



[n] n∈N Di

= X ∗ \ Ai .

Hi[n] = Di[n] \ int Di[n−1] , i = 1, 2, where Di[0] = ∅. Then, it follows that 

Hi[n] =

n∈N

X∗ \ Di[n]



Di[n] = X∗ \ Ai ,

n∈N

⊂ Ai ∪



Hi[m] and

m>n

Hi[n] ∩ Hi[n+1] = bd Di[n] for each n ∈ N. Moreover, Hi[m] ∩Hi[n] = ∅ if |m−n| > 1. Since pn (bd Di[n] ) and pn (bd Di[n−1] ) are disjoint closed sets in pn (Hi[n] ), we can take maps τi[n] : pn (Hi[n] ) → [2−n−1 , 2−n ], i = 1, 2, such that τi[n] (pn (bd Di[n] )) = 2−n−1 and τi[n] (pn (bd Di[n−1] )) = 2−n . We define τi : X∗ → I, i = 1, 2, as follows: τi (Ai ) = 0 and τi |Hi[n] = τi[n] pn |Hi[n] . It is easy to see that τi is continuous at each point of X∗ \Ai . To verify the continuity of τi at x ∈ Ai , for each ε > 0, choose n ∈ N so that 2−n < ε. Then, U = X∗ \ Di[n]  is a neighborhood of x in X∗ . Note that U \ Ai ⊂ m>n Hi[m] . If y ∈ U \ Ai , then y ∈ Hi[m] for some m > n, hence τi (y) = τi[m] pm (y)  2−m < 2−n < ε. Therefore, τi (U ) ⊂ [0, ε). It follows from the definitions that τi and (Di[n] )n∈N satisfy the conditions of Lemma 2.3.1. Moreover, τi−1 (0) = Ai . For each n ∈ N, let D0[n] = D1[n] ∪ D2[n] . Define τ0 : X∗ → I by τ0 (x) = max{τ1 (x), τ2 (x)}. Then, τj and (Dj[n] )n∈N , j = 0, 1, satisfy the required conditions. Indeed, τ0−1 (0) = τ1−1 (0) ∩ τ2−1 (0) = A1 ∩ A2 = A0 .

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2 Fundamental Results on Infinite-Dimensional Manifolds

Since X∗ \ W ⊂ A2 = τ2−1 (0), τ0 (x) = τ1 (x) for each x ∈ X∗ \ W , that is, τ0 |X∗ \ W = τ1 |X∗ \ W . Thus, those satisfy condition (0). Finally, using the fact that τi and (Di[n] )n∈N , i = 1, 2, satisfy the conditions of Lemma 2.3.1, we can easily " ! show that τ0 and (D0[n] )n∈N also satisfy the same conditions. Remark 2.3 In the above proof, each x ∈ X∗ \ A1 is contained in some D1[n] . Then, m ⊂ p (B [m] ), hence x ∈ B [m] . there is some m  n such that pm (x) ∈ D1,n m 1 1 Thus, x is contained in some m-basic B ∈ B1 . Since τ1 (x)  2−m , it follows from () that pm θtEτ1 (x) (x, z) = pm (x), so θtEτ1 (x) (x, z) ∈ B for every z ∈ E and t ∈ I, where θ0E = prX∗ . By choosing B1 to refine a given U ∈ cov(X∗ \ A1 ), the homeomorphism θτE1 : (X∗ × E)A1 → X∗ is U-homotopic to prX∗ by the homotopy θtEτ1 , t ∈ I. Recall that we have the natural map qA1 : X∗ × E → (X∗ × E)A1 defined by qA1 |(X∗ \ A1 ) × E = id and qA1 |A1 × E = prA1 . Then, f = θτE1 qA1 : X∗ × E → X∗ is a map such that f |A1 × E = prA1 and f |(X∗ \ A1 ) × E is a homeomorphism onto X∗ \ A1 that is U-homotopic to prX∗ \A1 . In this remark, for an open set U ⊂ X, let A1 = X \ U . Then, we have the following: Proposition 2.3.3 Let X be an E-stable perfectly normal space, A be an Edeficient set in X, and U be an open set in X. Then, for each open cover U of U , there exists a map f : X × E → X such that f (x, 0) = x for each x ∈ A ∩ U , f |(X \ U ) × E = prX\U and f |U × E is a homeomorphism onto U that is Uhomotopic to prU . " ! Corollary 2.3.4 Let X be an E-stable perfectly normal space and A be an Edeficient set in X. Then, the following hold: (1) (2) (3) (4)

Every open set in X is E-stable. For each open set U in X, A ∩ U is E-deficient in U . The projection prX : X × E → X is a near-homeomorphism. For any open cover U of X, there is a homeomorphism f : X × E → X such that f (x, 0) = x for each x ∈ A and f U prX . " !

Replacing E = (E, 0) with a pointed space L = (L, ∗), the L-stability and the L-deficiency are similarly defined, that is, a space X is L-stable if X × L ≈ X, and a subset A of X is L-deficient if there is a homeomorphism h : X → X × L such that h(A) ⊂ X × {∗}. Corollary 2.3.5 Let X be an E-stable perfectly normal paracompact space, A be an E-deficient set in X, and L = (L, ∗) be a pointed space such that E × L ≈ E. Then, for each open cover U of X, there exists a homeomorphism h : X × L → X such that h(x, ∗) = x for each x ∈ A and h U prX . Consequently, the projection prX : X × L → X is a near-homeomorphism and every E-deficient set in X is L-deficient.

2.3 Stability and Deficiency

115

Proof Let ϕ : E × L → E be a homeomorphism with ϕ(0, ∗) = 0. Since X is paracompact, there exists V ∈ cov(X) such that st V ≺ U. By Corollary 2.3.4(4) above, we have a homeomorphism f : X × E → X such that f (x, 0) = x for each x ∈ A and f V prX . Then, the desired homeomorphism h : X × L → X can be defined as the composition of the following homeomorphisms: f −1 ×idL

X×L

X×E×L

idX ×ϕ

X×E

f

X

Indeed, it is easy to see that h(x, ∗) = x for each x ∈ A and h V prX (idX × ϕ)(f −1 × idL ) = prX (f −1 × idL ) = prX f −1 prX , where the same notation prX stands for three different projections. Since prX f −1 V ff −1 = idX , it follows that h st V prX , hence h U prX . " ! It is said that X is locally E-stable if each x ∈ X has an E-stable neighborhood in X. Applying Michael’s Theorem 1.3.20 on local property, we prove the following theorem: Theorem 2.3.6 For every perfectly normal paracompact space X, if X is locally E-stable, then X is E-stable and the projection prX : X × E → X is a nearhomeomorphism. Proof Let P be the property for open sets in X to be E-stable. The result follows from Michael’s Theorem 1.3.20 on local properties if P is G-hereditary. The condition (G-1) is Corollary 2.3.4(1), and (G-3) is obvious. It remains to show (G2). (G-2): For any E-stable open sets U1 and U2 in X, let U = U1 ∪ U2 . Since U is normal, we can choose open sets V1 and V2 in U so that clU V1 ∩ clU V2 = ∅, U \ U2 ⊂ V1 , and U \ U1 ⊂ V2 . Applying Lemma 2.3.2 to A0 = ∅, A1 = clU V1 , W = U1 \ clU V2 , and X = U1 , we have a homeomorphism h1 : U1 × E → (U1 × E)clU V1 such that h1 |(U1 ∩ clU V2 ) × E = id. Then, h1 can be extended to a homeomorphism h¯ 1 : U × E → (U × E)clU V1 by h¯ 1 |(U \ U1 ) × E = id (Fig. 2.6). (U 1

clU V 2 ) × E

h¯ 1

U2

h¯ 2

clU V 1 U

U1

V2

U1

Fig. 2.6 Homeomorphisms h¯ 1 and h¯ 2

V1

V2

U2

V1

U2

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Next, applying Lemma 2.3.2 to A0 = U2 ∩ clU V1 , A1 = U2 , and W = X = U2 , we have a homeomorphism h2 : (U2 ×E)U2 ∩clU V1 → U2 such that h2 |U2 ∩clU V1 = id. Then, h2 can be extended to a homeomorphism h¯ 2 : (U × E)clU V1 → U by h¯ 2 |U \ U2 = id. Thus, we have a homeomorphism h¯ 2 h¯ 1 : U × E → U , that is, U is E-stable. " ! Note that every E-manifold is perfectly normal and paracompact because it is metrizable. Since each open set in E is E-stable by Corollary 2.3.4(1), every Emanifold is locally E-stable. Thus, as a corollary of Theorem 2.3.6 above, we have the following STABILITY THEOREM: Theorem 2.3.7 (STABILITY) Every E-manifold M is E-stable and the projection prM : M × E → M is a near-homeomorphism. " ! It is said that a subset A of an E-stable space X is locally E-deficient in X if each a ∈ A has a neighborhood U in X such that A ∩ U is E-deficient in U . By making small changes in the proof of Theorem 2.3.6, we can prove the following: Proposition 2.3.8 Let X be an E-stable perfectly normal paracompact space. Then, every locally E-deficient closed set A in X is E-deficient in X. As the result, if a closed set A ⊂ X is E-deficient in an open set U in X, then A is E-deficient in X. Proof Let P be the property for open sets U in M such that A ∩ U is E-deficient in U . This property implies that U is E-stable. Actually, an open set U with A ∩ U = ∅ has property P if U is E-stable. Then, X has P locally.15 In the proof of Theorem 2.3.6, applying Corollary 2.3.4(2) instead of (1) gives a homeomorphism h : X × E → X such that h(x, 0) = x for each x ∈ A. " ! Now, we consider Q-stability. Because [−1, 1]2 ≈ B2 , there exists an isotopy ϕ : [−1, 1]2 × I → [−1, 1]2 such that ϕ0 = id, ϕt (0, 0) = (0, 0) for each t ∈ I, and ϕ1 (x, y) = (−y, x) for each (x, y) ∈ [−1, 1]2, that is, ϕ is induced by the anti-clockwise rotation of B2 . We define the isotopy ϕ Q : Q2 × I → Q2 as follows:   Q ϕt (x, y) = ψ (ϕt (x(i), y(i)))i∈N ,  N where ψ : [−1, 1]2 → Q2 is the homeomorphism defined by     ψ (x(i), y(i))i∈N = (x(i))i∈N, (y(i))i∈N .

15 When A is not closed in M, it is unknown whether a point a ∈ cl A\A has an open neighborhood U such that A ∩ U is E-deficient in U .

2.3 Stability and Deficiency Q

Q

117 Q

Then, ϕ0 = id, ϕt (0, 0) = (0, 0) for each t ∈ I, and ϕ1 (x, y) = (−y, x) for each (x, y) ∈ Q2 . Using this isotopy, we can define the coordinate-switching pseudoisotopy θ Q : X × QN × Q × I → X × QN in the same way as θ E : X∗ × E × I → X∗ (X∗ = X × E N ). Then, θ Q satisfies the following conditions: () ()

Q

θt (x, 0) = x for each x ∈ X × {0} ⊂ X × QN and t ∈ I. Q t  2−n ⇒ pn θt = pn prX×QN ,

where pn : X × QN → X × Qn is the projection. Note that QN ≈ Q. By the same arguments as above, we have the following Q-stable versions of Proposition 2.3.3 and Corollary 2.3.4: Proposition 2.3.9 Let X be a Q-stable perfectly normal space, A be a Q-deficient set in X, and U be an open set in X. Then, for each open cover U of U , there exists a map f : X × Q → X such that f (x, 0) = x for each x ∈ A ∩ U , f |(X \ U ) × Q = prX\U and f |U × Q is a homeomorphism onto U that is Uhomotopic to prU . " ! Corollary 2.3.10 Let X be a Q-stable perfectly normal space and A be a Qdeficient set in X. Then, the following hold: (1) (2) (3) (4)

Every open set in X is Q-stable. For each open set U in X, A ∩ U is Q-deficient in U . The projection prX : X × Q → X is a near-homeomorphism. For any open cover U of X, there is a homeomorphism f : X × Q → X such that f (x, 0) = x for each x ∈ A and f U prX . " !

Since Q × In ≈ Q for every n ∈ N, the following can be obtained in the same way as Corollary 2.3.5: Corollary 2.3.11 For every Q-stable perfectly normal paracompact space X and " ! n ∈ N, the projection prX : X × In → X is a near-homeomorphism. It is said that X is locally Q-stable if each x ∈ X has a Q-stable neighborhood in X. The following is the Q-manifold version of Theorem 2.3.6: Theorem 2.3.12 For every perfectly normal paracompact space X, if X is locally Q-stable, then X is Q-stable and the projection prX : X × Q → X is a nearhomeomorphism. " ! Thus, we have the following Q-manifold version of Theorem 2.3.7: Theorem 2.3.13 (STABILITY FOR Q-MANIFOLDS) Every Q-manifold M is Qstable and the projection prM : M × Q → M is a near-homeomorphism.

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2 Fundamental Results on Infinite-Dimensional Manifolds

Additionally, for each n ∈ N, the projection prM : M × In → X is also a nearhomeomorphism. " ! In the same way, locally Q-deficient sets and Q-deficient embeddings are defined. Then, the following can also be obtained: Proposition 2.3.14 Let X be a perfectly normal paracompact space. Then, every locally Q-deficient closed set in X is Q-deficient in X. " ! Note In the above, to prove Proposition 2.3.9, we translated the arguments for E to Q, where QN was used instead of E N . However, because Q itself has infinitely many coordinates, it is possible to define the coordinate-switching pseudo-isotopy θQ : X × Q × Q × I → X × Q Q

Q

so that θ0 is the projection, θt is a homeomorphism for t > 0, and ()

Q

θt (x, 0) = x for each x ∈ X × {0} ⊂ X × Q and t ∈ I.

Here, we use an isotopy ϕ : [−1, 1]2 × I → [−1, 1]2 such that ϕ0 = id, ϕt (0, 0) = (0, 0) for each t ∈ I, and ϕ1 (x, y) = (y, −x) for each (x, y) ∈ [−1, 1]2 , that is, ϕ is induced by the clockwise rotation of B2 ≈ [−1, 1]2 . Then, we define θ Q as follows: For (x, y, z) ∈ X × Q × Q, Q

θ1 (x, y, z) = (x, z(1), y(1), z(2), y(2), z(3), y(3), . . . ) Q

θ2−1 (x, y, z) = (x, y(1), −z(1), y(2), −z(2), y(3), −z(3), . . . ), .. . Q

θ2−n (x, y, z) = (x, y(1), . . . , y(n), (−1)n z(1), y(n + 1), (−1)n z(2), y(n + 2), . . . ), Q

θ2−(n+1) (x, y, z) = (x, y(1), . . . , y(n), y(n + 1), (−1)n+1 z(1), y(n + 2), (−1)n+1 z(2), . . . ), .. . Q

θ0 (x, y, z) = (x, y) = (x, y(1), y(2), y(3), . . . ) and for 2−(n+1)  t  2−n , Q

θt (x, y, z) = (x, y(1), . . . , y(n), ϕ2−2n+1 t ((−1)n z(1), y(n + 1)), ϕ2−2n+1 t ((−1)n z(2), y(n + 2)), . . . ).

2.3 Stability and Deficiency

119

Then, () and the following are satisfied: ()

Q

t  2−n ⇒ pn θt = pn prX×Q ,

where pn : X × Q → X × [−1, 1]n is the projection. Let d ∈ Metr(Q) be the metric defined by d(x, y) = sup 2−n |x(n) − y(n)|. n∈N

Taking any d ∈ Metr(X), we define ρ ∈ Metr(X × Q) as follows: ρ((x, y), (x  , y  )) = dX (x, x  ) + d(y, y  ). Q

Then, ρ(θt , prX×Q )  2−n if t  2−n . For the space s ≈ RN , we can analogously define the coordinate-switching pseudoisotopy θ s : X × s × s × I → X × s.

An embedding f : Y → X is said to be E-deficient if f (Y ) is E-deficient in X. Recall that FE is the class of spaces which can be embedded in E as closed sets. Applying the Stability Theorem 2.3.7, we can prove the following theorem, which is also known as the Closed Embedding Approximation Theorem: Theorem 2.3.15 (STRONG UNIVERSALITY OF E-MANIFOLDS) Let M be an Emanifold and A ∈ FE , with a closed set B ⊂ A and a map f : A → M, such that f |B is an E-deficient closed embedding. Then, for each U ∈ cov(M), there exists an E-deficient closed embedding h : A → M such that h|B = f |B and h U f . Proof First, note that E 3 ≈ E and M is an ANR. By Corollary 2.3.4(4), we have a homeomorphism g : M × E 3 → M such that g(f (x), 0, 0, 0) = f (x) for each x ∈ B and g U prM . From the assumption, there is a closed embedding j : A → E. Take d ∈ Metr(A) and dM ∈ Metr(M) such that d, dM  1/2, and define dA ∈ Metr(A) as follows: dA (x, y) = d(x, y) + dM (f (x), f (y)) for each x, y ∈ A. Then, dM (f (x), f (y))  dA (x, y)  1 for each x, y ∈ A. Let k : A → I be the map defined by k(x) = dA (x, B). Let e = 0 ∈ E be fixed. The desired embedding h : A → M can be defined by h(x) = g(f (x), k(x)e, k(x)j (x), 0). Indeed, h|B = f |B by definition, and h(A) is E-deficient in M because M × E 2 ≈ M. Since g U prM , it follows that h U f . If x = y ∈ A, then k(x) = k(y),

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2 Fundamental Results on Infinite-Dimensional Manifolds

k(x)j (x) = k(y)j (y), or k(x) = k(y) = 0. In the third case, x, y ∈ B, which implies f (x) = f (y). Thus, h is injective. It remains to see that h is closed. Let xn ∈ A, n ∈ N, such that (h(xn ))n∈N is convergent in M. Then, it is enough to show that (xn )n∈N is convergent. The sequences (f (xn ))n∈N , (k(xn ))n∈N , and (k(xn )j (xn ))n∈N are convergent. When k(xn ) → t = 0, k(xn ) > 0 for sufficiently large n and k(xn )−1 → t −1 , so it follows that (j (xn ))n∈N is convergent. Since j is a closed embedding, (xn )n∈N is convergent. When k(xn ) → 0, we can choose yn ∈ B so that dA (xn , yn ) → 0, hence dM (f (xn ), f (yn )) → 0 by the definition of dA . Then, (f (yn ))n∈N is convergent. Since f |B is a closed embedding, (yn )n∈N is convergent, and hence so is (xn )n∈N . " ! Note that the class M0 of compacta (= compact metrizable spaces) is equal to the class FQ of spaces which can be embedded into Q. By the same proof as Theorem 2.3.15 above, we can prove the following: Theorem 2.3.16 (STRONG UNIVERSALITY OF Q-MANIFOLDS) Let M be a Qmanifold and A be a compactum, with a closed set B ⊂ A and a map f : A → M, such that f |B is a Q-deficient closed embedding of a closed set B ⊂ A. Then, for each U ∈ cov(M), there exists a Q-deficient closed embedding h : A → M such that h|B = f |B and h U f . " ! The following is a non-compact version of the above: Theorem 2.3.17 (CLOSED EMBEDDING APPROXIMATION) Let M be a Qmanifold, and A be a locally compact separable metrizable space, with a closed set B ⊂ A and a proper map f : A → M, such that f |B is a Q-deficient closed embedding. Then, for each U ∈ cov(M), there exists a Q-deficient closed p embedding h : A → M such that h|B = f |B and h U f .16 Proof Replacing U by a refinement, we can assume that U is locally finite and every U ∈ U has the compact closure in M. By Corollary 2.3.10(4), we have a homeomorphism g : M × Q2 → M such that g(f (x), 0, 0) = f (x) for each x ∈ B 17 and g U prM . We have an embedding j : A → Q with j (A) ⊂ pr−1 1 (1) (≈ Q), where pr1 : Q → [−1, 1] is the projection of the first factor. Let k : A → I be a map with k −1 (0) = B. The desired embedding h : A → M can be defined as follows: h(x) = g(f (x), k(x)j (x), 0).

p

that h U f means that h is properly U-homotopic to f . 17 This embedding is not closed unless A is compact. An embedding of A can be obtained by restricting an embedding of the one-point compactification αA = A ∪ {∞}. 16 Recall

2.4 Negligibility and Deficiency

121

Indeed, by definition, h|B = f |B and h(A) is Q-deficient in M. Let x = x  ∈ A. If k(x) = k(x  ), then k(x)j (x) = k(x  )j (x  ) because pr1 (k(x)j (x)) = k(x) = k(x  ) = pr1 (k(x  )j (x  )). If k(x) = k(x  ) > 0, then k(x)j (x) = k(x  )j (x  ) because j (x) = j (x  ). If k(x) = k(x  ) = 0, then x, x  ∈ B, so h(x) = f (x) = f (x  ) = h(x  ) because f |B is an embedding. In any case, h(x) = h(x  ). Thus, h is injective. Since g U prM , it easily follows that h U f . Then, because f is proper, it p follows from Proposition 1.3.8 that h U f . In particular, h is proper, hence it is closed (cf. Proposition 1.3.7(c)). Therefore, h is a closed embedding. " ! Remark 2.4 In Theorem 2.3.15, it suffices to assume that M is E-stable like Proposition 2.3.8. In Theorems 2.3.16 and 2.3.17, it suffices to assume that M is Q-stable like Proposition 2.3.14.

2.4 Negligibility and Deficiency In this section, we will prove that if A is an E-deficient closed18 set in an E-stable space X, then X \ A ≈ X. We will also prove that E is homeomorphic to its open cone when E is a normed linear space. Additionally, it is shown that every finite union of E-deficient sets is also E-deficient. First, we prove the following: Proposition 2.4.1 There exists an I-preserving homeomorphism    N  N ζ : RN × I \ {(0, 0)}, RN f × I \ {(0, 0)} → R × I, Rf × I . Proof For each n ∈ N, let ϕ (n) : R2 × I → R2 be an ambient invertible isotopy (as illustrated in Fig. 2.7) such that ϕ0 = id, ϕ1 (n − 1, y) = (n + y, 0) if y  −2−n and (n)

(n)

ϕt |(−∞, (n − 1) − 2−(n−1) ] × R = id for each t ∈ I. (n)

(n)

The second condition above means that ϕ1 maps the vertical ray {n − 1} × [−2−n , ∞) onto the horizontal ray [n − 2−n , ∞) × {0} isometrically.

18 The

condition being closed is weakened to being locally closed (see Theorem 2.4.3).

122

2 Fundamental Results on Infinite-Dimensional Manifolds (− , (n − 1) − 2−( n −1) ] × R {n − 1} × [−2− n , ) n

n−1

[n − 2− n ,

) × {0}

2− n

2−( n −1) Fig. 2.7 The ambient invertible isotopy ϕ (n)

We define an ambient invertible isotopy ψ (n) : RN × I → RN , (x, t) → y = as follows:

ψt(n) (x),

y(i) = x(i) if i = 1, n + 1, and (n)

(y(1), y(n + 1)) = ϕt γn (x)(x(1), x(n + 1)), where γn : RN → I is a map defined by γ1 (x) = 1 and, for n  2,   γn (x) = max 0, 1 − 2n−1 max{|x(2)|, . . . , |x(n)|} . It should be remarked that 2−n+1 γn (x) is the distance of (x(2), . . . , x(n)) ∈ Rn−1 from the complement of [−2−n+1 , 2−n+1 ]n−1 (the 2−n+1 -neighborhood of 0 ∈ Rn−1 ) with respect to the norm y∞ = max{|y(1)|, . . . , |y(n − 1)|}.19 For each n ∈ N, let hn = ψ1(n) · · · ψ1(1) : RN → RN . For the sake of convenience, write h0 = idRN . If there exists a homeomorphism h∞ : RN \ {0} → RN such that each x ∈ RN \ {0} has a neighborhood U in RN \ {0} such that h∞ |U = ψt(n+1) hn |U = hn |U for a sufficiently large n ∈ N and any t ∈ I, and each y ∈ (n+1) −1 −1 RN has a neighborhood V such that h−1 ) |V = h−1 ∞ |V = hn (ψt n |V for a sufficiently large n ∈ N and any t ∈ I, then the desired homeomorphism ζ can be defined as follows:  (n) (ψ2−2 if 2−n  t  2−(n−1) n t hn−1 (x), t) ζ (x, t) = (h∞ (x), 0) if t = 0.

  n−1 n−1 2 1/2 or y = norm of Rn−1 can be replaced with y2 = 1 i=1 y(i) i=1 |y(i)|. In       n 2 1/2 or γ (x) = max 0, 1 − this case, γn is defined by γn (x) = max 0, 1 − 2n−1 x(i) n 2   2n−1 n2 |x(i)| .

19 This

2.4 Negligibility and Deficiency

123

To see the existence of h∞ , it suffices to prove the following two facts: (1) Each x ∈ RN \ {0} has a neighborhood U in RN \ {0} and n ∈ N such that (j ) ψt |hn (U ) = id for j > n; (2) Each y ∈ RN has a neighborhood V in RN and n ∈ N such that h−1 n (V ) ⊂ (j ) RN \ {0} and (ψt )−1 |V = id for j > n. (1): For each x ∈ RN \ {0}, let m = min{i ∈ N | x(i) = 0}. In other words, x(m) is the first nonzero coordinate of x. We write y = hm−1 (x) and z = hm (x) = (m) ψ1 (y). When m = 1, y = h0 (x) = x and z = ψ1(1) (x). Since x(1) = 0, it follows that (z(1), z(2)) ∈ [2−1 , ∞) × {0}, that is, z(1) < 2−1 or z(2) = 0. When m > 1, observe that hm−1 (x) = y = (y(1), 0, . . . , 0, y(m), x(m + 1), x(m + 2), . . . ) and hm−1 (x) = hm−1 (0, 0, . . . , 0, x(m + 1), x(m + 2), . . . ) = (m − 1, 0, . . . , 0, x(m + 1), x(m + 2), . . . ). Thus, we have y(1) = m − 1 or y(m) = 0. In the case y(1) = m − 1, since (z(1), z(m + 1)) ∈ [m − 2−m , ∞) × {0}, it follows that z(1) < m − 2−m or z(m + 1) = 0. If y(m) = 0, then z(m) = y(m) = 0. Anyway, we have z(1) < m − 2−m or max{|z(2)|, . . . , |z(m + 1)|} = 0. When z(1) < m − 2−m , V = {z ∈ RN | z (1) < m − 2−m } is a neighborhood (j ) of z in RN and ψt |V = id for every j > m. Then, U = h−1 m (V ) is the desired neighborhood of x in RN . When max{|z(2)|, . . . , |z(m + 1)|} = 0, choose n  m + 1 so that 2−n < max{|z(2)|, . . . , |z(m + 1)|}. Let z∗ = hn (x). Since z∗ (2) = z(2), . . . , z∗ (m + 1) = z(m + 1), we have max{|z∗ (2)|, . . . , |z∗ (n)|}  max{|z(2)|, . . . , |z(m + 1)|} > 2−n . So, x has a neighborhood U in RN such that z ∈ hn (U ) ⇒ max{|z (2)|, . . . , |z (n)|} > 2−n . Then, γn (z ) = 0 for each z ∈ hn (U ). Hence, ψt |hn (U ) = id for every j > n. (j )

(2): For each y ∈ RN , choose n ∈ N so that y(1) < n − 2−n . Then, V = {z ∈ | z(1) < n − 2−n } is a neighborhood of y in RN and hn (0) = (n, 0, 0, . . . ) ∈ (j ) N V , that is, h−1 n (V ) ⊂ R \ {0}. For each j > n, we have ψt |V = id, that is, (j ) (ψt )−1 |V = id. " !

RN

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Using Proposition 2.4.1 above, we prove the following: Proposition 2.4.2 Let X be an E-stable perfectly normal paracompact space and A be an E-deficient closed set in X. Then, for each open cover U of X, there exists an I-preserving homeomorphism h : X × I → X × I \ A × {0} such that each ht is U-close to idX . In particular, h0 : X → X \ A is a homeomorphism that is U-close to idX . N N Proof For simplicity, we write F = RN or RN f depending on E ≈ E or E ≈ Ef . Since E ≈ E × F by Lemma 2.1.9, X is F -stable and A is F -deficient in X (Corollary 2.3.5). Let ζ : F × I \ {(0, 0)} → F × I be the I-preserving homeomorphism obtained in Proposition 2.4.1. Then, we have an I-preserving homeomorphism

ϕ : (X × F × I) \ (A × {0} × {0}) → X × F × I defined by ϕ(x, y, t) = (x, ζmax{t,α(x)}(y), t), where α : X → I is a map such that α −1 (0) = A. Let V be an open star-refinement of U. By Corollary 2.3.4(4), we have a homeomorphism f : X × F → X such that f (x, 0) = x for each x ∈ A and f is V-close to prX , so prX f −1 is V-close to idX . The desired I-preserving homeomorphism h can be defined by h(x, t) = (f ϕt−1 f −1 (x), t). Indeed, each ht = f ϕt−1 f −1 is V-close to prX ϕt−1 f −1 = prX f −1 because ϕ preserves Xcoordinates. Then, it follows that ht is U-close to idX . " ! A set A in a space X is said to be negligible in X if X \ A ≈ X. It is said that A is locally closed in X if each x ∈ A has a neighborhood U in X such that A ∩ U is closed in U . It is easy to see that A is locally closed in X if and only if A = W ∩clX A for some open set W in X. Now, we have the following Negligibility Theorem: Theorem 2.4.3 (NEGLIGIBILITY) Let X be an E-stable metrizable space. If A is an E-deficient locally closed set in X, then the inclusion X \ A ⊂ X is a nearhomeomorphism, hence A is negligible in X. Proof Let U ∈ cov(X) and d ∈ Metr(X). Choose an open set W in X so that A ⊂ W and A is closed in W . Due to Corollary 2.3.4(1), (2), W is E-stable and A is E-deficient. By Proposition 2.1.17, W has an open cover V that is fitting in X and refines U. By Proposition 2.4.2, we have a homeomorphism h : W → W \ A that is V-close to id. Then, h can be extended to a homeomorphism h˜ : X → X \ A by ˜ \ W = id. The inverse homeomorphism h˜ −1 : X \ A → X is U-close to the h|X inclusion X \ A ⊂ X. " !

2.4 Negligibility and Deficiency

125

A locally closed set in a metrizable space X is Fσ in X, because it is the intersection of a closed set and an open set. Hence, an E-deficient locally closed set in X is a countable union of E-deficient closed sets in X, where it should be remarked that a countable union of E-deficient locally closed sets in X can be written as a countable union of E-deficient closed sets in X. In the case where X is completely metrizable, Theorem 2.4.3 can be strengthened slightly as follows: Theorem 2.4.4 (NEGLIGIBILITY) Let X be an E-stable completely metrizable space. If A is a countable union of E-deficient closed sets in X, then the inclusion X \ A ⊂ X is a near-homeomorphism, hence A is negligible in X.  An , where each An is an E-deficient closed set in X. For each Proof Let A = n∈N n ∈ N, let Xn = X \ ni=1 Ai . Note that Xn is open in X, and is hence completely metrizable. For each U ∈ cov(X), X has an admissible complete metric d0 such that {Bd0 (x, 1) | x ∈ X} ≺ U (cf. 1.3.22(1)). Since the inclusion X1 ⊂ X is a near-homeomorphism by Theorem 2.4.3, we have a homeomorphism f1 : X → X1 with d0 (f1 , id) < 2−1 (hence d0 (f1−1 , id) < 2−1 ). Choose an admissible complete metric d1 for X1 so that d1 (x, x  )  max{d0 (x, x  ), d0 (f1−1 (x), f1−1 (x  ))} for each x, x  ∈ X1 . Since A2 ∩ X1 is an E-deficient closed set in X1 and X2 = X1 \ A2 , the inclusion X2 ⊂ X1 is a near-homeomorphism by Theorem 2.4.3. Then, we have a homeomorphism f2 : X1 → X2 with d1 (f2 , id) < 2−2 (hence d1 (f2−1 , id) < 2−2 ). By induction, we can obtain homeomorphisms fn : Xn−1 → Xn , and admissible complete metrics dn for Xn , such that dn−1 (fn , id) < 2−n (hence dn−1 (fn−1 , id) < 2−n ) and dn (x, x  )  max{dn−1 (x, x  ), dn−1 (fn−1 (x), fn−1 (x  ))} for each x, x  ∈ Xn , where X0 = X. Then, (fn fn−1 · · · f1 )n∈N is a Cauchy sequence in C(X, X) with respect to the sup-metric induced by d0 , so it converges to a map f = lim fn fn−1 · · · f1 : X → X. n→∞

 Since d0 (f, id)  n∈N 2−n = 1, it follows that f is U-close to id. For each x ∈ X and i ∈ N, let y = fi · · · f1 (x) ∈ Xi . Then, (fn fn−1 · · · fi+1 (y))n>i is a Cauchy sequence in Xi with respect to di , which implies f (x) = lim fn fn−1 · · · fi+1 (y) ∈ Xi . n>i

Therefore, f (X) ⊂ homeomorphism.



i∈N Xi

= X \ A. It remains to show that f : X → X \ A is a

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2 Fundamental Results on Infinite-Dimensional Manifolds

To construct the inverse of f : X → X \ A, observe that for each n ∈ N and x ∈ X \ A, −1 −1 −1 d0 (f1−1 · · · fn−1 fn (x), f1−1 · · · fn−1 (x)) −1 −1 −1  d1 (f2−1 · · · fn−1 fn (x), f2−1 · · · fn−1 (x))  · · · −1 −1 −1  dn−2 (fn−1 fn (x), fn−1 (x))  dn−1 (fn−1 (x), x) < 2−n . −1 −1 fn |X \ A)n∈N is a Cauchy sequence in C(X \ A, X) with Hence, (f1−1 · · · fn−1 respect to the sup-metric induced by d0 , and this converges to a map −1 −1 fn : X \ A → X. g = lim f1−1 · · · fn−1 n→∞

For each x ∈ X \ A and ε > 0, choose n ∈ N so that −1 −1 d0 (fg(x), ff1−1 · · · fn−1 fn (x)) < ε/2 and d0 (f, fn fn−1 · · · f1 ) < ε/2,

which implies that d0 (fg(x), x) < ε. Hence, fg(x) = x for each x ∈ X \ A. Now, we shall show that gf (x) = x for each x ∈ X.20 For each ε > 0, choose n ∈ N so that −1 −1 fn |X \ A) < ε/3 and d0 (g, f1−1 · · · fn−1 −1 −1 −1 d0 (f1−1 · · · fn−1 fn |Xm , f1−1 · · · fm−1 fm−1 ) < ε/3 for every m  n. −1 −1 fn at f (x), we have δ > 0 such that By the continuity of f1−1 · · · fn−1

y ∈ Xn , dn (f (x), y) < δ −1 −1 −1 −1 fn f (x), f1−1 · · · fn−1 fn (y)) < ε/3. ⇒ d0 (f1−1 · · · fn−1

Choose m  n so that 2−m < δ. Then, dn (f (x), fm fm−1 · · · f1 (x)) 

∞ 

dm+i (fm+i , id) < 2−m < δ,

i=1

which implies that −1 −1 −1 −1 fn f (x), f1−1 · · · fn−1 fn fm fm−1 · · · f1 (x)) < ε/3. d0 (f1−1 · · · fn−1

20 Note

X \ A.

that g is defined on

 n∈N

Xn = X \ A but the image fn fn−1 · · · f1 (X) is not contained in

2.4 Negligibility and Deficiency

127

It should be remarked that −1 −1 d0 (f1−1 · · ·fn−1 fn fm fm−1 · · · f1 , id) −1 −1 −1 = d0 (f1−1 · · · fn−1 fn |Xm , f1−1 · · · fm−1 fm−1 ) < ε/3.

Then, it follows that −1 −1 fn f (x)) d0 (gf (x), x)  d0 (gf (x), f1−1 · · · fn−1 −1 −1 −1 −1 + d0 (f1−1 · · · fn−1 fn f (x), f1−1 · · · fn−1 fn fm fm−1 · · · f1 (x)) −1 −1 + d0 (f1−1 · · · fn−1 fn fm fm−1 · · · f1 (x), x)

< ε/3 + ε/3 + ε/3 = ε. Therefore, gf (x) = x. Thus, f is a homeomorphism with f −1 = g.

" !

We consider C(X) = (I × X){0} and C o (X) = ([0, 1) × X){0} ≈ ([0, ∞) × X){0} , where C o (X) is called the open cone over X. Proposition 2.4.5 E ≈ C o (E). Proof Let SE be the unit sphere of E. Then, we have homeomorphisms: f

E ≈ E \ {0} ≈ SE × (0, ∞) ≈ SE × R, where the first homeomorphism is obtained by the Negligibility Theorem 2.4.3 above (or Proposition 2.4.2) and the second homeomorphism f is defined by f (x) = (x−1 x, x). Note that E × R is E-stable and {0} × R is E-deficient in E × R. By the Negligibility Theorem 2.4.3, we have   E × R ≈ E × R \ {0} × (R \ {0}) = (E \ {0}) × R ∪ {(0, 0)}. The desired homeomorphism is obtained by the following diagram: E ≈ E × R ≈ (E \ {0}) × R ∪ {(0, 0)} ⏐ ⏐ ⏐ ⏐ ≈! !g   C o (E) ≈ C o (SE × R) ≈ [0, ∞) × (SE × R) {0} ,

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2 Fundamental Results on Infinite-Dimensional Manifolds

x r ·t , x x

s

= (z, s) (x, t )

r r SE

E

0 SE

x + |t | = r

x x

r SE ×

SE ×

  Fig. 2.8 (E \ {0}) × R ∪ {(0, 0)} ≈ [0, ∞) × (SE × R) {0}

where g is defined as follows:  g(0, 0) = 0 and g(x, t) = x + |t|,

 x + |t| x , ·t . x x

It is easy to see that g is continuous. The inverse of g can be defined by g

−1

(0) = (0, 0) and g

−1

 (r, z, s) =

 rs r2 · z, . r + |s| r + |s|

The continuity of g −1 at 0 comes from the following:    2    r  rs       r + |s| · z  r and  r + |s|   r for (r, z, s) ∈ (0, ∞) × SE × R. The proof is complete (Fig. 2.8).

" !

For a closed set X in E, the open cone C o (X) is closed in C o (E) ≈ E and the cone C(X) = (I × X){0} is closed in ([0, ∞) × E){0} ≈ C o (E) ≈ E. Thus, the following holds: Corollary 2.4.6 For every X ∈ FE , C o (X) ∈ FE and C(X) ∈ FE .

" !

2.4 Negligibility and Deficiency

129

Moreover, we have the following: Corollary 2.4.7 For every locally finite-dimensional simplicial complex K with card K (0)  w(E), the polyhedron |K|m with the metric topology can be embedded in E as a closed set, i.e., |K|m ∈ FE . Proof First, suppose that dim K < ∞. Then, by induction on dim K, we shall show that |K|m ∈ FE . Since E contains a discrete set D with card D = w(E) by Lemma 2.1.10, we have the case dim K = 0. Assuming the (n − 1)-dimensional case, we prove the case dim K = n. For each v ∈ K (0) , | Lk(v, K)|m ∈ FE and | St(v, K)|m is homeomorphic to the metrizable cone C(| Lk(v, K)|m ), hence | St(v, K)|m ∈ FE by Corollary 2.4.6 above. In the same way as Theorem 2.1.11, we can apply Michael’s Theorem on local properties (Corollary 1.3.21) to obtain |K|m ∈ FE . When K is locally finite-dimensional, it follows from the above finitedimensional case that | St(v, K)|m ∈ FE for each v ∈ K (0) . Again, we can apply Michael’s Theorem (Corollary 1.3.21) to obtain |K|m ∈ FE . " ! Corollary 2.4.8 E × [0, 1) ≈ E. Proof First, note that E × [0, 1) is E-stable. Then, by Lemma 2.3.2 and Proposition 2.4.5, we have E × [0, 1) ≈ ((E × [0, 1)) × E)E×{0} = E × ([0, 1) × E){0} = E × C o (E) ≈ E × E ≈ E.

" !

A subset A in a space X is said to be collared in X if there is an open embedding k : A × [0, 1) → X such that k(x, 0) = x for all x ∈ A. Such an embedding k is called a collar of A in X. The following can be easily observed: Fact Every collared set in X is locally closed in X. Using collars, we can characterize the E-deficiency as follows: Theorem 2.4.9 Let X be an E-stable perfectly normal paracompact space with A ⊂ X. Suppose that E is a normed linear space. Then, the following are equivalent: (a) A is E-deficient in X; (b) The closure cl A is contained in some collared set in X; (c) There is a homeomorphism h : X → X × [0, 1) such that h(A) ⊂ X × {0}. Proof The implication (c) ⇒ (b) is trivial, because h(cl A) = cl h(A) ⊂ X × {0} in the condition (c). (a) ⇒ (c): Since E × [0, 1) ≈ E (Corollary 2.4.8), we have a homeomorphism h : X → X × [0, 1) such that h(A) ⊂ X × {0} by Corollary 2.3.5. (b) ⇒ (a): Let C be a collared set in X with cl A ⊂ C. Then, C × [0, 1) is homeomorphic to an open set in X, hence it is E-stable by Corollary 2.3.4(1). It

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2 Fundamental Results on Infinite-Dimensional Manifolds

suffices to show that C × {0} is E-deficient in C × [0, 1). Indeed, this implies that cl A × {0} is E-deficient in C × [0, 1). Since every locally E-deficient closed set in X is E-deficient (Proposition 2.3.8), cl A is E-deficient in X, which implies (a). By Lemma 2.3.2, we have 

   C × [0, 1), C × {0} ≈ ((C × [0, 1)) × E)C×{0} , C × {0}   = C × ([0, 1) × E){0} , C × {0}

On the other hand, ([0, 1) × E){0} = C o (E) ≈ E ≈ [0, 1) × E by Proposition 2.4.5 and Corollary 2.4.8. Then, it follows from the homogeneity of E that 

   C × [0, 1), C × {0} ≈ C × [0, 1) × E, C × {0} × {0} ,

which implies that C × {0} is E-deficient in C × [0, 1).

" !

Corollary 2.4.10 C(E) ≈ E and E × I ≈ E. Proof By Proposition 2.4.5 and Corollary 2.4.8, C(E) = C o (E) ∪ (0, 1] × E is an E-manifold, hence it is E-stable (Theorem 2.3.7). Since a collared set in an E-stable space is negligible (Theorems 2.4.9 and 2.4.3), we have C(E) ≈ C(E) \ E × {1} = C o (E) ≈ E by Proposition 2.4.5. On the other hand, E × I is also an E-manifold by Corollary 2.4.8. Similarly, we have E × I ≈ E × [0, 1) ≈ E. " ! N We define Qf = [−1, 1]N f ⊂ [−1, 1] = Q.

Corollary 2.4.11 Depending on E ≈ E N or E ≈ EfN , E × Q ≈ E or E × Qf ≈ E.

" !

Remark 2.5 Since E × I ≈ E by Corollary 2.4.10, Theorem 2.4.9 is valid even if [0, 1) in (c) can be replaced with I. Moreover, under the assumption of Theorem 2.4.9, A is E-deficient in X if and only if A is Q-deficient (resp. Qf deficient) in X in the case that E N ≈ E (resp. EfN ≈ E) by Corollaries 2.3.5 and 2.4.11. As with the model space E (Corollary 2.4.10), the Hilbert cube Q is homeomorphic to the cone C(Q). This will be proved in a different way in Sect. 2.7. Moreover, the Q-deficient closed set in a Q-manifold can be characterized in the same way as in Theorem 2.4.9, where E and [0, 1) are replaced with Q and I. This will be shown in Theorem 2.10.9.

2.4 Negligibility and Deficiency

131

Combining the Stability Theorem 2.3.7 with Corollaries 2.4.8, 2.4.10, and 2.4.11, we have the following: Corollary 2.4.12 For every E-manifold M, M × [0, 1) ≈ M × I ≈ M. And, depending on E ≈ E N or E ≈ EfN , M × Q ≈ M or M × Qf ≈ M. " ! In the rest of this section, we shall prove that every finite union of E-deficient sets is also E-deficient. The following is a key lemma: Lemma 2.4.13 There exists a homeomorphism h : I × E → (I × E){0} such that h|(0, 1] × {0} = id and h(0, 0) = 0. Proof Since E ≈ C(E) = (I × E){0} by Corollary 2.4.10 and E is homogeneous, there is a homeomorphism f : E → (I × E){0} with f (0) = 0. Then, we have a homeomorphism f  : (I × E){0} → (I × (I × E){0} ){0} such that f  (0) = 0 and f  |(0, 1] × E = id × f . As is easily observed,    I2 ≈ x ∈ I2  x(1)  x(2) ≈ (I × I){0} . We can easily construct a homeomorphism g : I×I → (I×I){0} such that g(0, 0) = 0 and g|(0, 1] × {0} = id, which induces the homeomorphism g  : ((I × I) × E)I×{0} → ((I × I){0} × E){0}∪(0,1]×{0} . Then, g  |I × {0} = g|I × {0} (i.e., g  (0, 0) = 0 and g  |(0, 1] × {0} = id), and g  |(I × (0, 1]) × E = (g|I × (0, 1]) × idE . We define the desired homeomorphism h as the composition of three homeomorphisms in the following diagram: h

I×E idI ×f

f

1

(I × (I × E){0} ){0}

I × (I × E){0}

((I × I) × E)I×{0}

(I × E){0}

g

((I × I){0} × E){0}∪(0,1]×{0}

For each t ∈ (0, 1], since f  h(t, 0) = g  (t, f (0)) = g  (t, 0) = (t, 0) and f  (t, 0) = (t, f (0)) = (t, 0), it follows that h(t, 0) = (t, 0). Moreover, f  h(0, 0) = g  (0, f (0)) = g  (0, 0) = 0. " ! Proposition 2.4.14 Let X be an E-stable perfectly normal space. Every finite union of E-deficient sets in X is also E-deficient.

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2 Fundamental Results on Infinite-Dimensional Manifolds

Proof It suffices to show that the union of two E-deficient sets A and B in X is also E-deficient, where we may assume that A and B are closed in X because their closures in X are E-deficient. Since I×E ≈ E by Corollary 2.4.10, we have a homeomorphism f : X → X×I such that f (A) ⊂ X × {0} (Corollary 2.3.5). Let C = f −1 (X × {0}). Then, A ⊂ C and f induces the homeomorphism f  : (X × E)C → ((X × I) × E)X×{0} = X × (I × E){0} . Then, f  |C = f |C and f  |(X \ C) × E = (f |X \ C) × idE . By Lemma 2.4.13 above, we have a homeomorphism h : I × E → (I × E){0} such that h(t, 0) = (t, 0) for every t ∈ (0, 1] and h(0, 0) = 0. Let h : X × E → (X × E)C be the homeomorphism defined by the following diagram: X×E

h

f ×idE

(X × E)C f

X×I×E

idX ×h

1

X × (I × E){0}

For each x ∈ C, h (x, 0) = x because f (x) = f  (x) ∈ X ×{0}. For each x ∈ X \C, let f (x) = (x  , t). Then, t > 0. Since f  h (x, 0) = (x  , h(t, 0)) = (x  , t, 0) and f  (x, 0) = (f (x), 0) = (x  , t, 0), we have h (x, 0) = (x, 0). Applying Lemma 2.3.2 (A = B, A0 = C, A1 = X), we have a homeomorphism g : (X × E)C → X such that g(x, 0) = x for each x ∈ B \ C and g|C = id. Then, gh : X×E → X is a homeomorphism. For each x ∈ C, gh (x, 0) = g(x) = x. For each x ∈ B \ C, gh (x, 0) = g(x, 0) = x. Thus, gh (x, 0) = x for each x ∈ A ∪ B. " !

2.5 The Collaring and Unknotting Theorems In this section, we show that an E-manifold closed subset of an E-manifold is collared if it is E-deficient, that every E-manifold can be embedded in E as an open set, and that a homeomorphism between E-deficient closed sets in an E-manifold M can be extended to a homeomorphism of M onto itself if it is homotopic to id in M. We frequently use the following lemma in this section: Lemma 2.5.1 Let A be a closed set in a perfectly normal paracompact space X, a  0  b, and U be an open cover of A × [a, b] in X × R such that    {x} × [a, b]  x ∈ A ≺ U.

2.5 The Collaring and Unknotting Theorems

133

R b

0

A

N

X

a

U {x } × [a, b ]

Fig. 2.9 The maps α and β

Then, there exist maps α, β, γ : X → R such that: (i) (ii) (iii) (iv) (v)

a  α(x)  0  β(x)  b for x ∈ X, α(x) = a and β(x) = b if x ∈ A,  α(x) = β(x) = γ (x) = 0 if (x, 0) ∈ U, γ (x) > 0 if (x, 0) ∈ U,     {x} × [α(x) − γ (x), β(x) + γ (x)]  (x, 0) ∈ U ≺ U.

Proof First, observe that A is contained in the following open set:    N = x ∈ X  {x} × [a, b] ⊂ U for some U ∈ U . Let k : X → I be an Urysohn map with k(X \ N) = 0 and k(A) = 1. We define maps α, β : X → R by α(x) = ak(x) and β(x) = bk(x). Then, α and β satisfy (i), (ii), and (iii). For each x ∈ N, {x} × [α(x), β(x)] is contained in some U ∈ U (Fig. 2.9).   Let U0 = {x ∈ X | (x, 0) ∈ U}. Then, U0 is an open set in X and U ∩ (X × {0}) = U0 × {0}. We define γ  : U0 → (0, 1] as follows:   γ  (x) = sup t ∈ (0, 1)  ∃U ∈ U such that

 {x} × [α(x) − t, β(x) + t] ⊂ U .

Then, γ  is l.s.c. Indeed, if γ  (x) > t > s then {x}×[α(x)−t, β(x)+t] is contained in some U ∈ U, so x has a neighborhood V in X such that V × [α(x) − t, β(x) + t] ⊂ U. Since α and β are continuous, x has a neighborhood W in X such that W ⊂ V and |α(x) − α(y)|, |β(x) − β(y)| < t − s for y ∈ W . Hence, {y} × [α(y) − s, β(y) + s] ⊂ U for every y ∈ W,

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2 Fundamental Results on Infinite-Dimensional Manifolds

that is, γ  (y) > s for y ∈ W . Note that U0 is paracompact because X is hereditarily paracompact by Theorem 1.3.13. Applying Theorem 1.3.14, we have a map γ0 : U0 → (0, 1) such that 0 < γ0 (x) < γ  (x) for each x ∈ U0 . Since X is perfectly normal, X \ U0 = k0−1 (0) for some map k0 : X → I. We can define a map γ : X → I by γ (X \ U0 ) = {0} and γ (x) = k0 (x)γ0 (x) for each x ∈ U0 . It is easy to see that γ satisfies conditions (iii), (iv), and (v). " ! Remark 2.6 In Lemma 2.5.1 above, if X is not assumed to be perfectly normal, we define a map γ using k instead of k0 . This does not satisfy (iv), but satisfies the following weaker condition: (iv) γ (x) > 0 if α(x) < β(x) (equivalently α(x) < 0 or β(x) > 0). When a = b = 0 in Lemma 2.5.1, we have the following: Lemma 2.5.2 Let X be a perfectly normal paracompact space.21 For each open collection U  in X × I, there exists a map γ : X → I such that γ (x) > 0 if and only if (x, 0) ∈ U, and 

   {x} × [0, γ (x)]  x ∈ X, (x, 0) ∈ U ≺ U.

In particular, for an open set U in X × I, there is a map γ : X → I such that γ (x) > 0 if and only if (x, 0) ∈ U , and    (x, t) ∈ X × I  γ (x) > 0, t  γ (x) ⊂ U.

" !

In the above, if U is an open cover (or U is an open neighborhood) of X × {0} in X × I, without assuming the perfect normality of X, we can prove the same result, that is: Lemma 2.5.3 Let X be a paracompact space. For each open cover U of X × {0} in X × I, there exists a map γ : X → I such that γ (x) > 0 if and only if 

  {x} × [0, γ (x)]  x ∈ X, γ (x) > 0 ≺ U.

In particular, for an open neighborhood U of X × {0} in X × I, there is a map γ : X → I such that γ (x) > 0 if and only if (x, 0) ∈ U , and    (x, t) ∈ X × I  γ (x) > 0, t  γ (x) ⊂ U.

" !

Lemma 2.5.4 Let X be a hereditarily paracompact space, that is, every open set in X is paracompact (cf. Theorem 1.3.13). If A is a collared set in X and U is an open set in X with A ∩ U = ∅, then A ∩ U is collared in U . U is an open cover (or U is an open neighborhood) of X × {0} in X × I, the perfect normality of X need not be assumed.

21 If

2.5 The Collaring and Unknotting Theorems

135

Proof Let k : A × [0, 1) → X be a collar of A in X. Applying Lemma 2.5.3, we have a map γ : A ∩ U → (0, 1) such that    (x, t) ∈ (A ∩ U ) × [0, 1)  t < γ (x) ⊂ k −1 (U ). We define h : (A ∩ U ) × [0, 1) → U by h(x, t) = k(x, γ (x)t). Then, h is a collar of A ∩ U in U . " ! For a collared closed set A in a space X, a collar k : A × [0, 1) → X may have a defect such that k(A × [0, t]) need not be closed in X for 0 < t < 1. For example, (0, ∞) × {0} is a closed set in [0, ∞)2 \ {(0, 0)} that has the natural collar with the image (0, ∞) × [0, 1). On this account, we introduce a collar without such a defect. A closed collar of A in X is a closed embedding k : A × I → X such that k|A × [0, 1) is a collar of A in X. Proposition 2.5.5 Every collared closed set A in a paracompact space X has a closed collar in X. Moreover, for each open cover U of X, there is a closed collar k : A × I → X such that {k({x} × I) | x ∈ A} ≺ U. Proof Let h : A × [0, 1) → X be a collar of A in X. Choose an open set U in X so that A ⊂ U ⊂ cl U ⊂ h(A × [0, 1)). By Lemma 2.5.3, we have a map γ : A → (0, 1) such that    Aγ = (x, t) ∈ A × [0, 1)  t  γ (x) ⊂ h−1 (U ). Since Aγ is closed in A × [0, 1) and h(Aγ ) ⊂ cl U ⊂ h(A × [0, 1)), it follows that h(Aγ ) is closed in cl U , hence it is closed in X. Then, a closed collar k : A × I → X of A can be defined by k(x, t) = h(x, γ (x)t). When an open cover U of X is given above, γ can be taken by Lemma 2.5.3 so as to satisfy    {x} × [0, γ (x)]  x ∈ A ≺ h−1 (U). Then, we have {k({x} × I) | x ∈ A} ≺ U. Thus, the additional statement is also true. " ! Remark 2.7 A subset B of a space X is said to be bi-collared in X if there is an open embedding k : B × (−1, 1) → X such that k(x, 0) = 0 for every x ∈ B, where k is called a bi-collar of B in X. Then, B is collared in X \ k(B × (−1, 0)) and in X \ k(B × (0, 1)). Every collared closed set A in a paracompact space X has a closed collar k : A × I → X such that bd k(A × I) = k(A × {1}) is bi-collared in X. Indeed, using a closed collar k  : A × I → X obtained by Proposition 2.5.5, such a closed collar k : A × I → X can be defined by k(x, t) = k  (x, t/2).

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2 Fundamental Results on Infinite-Dimensional Manifolds

When X is metrizable, we can show the following: Lemma 2.5.6 Let A be a collared set in a metrizable space X and U be an open set in X such that A ⊂ U . Then, there is a closed map h : cl A×I → X such that h|A×I is an embedding into U , h|A × [0, 1) is a collar of A in X, and h({x} × I) = {x} for each x ∈ cl A \ A. Proof Since A is locally closed in X, we can assume that A = U ∩cl A by replacing U with a small open set. We have a collar k : A × [0, 1) → U by Lemma 2.5.4. Then, note that k(A × [0, 1)) ∩ (cl A \ A) = ∅. Take d ∈ Metr(X) and let    V = k(x, t)  (x, t) ∈ A × [0, 1), d(k(x, t), x) < 2−1 d(x, X \ k(A × [0, 1))) . Then, A ⊂ V ⊂ U and V is open in X because k(A × [0, 1)) is open in X. By Lemma 2.5.2, there is a map γ : A → (0, 1) (as illustrated in the left-hand side of Fig. 2.10) such that 

  (x, t) ∈ A × [0, 1)  t  γ (x) ⊂ k −1 (V ).

For each (x, t) ∈ A × I and y ∈ X \ k(A × [0, 1)), since k(x, tγ (x)) ∈ V , it follows that d(k(x, tγ (x)), x) < 2−1 d(x, y), hence d(x, y)  d(k(x, tγ (x)), x) + d(k(x, tγ (x)), y) < 2−1 d(x, y) + d(k(x, tγ (x)), y) and d(k(x, tγ (x)), y)  d(k(x, tγ (x)), x) + d(x, y) < 2−1 d(x, y) + d(x, y). Thus, we have the following: 2−1 d(x, y) < d(k(x, tγ (x)), y) < 2d(x, y).

1

k (A × [0, 1))

A × [0, 1) k −1 (V )

V k

0

X A

A h(A × I )

Fig. 2.10 Refining a collar k : A × I → X

2.5 The Collaring and Unknotting Theorems

137

By this inequality, the desired map h : cl A × I → X can be defined as follows (see Fig. 2.10): h(x, t) =

 k(x, tγ (x)) x

if x ∈ A, if x ∈ cl A \ A.

Indeed, the continuity of h at (x, t) ∈ (cl A \ A) × I can be seen as follows: Take (xn , tn ) ∈ A × I, n ∈ N, so that (xn , tn ) → (x, t) as n → ∞. Since x ∈ cl A \ A, it follows that d(h(xn , tn ), h(x, t)) = d(k(xn , tn γ (xn )), x) < 2d(xn , x) → 0. To show that h is closed, let (xn , tn ) ∈ cl A×I, n ∈ N, and assume h(xn , tn ) → y for some y ∈ X. Then, it suffices to show that (xn , tn ) has a convergent subsequence. Since I is compact, we may assume that tn → t for some t ∈ I. When y ∈ k(A × [0, 1)), since k is an open embedding, it follows that (xn , tn ) is convergent in A × [0, 1). When y ∈ k(A × [0, 1)), if (xn , tn ) ∈ A × [0, 1) for infinitely many n ∈ N, then (xn )n∈N has a subsequence that converges to y. Otherwise, we may assume that (xn , tn ) ∈ A × [0, 1) for every n ∈ N. Then, it follows that 2−1 d(xn , y) < d(k(xn , tn γ (xn )), y) = d(h(xn , tn ), y) → 0, hence xn → y. It is easy to see that h satisfies the other conditions.

" !

A subset A in a space X is said to be locally collared in X if each x ∈ A has a neighborhood in A that is collared in X. Obviously, a collared set is locally collared, but the converse is also true, that is, the following theorem holds: Theorem 2.5.7 (M. BROWN) Every locally collared set A in a metrizable space X is collared in X. Proof First, observe that A is locally closed in X, that is, closed in some open set in X. Since a collared set in an open set in X is collared in X, we may assume that A is closed in X. To apply Michael’s Theorem 1.3.20, it suffices to show the following: (G-1) (G-2) (G-3)

If U is open in A and collared in X, then each open set V in U is also collared in X; If U and V are open in A and collared in X, then U ∪ V is collared in X; If {Uλ | λ ∈ } is a discrete collection of open sets in A such that every Uλ  is collared in X, then λ∈ Uλ is also collared in X.

Note that (G-1) easily follows from Lemma 2.5.4. For (G-3), we have a discrete collection {Wλ | λ ∈ } of open sets in X such that Uλ ⊂ Wλ (cf. the definition of collectionwise normality and its remark before Proposition 1.3.12(1)). Then, (G-3) can be obtained by Lemma 2.5.4.

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2 Fundamental Results on Infinite-Dimensional Manifolds

To show (G-2), let U and V be open sets in A that are collared in X. Take an  in X so that U = A ∩ U  and apply Lemma 2.5.6 to obtain a closed map open set U , h|U × [0, 1) is a collar h : cl U × I → X such that h|U × I is an embedding into U of U in X, and h({x} × I) = {x} for x ∈ cl U \ U . Observe A ∩ h(cl U × I) = (A ∩ h(U × I)) ∪ (A ∩ cl U ) ) ∪ cl U = cl U, ⊂ (A ∩ U hence A ∩ h(cl U × I) = cl U . Then, we have the homeomorphism ϕ : X → X \ h(U × [0, 1/2)) ⊂ X defined by ϕ|X \ h(U × [0, 1)) = id and ϕ(h(x, t)) = h(x, t/2 + 1/2) for each (x, t) ∈ cl U × I, where ϕ(x) = h(x, 1/2) for each x ∈ cl U . We have a collar k : V × [0, 1) → X of V in X, where it should be noted that ϕkϕ −1 |ϕ(V ) × id[0,1) is a collar of ϕ(V ) in ϕ(X). Let α : U ∪ V → I be a map with α −1 (0) = V \ U and α −1 (1) = U \ V . Then, we can define a map k ∗ : (U ∪ V ) × [0, 1) → X as follows (see Fig. 2.11):  ⎧  ⎨h x, t if x ∈ U and t  α(x), 2α(x) k ∗ (x, t) = ⎩ ϕk(x, t − α(x)) if x ∈ V and t  α(x). Then, k ∗ is injective and k ∗ (x, 0) = x for every x ∈ U ∪ V . To see that k ∗ is a collar of U ∪ V in X, it remains to be verified that k ∗ is an open map. Evidently, k ∗ is continuous at every point of the following open sets: (U ∪ V ) × (0, 1), U × [0, 1), (V \ cl U ) × [0, 1). 1− 1 [0, 1)

1 1 2

U

V

0 h

k

U

k (V × [0, 1))

h(U × I)

U Fig. 2.11 A collar of U ∪ V

V

V

A

k

0

2.5 The Collaring and Unknotting Theorems

139

We shall show the continuity of k ∗ at (x, 0), where x ∈ V ∩ (cl U \ U ). Let W be a neighborhood of x = k ∗ (x, 0) in X. Since h({x} × I) = {x}, we have some neighborhood N of x in V such that h((N ∩cl U )×I) ⊂ W . Observe that ϕk(x, 0) = ϕ(x) = h(x, 1/2) = x. Then, we have a neighborhood N  of x in V with 0 < δ < 1 such that ϕk(N  × [0, δ)) ⊂ W . It follows that k ∗ ((N ∩ N  ) × [0, δ)) ⊂ h((N ∩ cl U ) × [0, 1/2]) ∪ ϕk(N  × [0, δ)) ⊂ W. Finally, we shall prove that k ∗ is open. Let W be a neighborhood of (x, t) ∈ (U ∪ V ) × [0, 1). When t < α(x), it follows that x ∈ U . Then, W − = {(x  , t  /2α(x  )) | (x  , t  ) ∈ W, t  < α(x  )} is a neighborhood of (x, t/2α(x)) in U × [0, 1/2) and h(W − ) ⊂ k ∗ (W ). Hence, k ∗ (W ) is a neighborhood of h(x, t/2α(x)) = k ∗ (x, t) in X. When t > α(x), we have x ∈ V . Then, W + = {(x  , t  − α(x  )) | (x  , t  ) ∈ W, t  > α(x  )} is a neighborhood of (x, t − α(x)) in V × (0, 1) and ϕk(W + ) ⊂ k ∗ (W ). Hence, k ∗ (W ) is a neighborhood of ϕk(x, t − α(x)) = k ∗ (x, t) in X. If t = α(x) > 0, then x ∈ U ∩ V . Let W− = {(x  , t  /2α(x  )) | (x  , t  ) ∈ W, t   α(x  )} and W+ = {(x  , t  − α(x  )) | (x  , t  ) ∈ W, t   α(x  )}. Then, h(W− ) is a neighborhood of k ∗ (x, t) in h(cl U × [0, 1/2]) and ϕk(W+ ) is a neighborhood of k ∗ (x, t) in ϕ(X) = X \ h(U × [0, 1/2)). Hence, k ∗ (W ) = h(W− ) ∪ ϕk(W+ ) is a neighborhood of k ∗ (x, t) in X. When t = α(x) = 0 and x ∈ V \ cl U , we have (x, 0) ∈ (ϕk)−1 h(cl U × I) because h(cl U × I) ∩ A = cl U . Choose a neighborhood N of x in V \ cl U and 0 < δ < 1 so that   W0 = N × [0, δ) ⊂ (V × [0, 1)) \ (ϕk)−1 h(cl U × I) ∩ W. Then, k(W0 ) = ϕk(W0 ) is a neighborhood of k(x, 0) = k ∗ (x, t) in the open set k((V \ cl U ) × [0, 1)) in X, hence it is a neighborhood of k ∗ (x, t) in X. Since ϕk(W0 ) ⊂ k ∗ (W ), k ∗ (W ) is also a neighborhood of k ∗ (x, t) in X. When t = α(x) = 0 and x ∈ V ∩ bd U , choose an open neighborhood N of x in V and 0 < δ < 1 so that N × [0, δ) ⊂ W and α(N) ⊂ [0, δ). Then, h((N ∩ cl U ) × [0, 1/2]) is a neighborhood of x in h(cl U × [0, 1/2]). Observe that M = {(x  , t  − α(x  )) | x  ∈ N, α(x  )  t  < δ}

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2 Fundamental Results on Infinite-Dimensional Manifolds

is a neighborhood of (x, 0) in V × [0, 1), hence ϕk(M) is a neighborhood of x in ϕ(X) = X \ h(U × [0, 1/2)). Since k ∗ (W ) ⊃ h((N ∩ cl U ) × [0, 1/2]) ∪ ϕk(M), it follows that k ∗ (W ) is a neighborhood of x. Since obtained:

E2

" !

≈ E, the following version of Klee’s Trick 1.13.7 can be easily

Theorem 2.5.8 (KLEE’S TRICK) Every homeomorphism h : A → B between E-deficient closed sets in E can be extended to a homeomorphism h˜ : E → E. ! " Using this form of Klee’s Trick, we show the following: Theorem 2.5.9 (COLLARING) Let M and N be E-manifolds such that N is Edeficient closed in M. Then, N is collared in M. Moreover, for each open cover U of M, there is a closed collar k : N × I → M such that    k({x} × I)  x ∈ N ≺ U. Proof Since a locally collared set is collared (Theorem 2.5.7), it suffices to show that N is locally collared in M. As is easily observed, each x ∈ N has an open neighborhood U in M such that U ∩ N and U are homeomorphic to open sets in E. Let f : U ∩ N → E × {0} (⊂ E × [0, 1)) and g : U → E × [0, 1) be open embeddings. Choose an open neighborhood V of x in U so that cl V ⊂ U and f (N ∩ cl V ), g(cl V ) are closed in E × {0} and E × [0, 1), respectively. Since N ∩ cl V is E-deficient in U by Corollary 2.3.4(2), g(N ∩ cl V ) is an E-deficient closed set in E × [0, 1) by Proposition 2.3.8. On the other hand, f (N ∩ cl V ) is also an E-deficient closed set in E × [0, 1) by Theorem 2.4.9. Using Klee’s Trick above, f (g|N ∩cl V )−1 can be extended to a homeomorphism h : E ×[0, 1) → E ×[0, 1). Then, hg|V : V → E × [0, 1) is an open embedding such that hg(N ∩ V ) = f (N ∩ V ) is open in E × {0}. Since E × {0} is collared in E × [0, 1), hg(N ∩ V ) is also collared in hg(V ) by Lemma 2.5.4. Hence, N ∩ V is collared in V (Fig. 2.12). E × [0, 1)

1 h

g(U ) hg (U ) f g −1 g(N

0

E f (N

Fig. 2.12 A collar of U ∩ V

cl V )

cl V )

2.5 The Collaring and Unknotting Theorems

141

Due to Proposition 2.5.5, the additional statement follows from the fact that N is a collared closed set in M. " ! Now, we have the following: Theorem 2.5.10 (OPEN EMBEDDING) Every E-manifold M can be embedded in E as an open set.22 Moreover, if A is an E-deficient closed set in M, then there is an open embedding g : M → E such that g(A) is E-deficient and closed in E. Proof Note that M ∈ FE (Theorem 2.1.11). Then, by the Strong Universality Theorem 2.3.15, we have an E-deficient closed embedding h : M → E, hence h(M) has a collar k : h(M) × [0, 1) → E by the Collaring Theorem 2.5.9. On the other hand, by Theorem 2.4.9, we have a homeomorphism f : M → M × [0, 1) such that f (A) ⊂ M × {0}. Then, g = k(h × id)f : M → E is the desired open embedding because g(A) is closed in the closed subspace k(h(M) × {0}) = h(M) of E. " ! In the above proof, h(M) has a closed collar k : h(M) × I → E by Proposition 2.7. Furthermore, k can be taken so that bd k(h(M)×I) = k(h(M)×{1}) is bi-collared in E by Remark 2.7. Since M × I ≈ M (Corollary 2.4.12), we have a homeomorphism f : M → M × I Then, g = k(h × id)f : M → E is a closed embedding and bd g(M) = bd k(h(M) × I) = k(h(M) × {1}) is bi-collared in E. Thus, we have the following: Theorem 2.5.11 Every E-manifold M can be embedded in E as a closed set having " ! an E-manifold bi-collared boundary.23 Remark 2.8 Theorems 2.5.10 and 2.5.11 are valid for any E-manifold M such that w(M) = w(E); equivalently, M has at most w(E)-many components. Indeed, by Lemma 2.1.10, E has a discrete open collection U such that card U = w(E) and every U ∈ U is homeomorphic to E itself. Theorem 2.5.8 can be extended to E-manifolds as follows: Theorem 2.5.12 (UNKNOTTING) Let h : A → B be a homeomorphism between E-deficient closed sets in an E-manifold M and U be an open cover of M. If there is a homotopy f : A × I → M with f0 = id and f1 = h, then h can be extended to a homeomorphism h˜ : M → M that is ambiently invertibly U∗ -isotopic to id, where U∗ = {st(f ({x} × I), U) | x ∈ A}. Proof Take V ∈ cov(M) so that st3 V ≺ U and let V[B] = {V ∈ V | V ∩B = ∅}. First, we will construct an ambient invertible V[B]-isotopy ψ : M × I → M such that ψ0 = id and A ∩ ψ1 (B) = ∅. Let V be an open star-refinement of V. Since A ∪ B is E-deficient in M by Proposition 2.4.14, we apply Corollary 2.3.4(4) to obtain a homeomorphism k : M ×E → M such that k(x, 0) = x for each x ∈ A∪B

22 Recall 23 Recall

we assume that every E-manifold has the same density as E. our assumption that every E-manifold has the same density as E.

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2 Fundamental Results on Infinite-Dimensional Manifolds

and k is V -close to prM . Taking v ∈ E \ {0} and an Urysohn map α : M → I with α(k −1 (B)) = {1} and α(M \ k −1 (st(B, V ))) = {0}, we define an ambient invertible isotopy ψ  : M × E × I → M × E by ψ  (x, y, t) = (x, y + tα(x)v) for each (x, y, t) ∈ M × E × I. The isotopy ψ can be defined by ψt = kψt k −1 for each t ∈ I. To verify that ψ is a V[B]-isotopy, for each x ∈ M, let k −1 (x) = (x  , y). If  x ∈ st(B, V ), then α(x  ) = 0, hence ψt (x) = kψ  (x  , y, t) = k(x  , y) = x for every t ∈ I, that is, ψ({x} × I) = {x}. When x  ∈ st(B, V ), we have V  ∈ V such that x  ∈ V  and V  ∩ B = ∅. Since st V ≺ V, there is some V ∈ V such that st(V  , V ) ⊂ V , where V ∩ B = ∅, that is, V ∈ V[B]. Because k is V -close to prM , it follows that ψ({x} × I) ⊂ k({x  } × E) ⊂ st(x  , V ) ⊂ st(V  , V ) ⊂ V . It should be remarked that ψt−1 = kψt −1 k −1 for each t ∈ I, where ψt

−1

(x, y) = (x, y − tα(x)v) for each (x, y, t) ∈ M × E × I.

Then, by the same argument as above, the isotopy defined by ψt−1 , t ∈ I, is a V[B]isotopy.24 Let f  : A × I → M be the homotopy defined by  ft

=

f2t

if 0  t  1/2,

ψ2t −1 h if 1/2  t  1.

Since f0 = id and f1 = ψ1 h, f  |A × {0, 1} is an E-deficient closed embedding into M. Moreover, f  ({x} × I) ⊂ st(f ({x} × I), V) for each x ∈ A. By the Strong Universality Theorem 2.3.15, there is an E-deficient closed embedding f  : A × I → M such that f  |A × {0} = f  |A × {0} = id, f  |A × {1} = f  |A × {1} = ψ1 h and f  is V-close to f  . Then, f  ({x} × I) ⊂ st(f  ({x} × I), V) for each x ∈ A.

24 Cf.

Proposition 2.1.15.

2.5 The Collaring and Unknotting Theorems

143 W (x )

R

f (A × I)

j(A) × I

1 g A f ({x } × I)

M

j(A)

0

E

g(M ) {j(x )} × I

Fig. 2.13 The open embedding g and the open collection W

On the other hand, we have an E-deficient closed embedding j : A → E. Then, j (A) × I is an E-deficient closed set in E × R. Using the Open Embedding Theorem 2.5.10 and Klee’s Trick 2.5.8, we can construct an open embedding g : M → E × R such that g(f  (x, t)) = (j (x), t) for (x, t) ∈ A × I (cf. Fig. 2.13). For each x ∈ A, define W (x) = g(st(f  ({x} × I), V)) ⊂ g(M) and let W = {W (x) | x ∈ A}. Note that {j (x)} × I = gf  ({x} × I) ⊂ W (x) for each x ∈ A. By Lemma 2.5.1, we have maps β, γ : E → R such that: (i) (ii) (iii) (iv) (v)

β(x)  0 for x ∈ E, β(x) = 1 and γ (x) > 0 if x ∈j (A), β(x) = γ (x) = 0 if (x, 0) ∈ x∈A W (x), γ (x) > 0 if β(x) > 0,  {{x} × [−γ (x), β(x) + γ (x)] | (x, 0) ∈ x∈A W (x)} ≺ W.

On the right-hand side of Fig. 2.13, we have an ambient invertible W-isotopy ϕ  : E × R × I → E × R sliding points vertically so as to move {0} × E to the graph of β. The following is the precise definition of ϕ  :  ⎧ ((1 − t)β(x) + γ (x))s ⎪ ⎪ x, tβ(x) + if 0  s  β(x) + γ (x), ⎪ ⎪ β(x) + γ (x) ⎪ ⎨  tβ(x)(s + γ (x)) ϕt (x, s) = ⎪ x, s + if − γ (x)  s  0, ⎪ ⎪ γ (x) ⎪ ⎪ ⎩ (x, s) otherwise. Then, ϕ0 = id and ϕ1 g|A = gψ1 h. Indeed, for every x ∈ A, ϕ1 g(x) = ϕ1 gf  (x, 0) = ϕ1 gf  (x, 0) = ϕ1 (j (x), 0) = (j (x), 1) = gf  (x, 1) = gf  (x, 1) = gψ1 h(x).

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2 Fundamental Results on Infinite-Dimensional Manifolds

Since ϕt (g(M)) = g(M) for each t ∈ I, we have an ambient invertible g −1 (W)isotopy ϕ : M × I → M defined by ϕt = g −1 ϕt g. Then, ϕ0 = id and ϕ1 |A = g −1 ϕ1 g|A = ψ1 h. Thus, h is extended to a homeomorphism h˜ = ψ1−1 ϕ1 : M → M. Moreover, we can define an ambient invertible isotopy η : M × I → M by ηt =

 ϕ2t −1 ψ2t −1 ϕ1

if 0  t  1/2, if 1/2  t  1.

˜ Then, η0 = id and η1 = ψ1−1 ϕ1 = h. ∗ To verify that η is a U -isotopy, let x ∈ M. When ϕ({x} × I) is not contained in g −1 (W (y)) for any y ∈ A, since ϕ is a g −1 (W)-isotopy, it follows that ϕ({x}×I) = {x}. Then, x ∈



g −1 (W (y)) = st(f  (A × I), V),

y∈A  (A), V) = st(B, V). In this case, for each t ∈ I, ψ (x) = x, that hence x ∈ st(f1/2 t

is, ψt−1 (x) = x. Thus, we have η({x} × I) = {x}. When ϕ({x} × I) is contained in g −1 (W (y)) for some y ∈ A, since the isotopy defined by ψt−1 , t ∈ I, is a V[B]-isotopy, it follows that η({x} × I) ⊂ st(g −1 (W (y)), V) = st(st(f  ({y} × I), V), V) ⊂ st(f ({y} × I), st V) ⊂ st(f ({y} × I), U) ∈ U∗ . Therefore, η is a U∗ -isotopy.

" !

The Open Embedding Theorem 2.5.10 can be generalized as follows: Theorem 2.5.13 (OPEN EMBEDDING APPROXIMATION) Let M and N be Emanifolds, A be an E-deficient closed set in M and f : M → N be a map such that f |A is an E-deficient closed embedding into N. Then, for each open cover U of N, there exists an open embedding g : M → N such that g|A = f |A and g is U-homotopic to f . Proof Let V ∈ cov(N) such that st V ≺ U. Recall M ∈ FE (Theorem 2.1.11). By the Strong Universality Theorem 2.3.15, we have an E-deficient closed embedding h : M → N such that h|A = f |A and h is V-homotopic to f . By Corollary 2.3.5, we have a homeomorphism ϕ : M × [0, 1) → M such that ϕ(x, 0) = x for each x ∈ A and ϕ is h−1 (V)-homotopic to prM . Since h(M) is collared in N by the Collaring Theorem 2.5.9, we have an open embedding k : M × [0, 1) → N such that k(x, 0) = h(x) for each x ∈ M. By Lemma 2.5.2, we can require k to satisfy the condition that    k({x} × [0, 1))  x ∈ M ≺ V,

2.6 Classification of E-Manifolds

145

which implies that k is V-homotopic to h◦prM . Since h◦prM is V-homotopic to hϕ and h is V-homotopic to f , k is st V-homotopic, so U-homotopic to f ϕ. Thus, we have an open embedding g = k◦ϕ −1 : M → N which is U-homotopic to f . Moreover, g(x) = kϕ −1 (x) = k(x, 0) = h(x) = f (x) for each x ∈ A, that is, g|A = f |A.

" !

2.6 Classification of E-Manifolds In this section, we establish the following: Theorem 2.6.1 (CLASSIFICATION) Two E-manifolds are homeomorphic if they have the same homotopy type. As a corollary, we have the following: Corollary 2.6.2 Every contractible E-manifold is homeomorphic to E.

" !

Recall that a map f : X → Y is a U-homotopy equivalence for U ∈ cov(Y ) if there is a map g : Y → X (called a U-homotopy inverse of f ) such that fg is U-homotopic to idY and gf is f −1 (U)-homotopic to idX . A fine homotopy equivalence f : X → Y is a U-homotopy equivalence for every U ∈ cov(Y ). The Classification Theorem 2.6.1 is a direct consequence of the following Homeomorphism Approximation Theorem: Theorem 2.6.3 (HOMEOMORPHISM APPROXIMATION) Let M and N be Emanifolds and U be an open cover of N. Then, every U-homotopy equivalence f : M → N is st3 U-close (st6 U-homotopic) to a homeomorphism. In particular, every fine homotopy equivalence of M to N is a near-homeomorphism. We adjust the mapping cylinder for use in the proof. For each map f : X → Y and a < b < c, the fat mapping cylinder Mab,c (f ) of f is defined as follows: Mab,c (f ) = X × [a, b) ∪ Y × [b, c] with the topology generated by open sets in the product spaces X × [a, b) and Y × [b, c], and the following sets: (f −1 (U ) × (b − ε, b)) ∪ (U × [b, b + ε)), where U is open in Y and 0 < ε < min{b − a, c − b}. That is the copy of the following space: M(f ) ∪Y =Y ×{0} Y × I,

146

2 Fundamental Results on Infinite-Dimensional Manifolds N

M

N M (f ) f (M ) 0

f (M ) 1

2 k ( f (M ) × I)

1

2 k ( f (M ) × [1/ 2, 1]) (M f ) = k ( f (M ) × [0, 1/ 2])

Fig. 2.14 The homeomorphism ϕ

where Y ⊂ M(f ) is identified with Y × {0} ⊂ Y × I. Lemma 2.6.4 Let M and N be E-manifolds and f : M → N a closed embedding. Then, for each V ∈ cov(N × I), there exists a homeomorphism ϕ : M01,2 (f ) → N × [1, 2] such that ϕ(x, 0) = (f (x), 1) for each x ∈ M, ϕ|N × {2} = id and ϕ V r rel. M × {0} ∪ N × {2}, where r : M01,2 (f ) → N × [1, 2] is the retraction defined by r(x, t) = (f (x), 1) for each (x, t) ∈ M × [0, 1) and r|N × [1, 2] = id. Proof It follows from Theorem 2.4.9 that f (M) × {1} is an E-manifold that is Edeficient and closed in N × [1, 2]. By the Collaring Theorem 2.5.9, f (M) has a closed collar k : f (M) × I → N × [1, 2] such that {k({f (x)} × I) | x ∈ M} ≺ V. We can define a V-homotopy h : M01,2 (f ) × I → N × [1, 2] as follows:

ht (x, s) =

⎧ ⎪ ⎪ ⎨k(f (x), st/2) k(f (x  ), (s 

⎪ ⎪ ⎩(x, s)

+ t)/2)

if (x, s) ∈ M × [0, 1), if (x, s) = k(f (x  ), s  ) ∈ k(f (M) × [0, t]), if (x, s) ∈ (N × [1, 2]) \ k(f (M) × [0, t]).

Refer to Fig. 2.14. Then, h0 = r, ht |M × {0} = f × id{0} and ht |N × {2} = id for each t ∈ I, and ht is a homeomorphism for t > 0. Therefore, ϕ = h1 is the desired homeomorphism. " ! Lemma 2.6.5 Let M and N be E-manifolds, and let f : M → N and g : N → M be closed embeddings. Suppose that there exists a homotopy h : M × I → M with h0 = idM and h1 = gf . Then, for each U ∈ cov(M), there exists a homeomorphism ψ : M01,2 (f ) ∪ M23,4 (g) → M × [0, 4]

2.6 Classification of E-Manifolds

147

such that ψ|M × {0, 4} = id and prM ψ U∗ θ , where M01,2 (f ) ∪ M23,4 (g) is the space obtained by pasting M01,2 (f ) and M23,4 (g) along N × {2}, and θ : M01,2 (f ) ∪ M23,4 (g) → M is the map defined as follows: θ |M × [0, 1) = gf prM , θ |N × [1, 3) = gprN , and θ |M × [3, 4] = prM , and U∗ = {st(h({x} × I), U) | x ∈ M}. Proof Let η : M ×[3, 4] → M ×[0, 4] be the homeomorphism defined by η(x, t) = (x, 4t − 12). Take V ∈ cov(M × [0, 4]) and W ∈ cov(M × [3, 4]) so that st V ≺ −1 pr−1 M (U) and st W ≺ η (V). Observe that (N × [1, 2]) ∪ M23,4 (g) = M13,4 (g). Applying Lemma 2.6.4 to g, we can obtain a homeomorphism ϕ  : M13,4 (g) → M × [3, 4] such that ϕ  (x, 1) = (g(x), 3) for each x ∈ N, ϕ  |M × {4} = id, and ϕ  W r  rel. N × {1} ∪ M × {4}, where r  : M13,4 (g) → M × [3, 4] is the retraction defined by r  (x, t) = (g(x), 3) for each (x, t) ∈ N × [1, 3) and r  |M × [3, 4] = id. Extending the homeomorphism obtained in Lemma 2.6.4, we can obtain a homeomorphism ϕ  : M01,2 (f ) ∪ M23,4 (g) → (N × [1, 2]) ∪ M23,4 (g) = M13,4 (g) such that ϕ  (x, 0) = (f (x), 1) for each x ∈ M, ϕ  |M23,4 (g) = id, and ϕ  ϕ  −1 (W) r  rel. M × {0} ∪ M23,4 (g), where r  : M01,2 (f ) ∪ M23,4 (g) → M13,4 (g) is the retraction defined by r  (x, t) = (f (x), 1) for (x, t) ∈ M × [0, 1) and r  |M13,4(g) = id. Now, we have the following homeomorphism: ϕ = ϕ  ϕ  : M01,2 (f ) ∪ M23,4 (g) → M × [3, 4]. Refer to Fig. 2.15. Then, ϕ(x, 0) = (gf (x), 3) for each x ∈ M, ϕ|M × {4} = id, and ϕ st W r = r  r  rel. M × {0} ∪ M × {4},

148

2 Fundamental Results on Infinite-Dimensional Manifolds N

M

N

M (f ) f (M )

M (g) 1

0

M

2

3

N

M

4

g(N )

N

M

M (g) g(N )

f (M ) 1

2

3

4

the image of M ( f )

3

g f (M )

the image of N × [1, 2]

4 M (g)

Fig. 2.15 The homeomorphisms ϕ  , ϕ  , and ϕ

where r(x, t) = (gf (x), 3) for (x, t) ∈ M × [0, 1), r(x, t) = (g(x), 3) for each (x, t) ∈ N × [1, 3), and r|M × [3, 4] = id. It should be remarked that ηϕ|M × {0} = gf × id{0} , ηϕ|M × {4} = id, η(st W) ≺ V and prM ηr = prM r = θ. Recall that h : M × I → M is a homotopy with h0 = idM and h1 = gf . By Theorem 2.4.9, gf (M) × {0}, M × {0}, and M × {4} are E-deficient in M × [0, 4]. So, we can apply the Unknotting Theorem 2.5.12 to obtain a homeomorphism ζ : M × [0, 4] → M × [0, 4] such that ζ |M × {0} = gf × id{0} , ζ |M × {4} = id, and ζ is V∗ -isotopic to id, where    V∗ = st(h({x} × I) × {0}, V)  x ∈ M . Then, the following is the desired homeomorphism: ψ = ζ −1 ηϕ : M01,2 (f ) ∪ M23,4(g) → M × [0, 4]. Indeed, ψ|M × {4} = ζ −1 ηϕ|M × {4} = id and ψ|M × {0} = ζ −1 ηϕ|M × {0} = ζ −1 (gf × id{0} ) = ζ −1 ζ |M × {0} = id.

2.6 Classification of E-Manifolds

149

Since ψ = ζ −1 ηϕ V∗ ηϕ V ηr and    prM (st(V∗ , V)) = prM (st2 (h({x} × I) × {0}, V))  x ∈ M    ≺ prM (st(h({x} × I) × {0}, st V))  x ∈ M    ≺ st(h({x} × I), U))  x ∈ M = U∗ , it follows that prM ψ U∗ prM ηr = θ .

" !

We now prove Theorem 2.6.3: Proof (Theorem 2.6.3) Since f is a U-homotopy equivalence, f has a Uhomotopy inverse g : N → M, that is, fg U idN and gf f −1 (U) idM . Take V, W, W ∈ cov(N) so that st V ≺ U, st2 W ≺ V, W ≺ W, and g(W ) ≺ f −1 (W). By the Strong Universality Theorem 2.3.15, there exist closed embeddings f0 : M → N and g0 : N → M such that f0 W f and g0 f −1 (W) g. Then, it follows that f0 g0 W fg0 W fg U idN and g0 f0 f −1 (W) gf0 g(W ) gf f −1 (U) idM . Thus, we have f0 g0 st(U,st W) idN and g0 f0 f −1 (st(U,st W)) idM . For simplicity, we write as follows: 2n−3/2,2n−1

X2n−1 = M2n−2

2n−1/2,2n

(f0 ), X2n = M2n−1

(g0 ), n ∈ N.

Then, each X2n−1 and X2n are copies of M(f0 )  ∪N=N×{0} (N × I) and M(g0 ) ∪M=M×{0} (M × I), respectively. We define X = n∈N Xn , that is illustrated as the middle space in Fig. 2.16. Observe that st(st(U, st W), W) ≺ st(U, st2 W) ≺ st(U, V) and st(f −1 (st(U, st W)), f −1 (W)) ≺ f −1 (st(U, st V)). For each n ∈ N, applying Lemma 2.6.5, we obtain homeomorphisms ϕn : X2n−1 ∪ X2n → M × [2n − 2, 2n] and ψn : X2n ∪ X2n+1 → N × [2n − 1, 2n + 1]

150

2 Fundamental Results on Infinite-Dimensional Manifolds

M × [0, ) 0

2

N

M M ( f0 )

X 0

1

N

M M ( f0 )

M (g 0 )

1/ 2 X1

4

3/ 2 X2

2

M

M (g 0 )

5/ 2 X3

3

7/ 2 X4

N × [0, ) 0

1

3

Fig. 2.16 The homeomorphisms ϕ and ψ

satisfying the following properties: ϕn |M × {2n − 2, 2n} = id, ψn |N × {2n − 1, 2n + 1} = id, prM ϕn f −1 (st(U,V)) θn and prN ψn st(U,V) ϑn , where θn and ϑn are defined as follows: ⎧ ⎪ ⎪ ⎨θn |M × [2n − 2, 2n − 3/2) = g0 f0 prM , θn |N × [2n − 3/2, 2n − 1/2) = g0 prN , ⎪ ⎪ ⎩θ |M × [2n − 1/2, 2n] = pr , n

M

⎧ ⎪ ⎪ ⎨ϑn |N × [2n − 1, 2n − 1/2) = f0 g0 prN ,

ϑn |M × [2n − 1/2, 2n + 1/2) = f0 prM , ⎪ ⎪ ⎩ϑ |N × [2n + 1/2, 2n + 1] = pr . n

N

4

2.6 Classification of E-Manifolds

151

Additionally, applying Lemma 2.6.4, we have a homeomorphism ψ0 : X1 → N × [0, 1] such that ψ0 |M × {0} = f0 × id{0} , ψ0 |N × {1} = id, and prN ψ0 V ϑ0 , where ϑ0 is defined by ϑ0 |M × [0, 1/2) = f0 prM and ϑ0 |N × [1/2, 1] = prN . Then, we can define homeomorphisms ϕ : X → M × [0, ∞) and ψ : X → N × [0, ∞) by ϕ|X2n−1 ∪ X2n = ϕn , ψ|X2n ∪ X2n+1 = ψn for each n ∈ N, and ψ|X1 = ψ0 (see Fig. 2.16). Dividing the domain into the following three parts: 2n−1/2,2n

(1) X2n = M2n−1

(g0 ), 2n+1/2,2n+2

n ∈ N,

(2) X2n+1 = M2n (f0 ), n ∈ N, and 1/2,1 (3) X1 = M0 (f0 ), we will prove that prN ψ is st(st2 U, V)-close to f prM ϕ, which means that prN ψϕ −1 k −1 is st(st2 U, V)-close to f prM k −1 (= f prM ϕϕ −1 k −1 ). (1): The restriction f prM ϕ|X2n = f prM ϕn |X2n is st(U, V)-close to f θn |X2n and prN ψ|X2n = prN ψn |X2n is st(U, V)-close to ϑn |X2n . Since f is V-close to f0 , it follows that f θn |N × [2n − 1, 2n − 1/2) = fg0 prN and f θn |M × [2n − 1/2, 2n] = f prM are V-close to ϑn |N × [2n − 1, 2n − 1/2) = f0 g0 prN and ϑn |M ×[2n−1/2, 2n] = f0 prM , respectively. Thus, f θn |X2n is V-close to ϑn |X2n . Because st V ≺ U, we have st(V, st(U, V)) ≺ st(st U, V). Therefore, f prM ϕ|X2n is st(st U, V)-close to prN ψ|X2n . (2): The restriction f prM ϕ|X2n+1 = f prM ϕn+1 |X2n+1 is st(U, V)-close to f θn+1 |X2n+1 and prN ψ|X2n+1 = prN ψn |X2n+1 is st(U, V)-close to ϑn |X2n+1 . Since fg0 is st(U, V)-close to idN , it follows that f θn+1 |M × [2n, 2n + 1/2) = fg0 f0 prM and f θn+1 |N × [2n + 1/2, 2n + 1] = fg0 prN are st(U, V)-close to ϑn |M ×[2n, 2n+1/2) = f0 prM and ϑn |N ×[2n+1/2, 2n+1] = prN , respectively. Thus, f θn |X2n+1 is st(U, V)-close to ϑn |X2n+1 . Because st V ≺ U, we have st(st(U, V), st(U, V)) ≺ st(st(st U, U), V) = st(st2 U, V). Therefore, f prM ϕ|X2n+1 is st(st2 U, V)-close to prN ψ|X2n+1 . (3): The restriction f prM ϕ|X1 = f prM ϕ1 |X1 is st(U, V)-close to f θ1 |X1 and prN ψ|X1 = prN ψ0 |X1 is V-close to ϑ0 |X1 . Since fg0 is st(U, V)-close to idN , it follows that f θ1 |M × [0, 1/2) = fg0 f0 prM and f θ1 |N × [1/2, 1] = fg0 prN

152

2 Fundamental Results on Infinite-Dimensional Manifolds

are st(U, V)-close to ϑ0 |M × [0, 1/2) = f0 prM and ϑ0 |N × [1/2, 1] = prN , respectively. Because st V ≺ U, we have st(st(st(st V, U), V), V) ≺ st(st(U, U), st V) ≺ st(st U, U) = st2 U. Hence, f prM ϕ|X1 is st2 U-close to prN ψ|X2n+1 . Since [0, ∞) × E ≈ E, we can apply the Stability Theorem 2.3.7 to obtain homeomorphisms k : M × [0, ∞) → M and h : N × [0, ∞) → N such that k f −1 (V) prM and h V prN . Then, hψϕ −1 k −1 : M → N is the desired homeomorphism. Indeed, hψϕ −1 k −1 is V-close to prN ψϕ −1 k −1 , prN ψϕ −1 k −1 is st(st2 U, V)-close to f prM k −1 as we saw above, and f prM k −1 is V-close to f kk −1 = f . Since st V ≺ U, we have st(st(st2 U, V), V) ≺ st(st2 U, st V) ≺ st3 U. Hence, hψϕ −1 k −1 is st3 U-close to f . Furthermore, let r : M ×[0, ∞) → M ×{0} be the retraction defined by r(x, t) = (x, 0) for each (x, t) ∈ M × [0, ∞). Then, we have prN ψϕ −1 r = (prN ψϕ −1 |M × {0})r = (prN ψ|M × {0})r = (f0 prM |M × {0})r = f0 prM r = f0 prM ;       prN ψϕ −1 ({x} × [0, ∞))  x ∈ M ≺ st(f (x), st(st2 U, V))  x ∈ M ≺ st(st5 U, V). Now, it follows that hψϕ −1 k −1 V prN ψϕ −1 k −1 st(st5 U,V) prN ψϕ −1 rk −1 = f0 prM k −1 V f prM k −1 V f kk −1 = f. Observe that st2 (st(st5 U, V), V) = st(st5 U, st V) ≺ st6 U. Therefore, hψϕ −1 k −1 st6 U f .

" !

Remark 2.9 Later in Sect. 2.11, the Q-manifold version of Theorem 2.6.3 will be proved. It will be helpful to review the above proof. We have first applied the Strong Universality Theorem 2.3.15 to approximate a U-homotopy equivalence and its Uhomotopy inverse with closed embeddings, and then used Lemmas 2.6.4 and 2.6.5 to construct the homeomorphism ψϕ −1 : M × R+ → N × R+ so that prN ψϕ −1 is st(st2 U, V)-close to f prM , where V is an open star-refinement of U. Finally,

2.7 The Bing Shrinking Criterion

153

using the fact that the projections prM : M × R+ → M and prN : N × R+ → N are near-homeomorphisms, we have obtained Theorem 2.6.3. To prove Lemma 2.6.4, we only use the fact that a closed submanifold is collared if it is contained in some collared set. This fact follows from Theorem 2.4.9 and the Collaring Theorem 2.5.9. Those theorems will be proved for Q-manifolds in Sects. 2.10 and 2.11. For Lemma 2.6.5, we use also the Unknotting Theorem 2.5.12, which will be proved for Q-manifolds in Sect. 2.11. On the other hand, the Qmanifold version of the Strong Universality Theorem 2.3.15 has been obtained as the Closed Embedding Approximation Theorem 2.3.17. Because 1 () ≈ 2 () (Theorem 1.2.3), Corollary 2.2.13 is restated that 2 ()N ≈ 2 (). Every 2 ()-manifold M is an ANR with w(M) = card  = τ , which has the homotopy type of some locally finite-dimensional simplicial complex K with card K (0)  τ by Theorem 1.13.22. Due to Corollary 2.2.15, |K|m × 2 () is an 2 ()-manifold. By virtue of the Classification Theorem 2.6.1, we have M ≈ |K|m × 2 (). When M is an 2 -manifold, the above K is countable locally finite (Theorem 1.13.22), where |K|m = |K|. Then, M ≈ |K| × 2 . Thus, we have the following Triangulation Theorem for 2 ()-manifolds: Theorem 2.6.6 (TRIANGULATION) Each 2 ()-manifold is homeomorphic to |K|m × 2 () for some locally finite-dimensional simplicial complex K with card K (0)  card  = τ . In particular, each 2 -manifold is homeomorphic to |K| × 2 for some countable locally finite simplicial complex K. " ! Similarly, the Triangulation Theorem for 2f -manifolds can be obtained as a combination of the Classification Theorem 2.6.1 and Corollary 2.2.17: Theorem 2.6.7 (TRIANGULATION) Each 2f -manifold is homeomorphic to |K|× 2f for some countable locally finite simplicial complex K with card K (0)  card  = τ . " ! Remark 2.10 It is difficult to prove the Triangulation Theorem for Q-manifolds, which will be done in Chap. 4.

2.7 The Bing Shrinking Criterion In this section, we prove the following very useful criterion to examine whether maps are near-homeomorphisms: Theorem 2.7.1 (BING SHRINKING CRITERION) A map π : X → Y between completely metrizable spaces is a near-homeomorphism if π(X) is dense in Y and it satisfies the following condition: (Bi)

For each U ∈ cov(X) and V ∈ cov(Y ), there exist W ∈ cov(Y ) and a homeomorphism f : X → X such that πf is V-close to π and f (π −1 (W)) ≺ U:

154

2 Fundamental Results on Infinite-Dimensional Manifolds

π −1 (W) f shrinks

π

X V

∃f

Y

π

X

U

Proof For each V ∈ cov(Y ), we construct a homeomorphism g : X → Y that is V-close to π. Take complete metrics d ∈ Metr(X) and ρ ∈ Metr(Y ) so that diamd X  1/2 and {Bρ (y, 1) | y ∈ Y } ≺ V (cf. 1.3.22(1)). By induction, we construct Wn ∈ cov(Y ) and homeomorphisms fn : X → X, n ∈ N, so that (1) (2) (3) (4)

mesh Wn  2−n−1 ; mesh fn (π −1 (Wn ))  2−n−1 ; d(fn , fn−1 )  2−n ; −1 ρ(πfn−1 , πfn−1 )  2−n ,

where f0 = idX and W0 = {Bρ (y, 14 ) | y ∈ Y } ∈ cov(Y ). Assume that fn−1 and Wn−1 are defined. Take Wn ∈ cov(X) with mesh Wn  2−n−1 . We apply (Bi) to obtain Wn ∈ cov(Y ) and a homeomorphism h : X → X such that mesh Wn  −1 2−n−1 , h(π −1 (Wn )) ≺ fn−1 (Wn ), and h is π −1 (Wn−1 )-close to id. Let fn = fn−1 h and see the following diagram: π −1 (Wn ) h

X

−1 (Wn ) fn−1

π

h

shrinks fn

Wn−1

X fn−1

mesh Wn

2−n−1

π

Y

π

X

Then, Wn and fn satisfy (1) and (2). Since fn−1 satisfies (2), we have d(fn , fn−1 ) = d(fn−1 h, fn−1 )  mesh fn−1 (π −1 (Wn−1 ))  2−n . Since Wn−1 satisfies (1), we also have −1 ) = ρ(πh−1 , π)  mesh Wn−1  2−n . ρ(πfn−1 , πfn−1

Thus, (3) and (4) are also satisfied. Since X = (X, d) and Y = (Y, ρ) are complete, (fn )n∈N uniformly converges to a map f : X → X by (3) and (πfn−1 )n∈N uniformly converges to a map g : X → Y by (4). For each n ∈ N, d(f, fn )  2−n+1 by (3), hence mesh f (π −1 (Wn )) < 2−n+3 by (2). See the following diagram:

2.7 The Bing Shrinking Criterion

155

X

X π

fn−1

fn

X

f =lim fn

X

Y

g

g=lim πfn−1

Since π(X) is dense in Y and X = (X, d) is complete, we have a map g  : Y → X such that  {g  (y)} = cl f (π −1 (st(y, Wn ))) for each y ∈ Y. n∈N

The continuity of g  is easy to see. Observe that g  π = f . By (4), we have ρ(πfn−1 fm , π) = ρ(πfn−1 , πfm−1 )  2−n+1 for each n  m, which implies ρ(πfn−1 f, π)  2−n+1 for each n ∈ N, hence ρ(gf, π) = 0, i.e., gf = π. Since gg  π = gf = π and π(X) is dense in Y , it follows that gg  = id. Since g  gf = g  π = f and f (X) is dense in X, we have g  g = id. Therefore, g is a homeomorphism. Moreover, ρ(g, π) 

 n∈N

hence g is V-close to π.

−1 ρ(πfn−1 , πfn−1 )



2−n = 1,

n∈N

" !

Remark 2.11 In condition (Bi), a homeomorphism f can be replaced by a nearhomeomorphism. In fact, for each U ∈ cov(X) and V ∈ cov(Y ), take U ∈ cov(X) and V ∈ cov(Y ) such that st U ≺ U, U ≺ π −1 (V ), and st V ≺ V. Assume that there exists a near-homeomorphism f : X → X with W ∈ cov(Y ) such that πf is V -close to π and f (π −1 (W)) ≺ U (and f |A = id). Then, f is U -close to a homeomorphism h : X → X, which satisfies h(π −1 (W)) ≺ st(f (π −1 (W)), U ) ≺ st U ≺ U and πh is V -close to πf , so V-close to π. In the above theorem, let A ⊂ X. If f in (Bi) can be taken so as to satisfy the additional condition that f |A = id, then π can be approximated by a homeomorphism g : X → Y such that g|A = π|A. In fact, we can take fn in the proof so as to satisfy fn |A = id, hence πfn−1 |A = π|A. Taking the limit, g|A = π|A. Thus, we have the following: Theorem 2.7.2 Let π : X → Y be a map between completely metrizable spaces and A ⊂ X. Assume that π(X) is dense in Y and the following condition:

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2 Fundamental Results on Infinite-Dimensional Manifolds

For each U ∈ cov(X) and V ∈ cov(Y ), there exist W ∈ cov(Y ) and a homeomorphism f : X → X such that f |A = id, πf is V-close to π, and f (π −1 (W)) ≺ U:

(BiA )

π −1 (W) f

A

U

V

∃f

shrinks

A

⊂

π

X

⊂

Y

π

X

Then, for each V ∈ cov(Y ), π is V-close to a homeomorphism g : X → Y such that g|A = π|A. " ! Remark 2.12 When π is a closed map, for each open neighborhood U of π −1 (y) in X, y has an open neighborhood V in Y such that π −1 (V ) ⊂ U . In Theorems 2.7.1 and 2.7.2 above, the condition f (π −1 (W)) ≺ U in (Bi) and (BiA ) can be weakened to the condition {f π −1 (y) | y ∈ Y } ≺ U, that is, each f π −1 (y) is contained in some U ∈ U. For incomplete spaces, we have the following: Theorem 2.7.3 Let π : X → Y be a map between metrizable spaces and A a closed set in X such that B = π(A) is closed in Y . Assume that π(X) = Y and the following condition: (BiA )

For each U ∈ cov(X) and V ∈ cov(Y ), there exist W ∈ cov(Y ) and a homeomorphism f : X → X such that f |A ∪ (X \ π −1 (st(B, V))) = id, πf is V-close to π, and f (π −1 (W)) ≺ U:

π −1 (W) f

U

A ∪ (X \ π −1 (st(B, V)))

⊂

∃f

shrinks

A ∪ (X

\ π −1 (st(B, V)))

⊂

π

X

X

V

Y

π

B = π(A)

Then, for each V ∈ cov(Y ), π is V-close to a homeomorphism g : X → Y such that g|A ∪ (X \ π −1 (st(B, V))) = π|A ∪ (X \ π −1 (st(B, V))). Proof In the proof of Theorem 2.7.1, the homeomorphisms fn , n ∈ N, can be taken so as to satisfy the following condition in addition to (1)–(4): (5) fn |A ∪ (X \ π −1 (st(B, Wn−1 ))) = fn−1 |A ∪ (X \ π −1 (st(B, Wn−1 ))), from which we have fn |A ∪ (X \ π −1 (st(B, V))) = id. Using this additional condition (5) instead of the completeness, we show that (fn )n∈N and (πfn−1 )n∈N converge to maps f and g, respectively. Since (fn )n∈N and (πfn−1 )n∈N are uniformly Cauchy

2.7 The Bing Shrinking Criterion

157

by (3) and (4), it suffices to show that (fn (x))n∈N and (πfn−1 (x))n∈N are convergent for each x ∈ X. First, we show that (fn (x))n∈N is convergent in X. Due to (1), each x ∈ X \ π −1 (B) is contained in X\π −1 (st(B, Wn−1 )) for sufficiently large n, which implies that the sequence (fn (x))n∈N is eventually constant. For each x ∈ π −1 (B), there is a ∈ A such that π(a) = π(x), hence (fn (x))n∈N converges to a by (2) and (5). Thus, we have f (x) = limn→∞ fn (x) ∈ X for every x ∈ X. Next, we show that (πfn−1 (x))n∈N is convergent in Y . By virtue of (5), for each x ∈ A and n ∈ N, fn (x) = x, so πfn−1 (x) = π(x). For each x ∈ X \ A, choose m ∈ N so that 2−m−1 < d(x, A). Then, x  = fm−1 (x) ∈ X \ π −1 (st(B, Wm )). Otherwise, we have W ∈ Wm and a ∈ A such that π(x  ), π(a) ∈ W , hence d(x, a)  d(fm (x  ), fm (a))  diam fm (π −1 (W ))  2−m−1 < d(x, A), which is a contradiction. By (5), fm+1 (x  ) = fm (x  ) = x. By induction, we have fn (x  ) = x for any n  m, hence (πfn−1 (x))n∈N is eventually equal to π(x  ). Thus, we can define g : Y → X by g(x) = limn→∞ πfn−1 (x) ∈ X for every x ∈ X. In the proof of Theorem 2.7.1, we used a map g  to show that g is a homeomorphism. In order to define g  without completeness, we need to assume that π(X) = Y . Indeed, f (π −1 (y)) = ∅ for every y ∈ Y . As observed in the proof of Theorem 2.7.1, f (π −1 (Wn )) < 2−n+3 . Then, g  can be defined by {g  (y)} = f (π −1 (y)). " ! For compact metric spaces, the Bing Shrinking Criterion 2.7.1 can be easily rewritten using Remark 2.12 as follows: Theorem 2.7.4 (BING SHRINKING CRITERION) A map π : X → Y between compact metric spaces is a near-homeomorphism if π(X) = Y and π satisfies the following condition: (Bi0 )

For each ε > 0, there is a homeomorphism f : X → X such that diam f (π −1 (y)) < ε for each y ∈ Y and πf is ε-close to π. " !

Remark 2.13 By the same argument as in Remark 2.11, a homeomorphism f in (Bi0 ) can be replaced by a near-homeomorphism. Furthermore, as Theorem 2.7.2, given A ⊂ X, if f in (Bi0 ) can be taken so as to satisfy the condition f |A = id, then π can be approximated by a homeomorphism g : X → Y such that g|A = π|A. Applying the above Bing Shrinking Criterion 2.7.4, we show the following: Theorem 2.7.5 The Hilbert cube Q is homeomorphic to its cone C(Q), which is homeomorphic to the usual cone CQ = (I × Q)/({0} × Q) because Q is compact.

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2 Fundamental Results on Infinite-Dimensional Manifolds

[−1, 1] n ×

Q

I Fig. 2.17 The homeomorphism ϕ

Proof Let π : I × Q → C(Q) = (I × Q){0} be the natural map. We use the metric d ∈ Metr(I × Q) defined by d((t, x), (s, y)) = |t − s| +



2−n |x(n) − y(n)|.

n∈N

For each ε > 0 choose n ∈ N and 0 < δ < 1 so that 2−n < ε/4 and (0, δ] × Q ⊂ B(0, ε/2) ⊂ C(Q), where C(Q) is any given admissible metric. Represent Q = [−1, 1]n × Q and let q : I × Q → Q be the projection. It is easy to construct ϕ ∈ Homeo(I × Q) (as illustrated in Fig. 2.17) such that ϕ|[δ, 1] × Q = id, qϕ = q and ϕ({0} × Q) = {0} × [−ε/4, ε/4]n × Q . Then, diam ϕ({0} × Q) < ε and πϕ is ε-close to π. By the Bing Shrinking Criterion 2.7.4, π is a near-homeomorphism. Thus, we have Q ≈ I × Q ≈ C(Q). " ! Combining Theorem 2.7.5 with the homogeneity of Q (Theorem 2.1.2), we have the following: Corollary 2.7.6 Q \ {0} ≈ Q × [0, 1).

" !

As another application of the Bing Shrinking Criterion 2.7.4 (with Remark 2.13), we can prove the following GENERALIZED SCHOENFLIES THEOREM, which will be used in the proof of Theorem 4.5.1 concerning the existence of an immersion of punctured n-torus into Rn : Theorem 2.7.7 (GENERALIZED SCHOENFLIES) Let h : Sn−1 → Sn be an embedding such that h(Sn−1 ) is bi-collared in Sn . Then, h extends to a homeomorphism h˜ : Sn → Sn , where Sn−1 is identified with Sn ∩ (Rn × {0}) ⊂ Sn . As

2.7 The Bing Shrinking Criterion

159

a consequence, the complement Sn \ h(Sn−1 ) has precisely two components whose closures are n-cells. As preparation of the proof of Theorem 2.7.7, we introduce cellular sets. A homeomorphic image of Bn is called a (topological) n-cell (or just a cell).25 When (C, D) ≈ (Bn , Sn−1 ), D is called the boundary of a cell C and denoted by ∂C. Let X be an n-dimensional compactum. It is said that A ⊂ X is cellular in Xif there exist n-cells C1 , C2 , . . . in X such that Ci+1 ⊂ Ci \ ∂Ci , i ∈ N, and A = i∈N Ci . This is equivalent to the condition that for each neighborhood U of A in X, there is an n-cell C such that A ⊂ C \ ∂C ⊂ C ⊂ U . Lemma 2.7.8 Let f : Bn → Sn be a map with A ⊂ Bn \ Sn−1 such that f (A) is a finite set, A = f −1 (f (A)), and f |Bn \A is injective. Then, there exists a component D of Sn \ f (Sn−1 ) such that f (Bn ) = f (Sn−1 ) ∪ D. Proof First, note that f (Bn ) ⊂ f (Sn−1 ). Indeed, since h = f |Sn−1 : Sn−1 → f (Sn−1 ) is a homeomorphism, we have a retraction r = h−1 f : Bn → Sn−1 , which contradicts the No Retraction Theorem 1.12.2. Hence, f (Bn ) meets a component D of Sn \ f (Sn−1 ). Since f (Sn−1 ) does not separate f (Bn ), it follows that f (Bn ) ⊂ cl D, so we have f (Bn ) ⊂ f (Sn−1 ) ∪ D. Note that (Bn \Sn−1 )\A is an open set in Rn and f |(Bn \Sn−1 )\A is a continuous injection. Since D is regarded as an open set in Rn , it follows from Theorem 1.12.4 that f |(Bn \ Sn−1 ) \ A is an open embedding. Hence, f ((Bn \ Sn−1 ) \ A) is open in D \ f (A), which is also closed in D \ f (A). When n > 1, any finite set does not separate D, so D \ f (A) is connected. Then, it follows that f ((Bn \ Sn−1 ) \ A) = D \ f (A), hence f (Bn ) ⊃ D. Thus, we have f (Bn ) = f (Sn−1 ) ∪ D. When n = 1, since f (B1 ) (= f ([−1, 1])) is an arc connecting f (−1) and f (1), it follows that f (B1 ) = f (S0 ) ∪ D. " ! Lemma 2.7.9 Let f : Bn → Sn be a map with A ⊂ Bn \ Sn−1 such that f (A) is a singleton, A = f −1 (f (A)), and f |Bn \ A is injective. Then, A is cellular in Bn . Proof By Lemma 2.7.8, there is a component D of Sn \ f (Sn−1 ) such that f (Bn ) = f (Sn−1 ) ∪ D. Let U be any open set in Bn such that A ⊂ U ⊂ Bn \ Sn−1 . Then, f (U ) is open in D and f (A) ⊂ f (U ). Since f (A) is a singleton, we can take a homeomorphism h : Sn → Sn and a neighborhood V of f (A) in Sn such that V ⊂ f (U ), h|V = id, and h(D) ⊂ f (U ). Applying Theorem 1.12.4 as in the proof of Lemma 2.7.8, we can see that g = f |Bn \ A : Bn \ A → f (Bn \ A) is a homeomorphism. Then, we can define a map ϕ : Bn → Bn as follows: ϕ(x) =

25 In

 x g −1 hf (x)

if x ∈ A, if x ∈ Bn \ A.

Chapter 1, a (convex) linear cell is called a cell (see Sect. 1.5, p. 27).

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2 Fundamental Results on Infinite-Dimensional Manifolds

Since f −1 (V ) is a neighborhood of A in Bn and g −1 hf |f −1 (V ) \ A = id, it follows that ϕ is a well-defined embedding and ϕ|f −1 (V ) = id. Because hf (Bn ) ⊂ f (U ), we have f −1 (V ) ⊂ ϕ(Bn ) ⊂ U . Consequently, A is cellular in Bn . " ! Lemma 2.7.10 Let f : Sn → Sn be a map with A, B ⊂ Sn such that f (A), f (B) are singletons, A = f −1 (f (A)), B = f −1 (f (B)), and f |Sn \ (A ∪ B) is injective. Then, both A and B are cellular in Sn . Proof Decomposing Sn into two n-cells C0 , C1 with the common boundary ∂C0 = ∂C1 = C0 ∩ C1 so that A ∪ B ⊂ Sn \ C0 = C1 \ ∂C1 , we can write Sn = C0 ∪ C1 . Applying Lemma 2.7.8 to f |C1 , we have a component D of Sn \ f (∂C0 ) such that f (C1 ) = f (∂C1 ) ∪ D. Since f (A) and f (B) are singletons contained in D, there is an open set U in D such that f (A) ⊂ U and cl U ∩ f (B) = ∅. Then, we can take a homeomorphism h : Sn → Sn and a neighborhood V of f (A) in Sn such that V ⊂ U , h|V = id, and h(f (C1 )) ⊂ U . Applying Theorem 1.12.4 as before, we can see that g = f |f −1 (U ) \ A : f −1 (U ) \ A → U \ f (A) is a homeomorphism. We define a map ϕ : C1 → Sn as follows: ϕ(x) =

 x

if x ∈ A,

g −1 hf (x)

if x ∈ C1 \ A.

Then, ϕ(B) = g −1 hf (B) is a singleton, ϕ −1 ϕ(B) = f −1 f (B) = B, and ϕ|C1 \B is injective. So, it follows from Lemma 2.7.9 that B is cellular in C1 . By the definition of a cellular set, B is also cellular in Sn . In the same way, it can be shown that A is cellular in Sn . " ! Proof (Theorem 2.7.7) By shrinking a bi-collar, we can obtain a closed bi-collar k : h(Sn−1 ) × [−1, 1] → Sn of h(Sn−1 ) in Sn . We define an embedding h∗ : Sn−1 × [−1, 1] → Sn by h∗ (x, t) = k(h(x), t)

for each (x, t) ∈ Sn−1 × [−1, 1].

Then, Sn \ h∗ (Sn−1 × (−1, 1)) has two components A and B with h∗ (Sn−1 × {1}) ⊂ A and h∗ (Sn−1 × {−1}) ⊂ B. Consider the suspension Sn−1 of Sn−1 defined as the following quotient space:26 Sn−1 = Sn−1 × [−1, 1]/{Sn−1 × {−1}, Sn−1 × {1}}, where Sn−1 × {−1}, Sn−1 × {1} are regarded as two distinct points a, b, respectively. Let π : Sn−1 × [−1, 1] → Sn−1 be the quotient map. We can define a map

26 We

change the coordinates in the definition of the suspension in [GAGT, p. 453].

2.7 The Bing Shrinking Criterion

ϕ : Sn → Sn−1 as follows: ⎧ −1 ⎪ ⎪ ⎨πh∗ (x) ϕ(x) = a ⎪ ⎪ ⎩b

161

if x ∈ h∗ (Sn−1 × [−1, 1]), if x ∈ A, if x ∈ B.

Identifying Sn−1 = Sn , we have a map ϕ : Sn → Sn such that ϕ(A) = {a}, ϕ(B) = {b}, ϕ −1 (a) = A, ϕ −1 (b) = B and ϕ|Sn \ (A ∪ B) is injective. Then, both A and B are cellular by Lemma 2.7.10. For each ε > 0, a and b have open neighborhoods U and V in Sn respectively such that U ∩ V = ∅ and diam U , diam V < ε. Since A and B are cellular, there exist n-cells Ci , Di , i = 0, 1 in Sn such that A ⊂ C0 \ ∂C0 ⊂ C0 ⊂ C1 \ ∂C1 ⊂ C1 ⊂ ϕ −1 (U ) and B ⊂ D0 \ ∂D0 ⊂ D0 ⊂ D1 \ ∂D1 ⊂ D1 ⊂ ϕ −1 (V ). Then, we can take a homeomorphism ψ : Sn → Sn such that ψ|Sn \ (C1 ∪ D1 ) = id; and diam ψ(C0 ), diam ψ(D0 ) < ε. Observe that ϕψ is ε-close to ϕ, diam ψ(ϕ −1 (a)), diam ψ(ϕ −1 (b)) < ε, and ψ|h(Sn−1 ) = id. Now, applying the Bing Shrinking Criterion 2.7.4 with Remark 2.13, we can obtain a homeomorphism g : Sn → Sn such that g|h(Sn−1 ) = ϕ|h(Sn−1 ), hence n−1 g(h(x)) = πh−1 , ∗ (h∗ (x, 0)) = π(x, 0) = x for each x ∈ S

where we identify (Sn , Sn−1 × {0}) = (Sn , Sn−1 ). Then, h˜ = g −1 : Sn → Sn ˜ is a homeomorphism and h(x) = g −1 (x) = h(x) for each x ∈ Sn−1 , that is, n−1 ˜ h|S = h. " ! In Theorem 2.7.7, decomposing Sn into two n-cells C and D, we can write Sn = C ∪ D, where h(Sn−1 ) = C ∩ D = ∂C = ∂D. Given a ∈ C and b ∈ D, h ˜ ˜ extends to a homeomorphism h˜ : Sn → Sn with h(a) = a and h(b) = b. Indeed, n n−1 for each x0 , x1 ∈ B \ S , there exists a homeomorphism h : Bn → Bn such that h|Sn−1 = id and h(x0 ) = x1 . Using this fact, we can arrange the homeomorphism ˜ ˜ h˜ obtained in Theorem 2.7.7 so as to satisfy the condition h(a) = a and h(b) = b. n n Since S is the one-point compactification of R , the following can be obtained: Corollary 2.7.11 Let h : Sn−1 → Rn be an embedding such that h(Sn−1 ) is bi-collared in Rn . Then, h extends to a homeomorphism h˜ : Rn → Rn . As a

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2 Fundamental Results on Infinite-Dimensional Manifolds

consequence, the complement Rn \ h(Sn−1 ) has precisely two components and the closure of the bounded component is an n-cell. " !

2.8 Z-Sets and Strong Z-Sets in ANRs A closed set A in a space X is called a Z-set in X if, for each U ∈ cov(X), there is a map f : X → X \ A that is U-close to id. If A ∩ cl f (X) = ∅ in the above, A is called a strong Z-set in X. A countable union of Z-sets (resp. strong Z-sets) is called a Zσ set (resp. strong Zσ -set). Every closed subset of a (strong) Z-set is also a (strong) Z-set and every Fσ -set contained in a (strong) Zσ -set is a (strong) Zσ -set. Obviously a strong Z-set is a Z-set. The converse is also true in a locally compact paracompact space, that is: Proposition 2.8.1 Every Z-set in a locally compact paracompact space X is a strong Z-set. In particular, every Z-set in a Q-manifold is a strong Z-set. Proof Each U ∈ cov(X) has a locally finite open refinement V ∈ cov(X) such that cl V is compact for each V ∈ V. Then, idX is V-close to a map f : X → X \ A. Since idX is a proper map, f is also proper by Proposition 1.3.8, hence it is perfect by Proposition 1.3.7. Thus, f (X) is closed in X, which implies that A ∩ cl f (X) = A ∩ f (X) = ∅. " ! In general, a (compact) Z-set is not a strong Z-set even in an AR. For example, let    X = I × {0} ∪ n−1  n ∈ N × I ⊂ R2 and A = {(0, 0)} ⊂ X. Then, X is an AR, in which A is a Z-set but not a strong Z-set (Fig. 2.18).

X ··· ··· ··· A (2n)−1 n −1

A 0

Fig. 2.18 A Z-set being not a strong Z-set

3−1 2−1

1

2.8 Z-Sets and Strong Z-Sets in ANRs

163

Indeed, for each n ∈ N, we have a map fn : X → X \ A defined as follows: ⎧ ⎪ ⎪(s, t) ⎪ ⎪ ⎨(s, 2t − n−1 ) fn (s, t) = ⎪ (2nts + (1 − 2nt)n−1 , 0) ⎪ ⎪ ⎪ ⎩ −1 (n , 0)

if s  n−1 or t  n−1 , if s < n−1 and (2n)−1  t  n−1 , if s < n−1 and 0 < t  (2n)−1 , if s  n−1 and t = 0,

√ where t > 0 implies s = m−1 for some m ∈ N. Then, fn is 2n−1 -close to idX with respect to the Euclidean metric. Hence, A is a Z-set in X. On the other hand, assume that a map f : X → X is U-close to idX , where      U = {n−1 } × (2−1 , 1]  n ∈ N ∪ X ∩ [0, 1)2 , (2−1 , 1] × [0, 1) ∈ cov(X). For each n ∈ N, let vn = (n−1 , 1) ∈ X. Since f (vn ) ∈ {n−1 } × (2−1 , 1] and f (v1 ) ∈ (2−1 , 1]×[0, 1), the image f (X) contains every point (n−1 , 0) ∈ X, which implies (0, 0) ∈ clX f (X). Thus, A is not a strong Z-set in X.

Lemma 2.8.2 Let A be a closed set in a metrizable space X. If A is contained in some collared set C in X, then A is a strong Z-set. Proof Let k : C × [0, 1) → X be a collar of C in X. Then, A has an open neighborhood W in X with cl W ⊂ k(C × [0, 1)). For each U ∈ cov(X), choose a map γ : C → I so that γ (x) > 0 for x ∈ A, γ (x) = 0 for x ∈ C \ W , and {{x} × [0, γ (x)] | x ∈ C ∩ W } ≺ k −1 (U|W ) (Lemma 2.5.2). Let V = {k(x, t) | t < γ (x)} ⊂ W . Observe that V is open in X, A ⊂ V , and cl V = {k(x, t) | t  γ (x)}. We can define a retraction f : X → X \ V as follows: f |X \ V = id and f (k(x, t)) = k(x, γ (x)) if t  γ (x). Then, f is U-close to id and f (X) = X \ V is a closed set in X that misses A. Hence, A is a strong Z-set in X. " ! In a metrizable space, every collared set is an Fσ -set because it is locally closed, that is, closed in an open set. Thus, by Lemma 2.8.2 above, we have the following: Proposition 2.8.3 Every collared set in a metrizable space is a strong Zσ -set.

" !

The following is a combination of Theorem 2.4.9 and Lemma 2.8.2: Proposition 2.8.4 Every E-deficient closed (or Fσ ) set in an E-stable space is a strong Z-set (or a strong Zσ -set). ! " Let n ∈ N. A closed set A in a space X is called a Zn -set if, for each U ∈ cov(X), every map f : In → X is U-close to a map f  : In → X \ A. For every k  n ∈ N, each Zn -set in X is a Zk -set in X. Indeed, in the case k < n, let p = prIk : In = Ik × In−k → Ik be the projection and i : Ik → In be the embedding defined by i(x) = (x, 0) ∈ Ik × In−k for each x ∈ Ik . For each map f : Ik → X and U ∈ cov(X), we have a map f  : In → X \ A that is U-close to fp. Then,

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2 Fundamental Results on Infinite-Dimensional Manifolds

f  i : Ik → X \ A is U-close to f = fpi: Ik p

f

i

In

X ∪

f

X\A

When A is a Zn -set in X for every n ∈ N, we call A a Z∞ -set in X. Proposition 2.8.5 Every locally finite union of Zn -sets (resp. Z∞ -sets) in a paracompact space X is also a Zn -set (resp. a Z∞ -set). Proof First of all, observe that if the Zn -set case holds for every n ∈ N, then the Z∞ -set case follows from the definition. By induction, the finite case reduces to showing that if A and B are Zn -sets in X then A ∪ B is also a Zn -set in X. For each U ∈ cov(X), let V ∈ cov(X) be a star-refinement of U. Each map f : In → X is V-close to a map f  : In → X \ A. Let W ∈ cov(X) be a common refinement of V and {X \ A, X \ f  (In )}. Then, f  is W-close to a map f  : In → X \ B. Since f  (In ) ⊂ X \ A, it follows that f  (In ) ⊂ X \ A. Note that f  is U-close to f . Thus,  A ∪ B is a Zn -set in X. We now show the locally finite case. Let A = λ∈ Aλ be a locally finite union of Zn -sets Aλ in X and U ∈ cov(X). For each map f : In → X, since f (In ) is compact, f (I n ) has an open neighborhood U that meets only finitely many Aλ , that is, there are λ1 , . . . , λn ∈  such that U ∩ A = U ∩ (Aλ1 ∪ · · · ∪ Aλn ). Let V ∈ cov(X) be a common refinement of U and {U, X \ f (In )}. Since Aλ1 ∪ · · · ∪ Aλn is a Zn -set in X, f is V-close to a map f  : In → X \ (Aλ1 ∪ · · · ∪ Aλn ). Then, f  (In ) ⊂ X \ A because f  (In ) ⊂ U . Hence, A is a Zn -set in X.

" !

In an ANR, Z-sets are characterized as follows: Theorem 2.8.6 For a closed set A in an ANR X, the following are equivalent: (a) A is a Z-set in X; (b) For each ε > 0, each map f : Q → X is ε-close to a map g : Q → X \ A with respect to any given metric d ∈ Metr(X); (c) A is a Z∞ -set in X; (d) For an arbitrary simplicial complex K and any U ∈ cov(X), each map f : |K| → X is U-close to a map g : |K| → X \ A; (e) For a finite simplicial complex K and any U ∈ cov(X), each map f : |K| → X is U-close to a map g : |K| → X \ A;

2.8 Z-Sets and Strong Z-Sets in ANRs

165

(f) The inclusion X \ A ⊂ X is a hereditary weak homotopy equivalence, i.e., for every open set U in X, the inclusion U \ A ⊂ U is a weak homotopy equivalence; (g) For every open set U in X, the inclusion U \ A ⊂ U is a homotopy equivalence; (h) X \ A is homotopy dense in X; (i) X \ A is dense in X and, for each a ∈ A, each neighborhood U of a in X contains a neighborhood V in X such that every map α : Sn−1 → V \ A is null-homotopic in U \ A for every n ∈ N. Proof The implications (a) ⇒ (b) ⇒ (c) and (d) ⇒ (e) ⇒ (c) are trivial. The equivalence (f) ⇔ (g) follows from the fact that every weak homotopy equivalence between ANRs is a homotopy equivalence (Corollary 1.13.30). The equivalences (f) ⇔ (h) ⇔ (i) are direct consequences of Corollary 1.15.5. By Hanner’s Characterization of ANRs (Theorem 1.13.20), for each U ∈ cov(X), we have a simplicial complex K with maps f : X → |K| and g : |K| → X such that gf is Uclose to idX . Then, it is easy to see the implication (d) ⇒ (a). We have to show the implication (c) ⇒ (d) and the equivalence (e) ⇔ (f). Refer to the following diagram of implications: (a)

(1.13.20)

(d)

triv.

(b)

triv.

(e)

triv. triv.

(c)

(f)

(1.15.5)

(1.15.5)

(h)

(i)

(1.13.30)

(g)

(c) ⇒ (d): For each U ∈ cov(X), we adopt a metric d ∈ Metr(X) such that {Bd (x, 1) | x ∈ X} ≺ U (1.3.22). Then, it suffices to show that each map f : |K| → X is 1-close to a map g : |K| → X \ A. We shall construct maps gn : |K| → X, n ∈ ω, such that gn (|K (n) |) ⊂ X \ A, gn ||K (n−1) | = gn−1 ||K (n−1)| and gn is 2−n−1 -close to gn−1 , where g−1 = f . Then, (gn )n∈N is uniformly convergent to the map g : |K| → X defined by g||K (n) | = gn ||K (n) | for each n ∈ ω, whence g is 1-close to f and g(|K|) =



gn (|K (n) |) ⊂ X \ A.

n∈ω

Assume that g−1 , g0 , . . . , gn−1 have been obtained. By Corollary 1.13.10, we have V ∈ cov(X) that is an h-refinement of an open cover of X with mesh < 2−n−1 . Then, any two V-close maps are 2−n−1 -homotopic. For each n-simplex σ ∈ K, choose ε > 0 so that {B(gn−1 (x), ε) | x ∈ σ } ≺ V and let   δ = min ε/2, dist(gn−1 (∂σ ), A) > 0.

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2 Fundamental Results on Infinite-Dimensional Manifolds

Since X is an ANR, we apply (c) to obtain a map gσ : σ → X \ A with gσ δ gn−1 |σ in X. It follows from the definition of δ that gσ |∂σ δ gn−1 |∂σ in X \ A. Since X\A is also an ANR, we can apply the Homotopy Extension Theorem 1.13.11 to extend gn−1 |∂σ to a map gσ : σ → X \ A with gσ δ gσ . Note that gσ and gn−1 |σ are V-close because d(gσ (x), gn−1 (x)) < 2δ  ε for each x ∈ σ . Then, we have a map gn : |K (n) | → X \ A defined by gn ||K (n−1) | = gn−1 ||K (n−1) | and gn |σ = gσ for each n-simplex σ ∈ K, whence gn is V-close to gn−1 ||K (n) |, which implies gn 2−n−1 gn−1 ||K (n) | in X. Applying the Homotopy Extension Theorem 1.13.11 again, we can extend gn to a map gn : |K| → X that is 2−n−1 close to gn−1 . This completes the induction. (e) ⇒ (f): For each open set U in X and i ∈ N, let β : Bi → U be a map with α = β|Si−1 : Si−1 → U \ A. Since U is an ANR and β(Bi ) is compact, we can choose ε > 0 so that a map of Bi to U extending α is homotopic to β rel. Si−1 whenever it is ε-close to β (Theorem 1.13.10). Let   δ = min ε/2, dist(β(Bi ), X \ U ), dist(α(Si−1 ), A) > 0. We apply (e) with Theorem 1.13.10 to obtain a map γ : Bi → X \ A that is δhomotopic to β in X, hence in U \A. Observe that γ |Si−1 is δ-homotopic to α in U \ A. Since U \ A is an ANR, we can apply the Homotopy Extension Theorem 1.13.11 to extend α to a map α¯ : Bi → U \ A that is δ-homotopic to γ . Since α¯ is ε-close to β, α¯  β rel. Si−1 . This means that the inclusion U \ A ⊂ U is a weak homotopy equivalence. (f) ⇒ (e): Let dim K = n. For each U ∈ cov(X), we take open covers of X as follows: ∗

∗

∗

∗

∗

U = Un  Un−1  · · ·  U1  U0  U−1 . For each map f : |K| → X, by subdividing K, we can assume that K ≺ f −1 (U−1 ), that is, f (K) = {f (σ ) | σ ∈ K} ≺ U−1 (Theorem 1.6.9). Let f−1 = f , and assume that we have a map fi−1 : |K| → X such that fi−1 (|K (i−1) |) ⊂ X \ A, fi−1 (K) ≺ Ui−1 and fi−1 Ui−1 f. For each i-simplex σ ∈ K, we can choose Uσ ∈ Ui−1 so that fi−1 (σ ) ⊂ Uσ , whence fi−1 (∂σ ) ⊂ Uσ \ A. Since the inclusion Uσ \ A ⊂ Uσ is an i-homotopy equivalence, there is a map fσ : σ → Uσ \ A such that fσ |∂σ = fi−1 |∂σ and fσ  fi−1 |σ rel. ∂σ in Uσ . We define a map fi : |K (i) | → X \ A by fi ||K (i−1)| = fi−1 ||K (i−1) | and fi |σ = fσ for each i-simplex σ ∈ K. Observe that fi Ui−1 fi−1 ||K (i) | rel. |K (i−1) | in X. Then, by the Homotopy Extension Theorem 1.13.11, we can extend fi to a map fi : |K| → X that is Ui−1 -homotopic to fi−1 , so Ui -homotopic to f . Moreover, fi (K) ≺ st(fi−1 (K), Ui−1 ) ≺ st Ui−1 ≺ Ui .

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167

By induction, we have a map fn : |K| = |K (n) | → X \ A that is U-close to f . Thus, (e) holds. " ! Remark 2.14 When X is an E-manifold, each neighborhood U of any point of X contains a homotopically trivial open neighborhood V , where V is said to be homotopically trivial if for each n ∈ ω, every map α : Sn → V is null-homotopic. Then, (i) is derived from the following condition, which is weaker than (f) (or (g)): (z)

If U is a homotopically trivial open set in X, then U \ A is also homotopically trivial.

Thus, the above (z) characterizes Z-sets in an E-manifold, and this was the first definition due to R.D. Anderson. Using the above characterization of Z-sets (2.8.6), the following is easily shown: Corollary 2.8.7 Let X be an ANR. (1) A locally finite union of Z-sets in X is a Z-set. (2) For each Z-set A in X and each open set U in X, A ∩ U is a Z-set in U . (3) A closed set A ⊂ X is a Z-set in X if each x ∈ A has a neighborhood U in X such that A ∩ U is a Z-set in U . ! " Proposition 2.8.8 A closed Zσ -set A in an ANR X is a Z-set if A is completely metrizable (i.e., absolutely Gδ ). In particular, every compact Zσ -set in an ANR is a Z-set. Proof Since a finite union of Z-sets is also a Z-set (Corollary 2.8.7(1)), we have a  tower A1 ⊂ A2 ⊂ · · · of Z-sets in X such that A = n∈N An . Since A is closed in X and A has an admissible complete metric, we apply Hausdorff’s Metric Extension Theorem 1.13.8 to obtain d ∈ Metr(X) such that d is complete on A. To see that A is a Z-set in X, we verify that A satisfies condition (b) in Theorem 2.8.6. Let f : Q → X be a map and 0 < ε < 1. Using the fact that each An is a Z-set in X, we can inductively construct maps fn : Q → X, n ∈ N, so as to satisfy the following conditions: (1) (2) (3) (4)

fn (Q) ∩ An = ∅, εn = min{dist(fn (Q), An ), εn−1 /2}, where ε0 = ε < 1, d(fn , fn−1 ) < εn−1 /2, where f0 = f , fn |Dn−1 = fn−1 |Dn−1 , where Dn−1 = {x ∈ Q | d(fn−1 (x), A)  2−(n−1) .

Suppose that f0 , . . . , fn−1 have been constructed, where ε0 , . . . , εn−1 are defined in (2) and D1 , . . . , Dn−1 are defined in (4). Take δ > 0 such that a map g : Q → X with d(g, fn−1 ) < δ is εn−1 /2-homotopic to fn−1 . Since An is a Z-set in X, we have a map g : Q → X \ An with d(g, fn−1 ) < δ. Then, we have an εn−1 /2homotopy h : Q × I → X such that h0 = fn−1 and h1 = g. Let    U = x ∈ Q  d(ht (x), A) > 2−n for every t ∈ I .

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2 Fundamental Results on Infinite-Dimensional Manifolds

Note that εn−1 /2 < 2−n because ε0 = ε < 1. For each x ∈ Dn−1 and t ∈ I, d(ht (x), A)  d(fn−1 (x), A) − d(ht (x), fn−1 (x) > 2−(n−1) − εn−1 /2 > 2−n , which implies that U is an open neighborhood of Dn−1 in Q. Let α : Q → I be an Urysohn map with α(Dn−1 ) = {0} and α(Q \ U ) = {1}. We define fn : Q → X by fn (x) = h(x, α(x)). Then, fn (Q) ∩ An = ∅, d(fn , fn−1 ) < εn−1 /2, and fn |Dn−1 = fn−1 |Dn−1 . Note that Dn−1 ⊂ Dn for each n ∈ N. Indeed, for each x ∈ Dn−1 , d(fn (x), A)  d(fn−1 (x), A) − d(fn (x), fn−1 (x)) > 2−(n−1) − εn−1 /2 > 2−n .  If Q = n∈N Dn , we have a map f˜ = limn→∞ fn : Q → X \ A (i.e., f˜|Dn = fn |Dn for each n ∈ N) and d(f˜, f ) < ε, where it should be noted that (f n )n∈N is uniformly Cauchy but d might be non-complete on X. To show that Q = n∈N Dn ,  suppose that there exists x0 ∈ Q \ n∈N Dn . Then, we can take yn ∈ A, n ∈ N, such that fn (x0 ), yn ) < 2−n . Since (fn (x0 ))n∈N is Cauchy, so is (yn )n∈N . By the completeness of A, (yn )n∈N converges to y0 ∈ A. Then, limn→∞ fn (x0 ) = y0 and y0 ∈ Am for some m ∈ N. For each n > m, d(fm (x0 ), y0 )  d(fm (x0 ), fn (x0 )) + d(fn (x0 ), y0 )
β  . Then, it follows that g −1 (N2 [0, 1]) ∩ X[β, 1] = ∅.

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2 Fundamental Results on Infinite-Dimensional Manifolds

Sliding the I-coordinates and using an Urysohn map (see Fig. 2.21), we can easily define a homeomorphism h : X[0, 1] → X[0, 1] such that prX ◦h = prX , h|X[0] ∪ X[1] ∪ (X \ N0 )[0, 1] = id and h(x, β(x, y), y) = (x, α(x), y) for each x ∈ N1 and y ∈ E. Let ϕ = hg −1 . For each W ∈ W, W ∗ = qA ((W ∩ N2 )[0, 1]) is open in qA (X[0, 1]) and there are some N ∈ N and U ∈ U such that ϕ(qA−1 (W ∗ )) = ϕ((W ∩ N2 )[0, 1]) ⊂ N[0, α] ⊂ U. Thus, ϕ is the desired homeomorphism with the open cover 

  W ∗  W ∈ W ∪ ϕ −1 (U|(X \ A)[0, 1]) ∈ cov(qA (X[0, 1])).

This completes the proof (Fig. 2.21).

" !

When X \ A is an E-manifold, the projection prX\A : (X \ A) × E → X \ A is a near-homeomorphism by the Stability Theorem 2.3.7. Take d ∈ Metr(X) and let    U = Bd (x, 2−1 d(x, A))  x ∈ X \ A ∈ cov(X \ A). Then, we have a homeomorphism h : (X\A)×E → X\A that is U-close to prX\A . Let pA : (X × E)A → X be the natural map, that is, pA |(X \ A) × E = prX\A and pA |A = id. As is seen in the proof of Lemma 2.1.16(i), U is fitting in X, ˜ = id. Then, h˜ is a hence h can be extended to a map h˜ : (X × E)A → X by h|A homeomorphism. To see this, it suffices to show that, for each open set U in X, the following is open in X:   h˜ ((U \ A) × E) ∪ (U ∩ A) = h((U \ A) × E) ∪ (U ∩ A). Assume that this is not open in X, that is, this is not a neighborhood of x0 ∈ U ∩ A in X because h((U \ A) × E) is open in X. Then, we have xn ∈ X \ (h((U \ A) × E) ∪ (U ∩ A)), n ∈ N, such that xn → x0 (n → ∞). For every n ∈ N, we may assume that xn ∈ X \ A, so h−1 (xn ) ∈ (X \ A) × E. Since h is U-close to prX\A , we have yn ∈ X \ A such that xn , prX\A h−1 (xn ) ∈ Bd (yn , 2−1 d(yn , A)). Then, observe that d(yn , A)  d(yn , x0 )  d(xn , yn ) + d(xn , x0 ) < 2−1 d(yn , A) + d(xn , x0 ),

2.9 Z-Sets and Strong Z-Sets in E-Manifolds

179

which implies that 2−1 d(yn , A) < d(xn , x0 ). Since d(prX\A h−1 (xn ), xn ) < d(yn , A) and xn → x0 (n → ∞), it follows that prX\A h−1 (xn ) → x0 (n → ∞). Note that prX\A h−1 (xn ) ∈ X \ A. For sufficiently large n ∈ N, prX\A h−1 (xn ) ∈ U \ A, hence xn ∈ h((U \ A) × E). This is a contradiction. Consequently, (X × E)A ≈ X. Thus, we have the following: Corollary 2.9.2 Let A be a strong Z-set in X. If X × E and X \ A are E-manifolds, then X is also an E-manifold. " ! Remark 2.18 In Corollary 2.9.2 and Proposition 2.9.1, “strong Z-set” cannot be replaced by “Z-set.” Indeed, consider the example defined on p. 162 (see Fig. 2.18), that is,    X = I × {0} ∪ n−1  n ∈ N × I ⊂ R2 and A = {(0, 0)} ⊂ X. Then, X is a separable completely metrizable AR, in which A is a Z-set but not a strong Z-set. Similarly, in the reduced product (X × 2 )A , A is a Z-set but not a strong Z-set. Due to the following theorem, every Z-set in an 2 -manifold is a strong Z-set. Therefore, (X × 2 )A is not an 2 -manifold. However, by Toru´nczyk’s Factor Theorem 2.2.12, (X × 2 )A × 2 ≈ 2 and (X × 2 )A \ A = (X \ A) × 2 ≈ 2 . Moreover, X × 2 ≈ 2 ≈ (X × 2 )A . For Z-sets in an E-manifold, we have the following characterization: Theorem 2.9.3 For a closed set A in an E-manifold M, the following are equivalent: (a) (b) (c) (d) (e)

A is a Z-set in M; A is a strong Z-set in M; A is E-deficient in M; There is a homeomorphism h : M → M × [0, 1) such that h(A) ⊂ M × {0}; A is contained in some collared set in M.

Proof The equivalence among (c), (d), and (e) is due to Theorem 2.4.9 and the implication (e) (or (d)) ⇒ (b) is due to Lemma 2.8.2. The implication (b) ⇒ (a) is obvious. Note that an E-manifold is E-stable. By Lemma 2.3.2, there is a homeomorphism f : (M × E)A → M such that f |A = id. Then, (b) ⇒ (c) can be obtained by Proposition 2.9.1. It remains to prove (a) ⇒ (b). (a) ⇒ (b): For each U ∈ cov(M), choose V ∈ cov(M) so that st3 V ≺ U. From Remark 1.12 on Hanner’s characterization of ANRs (Theorem 1.13.20), we have a locally finite-dimensional simplicial complex K with card K (0)  w(M) and maps f : M → |K| and g : |K| → M such that gf V idM , where K is the nerve of an open refinement of V and f is a canonical map. By Theorem 1.7.3, ϕ = id : |K| → |K|m is an OK -homotopy equivalence. Let ψ : |K|m → |K| be an OK -homotopy

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2 Fundamental Results on Infinite-Dimensional Manifolds

inverse of ϕ. Since f −1 (OK ) ≺ V, it follows that g(OK ) ≺ gf (V) ≺ st V, hence gψϕf st V gf V id. Since |K|m ∈ FE by Corollary 2.4.7, gψ is V-close to a closed embedding h : |K|m → M by the Strong Universality Theorem 2.3.15. Then, hϕf is st2 V-close to id. Thus, it suffices to construct a map k : |K|m → M such that k is V-close to h and A ∩ cl k(|K|) = ∅. Since h is a closed embedding, there exists W ∈ cov(M) such that W ≺ V and h−1 (st W) ≺ OK ≺ SK . Take open covers of M as follows: ∗

∗

∗

W  W0  W1  · · · . By induction, we shall construct E-deficient closed embeddings hn : |K|m → M, n ∈ ω, such that hn is Wn -close to hn−1 , hn ||K (n−1)| = hn−1 ||K (n−1)|, and hn (|K (n) |) ∩ A = ∅, where h−1 = h. Since K (0) is discrete in |K|m and h is a closed embedding, M has a discrete open collection {Uv | v ∈ K (0) } such that h(v) ∈ Uv . Choose W0 ∈ cov(M) so that st W0 ≺ W0 and W0 ≺ {Uv | v ∈ K (0)} ∪ {M \ h(K (0) )}. Then, {st(h(v), W0 ) | v ∈ K (0)} is discrete in M. Since A is a Z-set in M, h is W0 close to a map k0 : |K|m → M \A. Then, k0 (K (0)) is discrete in M. Since k0 (|K (0)|) is E-deficient in M by Proposition 2.3.8, it follows that k0 ||K (0)| is an E-deficient closed embedding into M. By the Strong Universality Theorem 2.3.15, we have an E-deficient closed embedding h0 : |K|m → M such that h0 ||K (0)| = k0 ||K (0)| and h0 is W0 -close to k0 , hence h0 is W0 -close to h. Now, assume that hn−1 has been constructed. Since hn−1 (|K (n−1) |) and A are disjoint closed sets in M, we have an open set U in M such that hn−1 (|K (n−1) |) ⊂ U ⊂ cl U ⊂ M \ A. For each n-simplex σ ∈ K, let ˆ ). Cσ = σ \ h−1 n−1 (U ) ⊂ rint σ ⊂ OSd K (σ Then, {Cσ | σ ∈ K (n) \ K (n−1) } is discrete in |K|m . Since hn−1 is a closed embedding, we can use a method similar to the above to choose an open starrefinement Wn of Wn such that 

  st(hn−1 (Cσ ), Wn )  σ ∈ K (n) \ K (n−1) is discrete in M.

Since A is a Z-set in M, we apply the Homotopy Extension Theorem 1.13.11 to −1 obtain a map kn : |K|m → M \ A such that kn |h−1 n−1 (cl U ) = hn−1 |hn−1 (cl U ) and

2.9 Z-Sets and Strong Z-Sets in E-Manifolds

181

kn Wn hn−1 . Then, kn (|K (n) |) is closed in M. Indeed, kn (|K (n) |) =



  kn (Cσ ) ∪ hn−1 |K (n) | ∩ h−1 n−1 (cl U )

σ ∈K (n) \K (n−1)

and {kn (Cσ ) | σ ∈ K (n) \ K (n−1) } is discrete in M. By the Strong Universality Theorem 2.3.15, we have an E-deficient closed embedding hn : |K|m → M such that hn ||K (n−1)| = hn−1 ||K (n−1) |, hn (|K (n) |) ∩ A = ∅, and hn is Wn -close to kn , hence hn is Wn -close to hn−1 . Since K is locally finite-dimensional, we have a map k : |K|m → M defined by k||K (n) | = hn ||K (n) | that is W-close (so V-close) to h. Each a ∈ A is contained in some W ∈ W. Since h−1 (st W) ≺ SK , we have v ∈ K (0) such that h−1 (st(W, W)) ⊂ | St(v, K)|. Because of the local finite-dimensionality of K, St(v, K) is contained in some K (n) . For each x ∈ k −1 (W ), we have h(x) ∈ st(W, W), so x ∈ h−1 (st(W, W)) ⊂ |K (n) |. Then, we have k(|K|) ∩ W ⊂ k(|K (n) |). Since k(|K (n) |) = hn (|K (n) |) is closed in M and hn (|K (n) |) ∩ A = ∅, it follows that V = W \ hn (|K (n) |) is an open neighborhood of a in M and V ∩ k(|K|) = ∅. Thus, A ∩ cl k(|K|) = ∅. This completes the proof. " ! For a closed set in a homotopy dense subset of an E-manifold, conditions (a) and (b) in Theorem 2.9.3 above are equivalent to each other, that is, Proposition 2.9.4 Let X be a homotopy dense subset of an ANR M. Suppose that every Z-set in M is a strong Z-set in M. Then, every Z-set in X is also a strong Z-set in X. Proof Let A ⊂ X be a Z-set in X. For each open cover U of X, we have a collection  of open sets in M such that U|X   is open in M and X U = U. Then U = U  Since X is homotopy dense in U . Let V ∈ cov(X) be a star-refinement of U. is homotopy dense in M, it is easy to see that clM A is a Z∞ -set in M, which means that it is a Z-set in M by Theorem 2.8.6, and hence it is a strong Z-set in M by the assumption. Then, clU A = U ∩ clM A is a strong Z-set in U by Proposition 2.8.14. Thus, there is a map f : U → U such that f is V-close to idU and clU f (U ) ∩ clU A = ∅. Choose an open refinement W of V such that if a map f  : U → U is W-close to f then clU f  (U ) ∩ clU A = ∅. Since X is homotopy dense in U , there exists a map g : U → X that is W-close to idU . Then, the map h = gf |X is U-close to idX . Since gf is W-close to f , we have clU gf (U ) ∩ clU A = ∅, hence clX h(X) ∩ A = ∅. Therefore, A is a strong Z-set in X. " ! An embedding h : A → X is called a Z-embedding (resp. a strong Zembedding) if h(A) is a Z-set (resp. a strong Z-set) in X. By Theorem 2.9.3, an E-deficient closed embedding into an E-manifold is simply a Z-embedding. A Z-submanifold of an E-manifold M is a Z-set in M that is an E-manifold.

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By Theorem 2.9.3, the Collaring Theorem 2.5.9 can be reformulated (in a slightly stronger manner) as follows: Theorem 2.9.5 (COLLARING) Let M and N be E-manifolds such that N ⊂ M. Then, N is a Z-submanifold of M if and only if N is a collared closed set in M. ! " Remark 2.19 Due to Theorem 2.9.5 above, a Z-submanifold of M may be considered as a boundary of M, but it is not unique. By Theorem 2.9.3, we can restate the Strong Universality Theorem 2.3.15 and the Unknotting Theorem 2.5.12 as follows: Theorem 2.9.6 (STRONG UNIVERSALITY) Let M be an E-manifold and A ∈ FE with a closed set B ⊂ A and a map f : A → M such that f |B is a Z-embedding. Then, for each U ∈ cov(M), there exists a Z-embedding h : A → M such that h|B = f |B and h is U-close to f . " ! Theorem 2.9.7 (Z-SET UNKNOTTING) Let A be a Z-set in an E-manifold M and h : A → M be a Z-embedding with h  id by a homotopy f : A × I → M. Then, for each open cover U of M, h can be extended to a homeomorphism h˜ : M → M that is ambiently invertibly U∗ -isotopic to id, where U∗ ={st(f ({x} × I), U) | x ∈ A}. " ! As a corollary of the Negligibility Theorem 2.4.3, we have the following: Theorem 2.9.8 (NEGLIGIBILITY OF Z-SETS) For every Z-set A in an Emanifold M, the inclusion M \ A ⊂ M is a near-homeomorphism. ! " This can also be obtained from Theorems 2.6.3 and 2.8.6. When E is a complete normed linear space, every E-manifold is completely metrizable. Hence, the following is obtained from Theorem 2.4.4: Theorem 2.9.9 (NEGLIGIBILITY OF Zσ -SETS) When E is complete, if A is a Zσ -set in an E-manifold M, the inclusion M \ A ⊂ M is a nearhomeomorphism. " ! Remark 2.20 In Q-manifolds, every Z-set is a strong Z-set because of the local compactness. The Q-manifold version of Theorem 2.9.3 will be proved in the next section. The results for Q-manifolds corresponding to Theorems 2.9.5, 2.9.6, and 2.9.7 will also be established in Sect. 2.11. On the other hand, Theorem 2.9.8 has no Q-manifold versions. In fact, Q × I ≈ Q × [0, 1) ≈ Q × (0, 1).

2.10 Z-Sets in the Hilbert Cube and Q-Manifolds In this section, we show that a closed set in Q is a Z-set if and only if it is Qdeficient, and that every homeomorphism between Z-sets in Q can be extended to a homeomorphism of Q onto itself. Although we have denoted by pri the projection of a product space onto the i-th factor, when several product spaces are discussed,

2.10 Z-Sets in the Hilbert Cube and Q-Manifolds

183 1 ai

1 ai i

i −1

−ai

−ai −1 − a 1 −1

t

−1 − a 1 −1

a1 1

a1 1

Fig. 2.22 The homeomorphism ϕi : [−1, 1]2 → [−1, 1]2

the same pri is used for deferent projections. So, in this section, to avoid confusion, the projection of Q = [−1, 1]N onto the i-th factor is denoted by πi instead of pri . For the Hilbert cube Q = [−1, 1]N, we use d ∈ Metr(Q) defined as follows: d(x, y) =



2−i |x(i) − y(i)|.

i∈N

 Then, Q is isometric to the subspace n∈N [−2−n , 2−n ] of the Banach space 1 by the natural bijection x → (2−i x(i))i∈N . Recall that I (Q) = (−1, 1)N is the pseudo-interior of Q. Lemma 2.10.1 For every compact set A ⊂ I (Q), there exists a homeomorphism h : Q → Q such that pr1 h(A) = {0} and h(I (Q)) = I (Q).  Proof Due to its compactness, A  ⊂ i∈N [−ai , ai ] for some 0 < ai < 1, i ∈ N. Then, we may assume that A = i∈N [−ai , ai ]. For each i > 1, as illustrated in Fig. 2.22, we have a homeomorphism ϕi : [−1, 1]2 → [−1, 1]2 with the following properties: pr1 ϕi = pr1 , ϕi |∂[−1, 1]2 = id and −1 diam ϕi ([−a1, a1 ] × [−ai , ai ]) ∩ pr−1 for each t ∈ [−1, 1]. 2 (t)  i

We define a map ϕ : Q → Q as follows: (pr1 ϕ(x), pri ϕ(x)) = ϕi (x(1), x(i)) for every i > 1. It is easy to see that ϕ is a homeomorphism and ϕ(I (Q)) = I (Q). We write Q = [−1, 1] × Q1 , where Q1 = [−1, 1]N\{1}. Let p : Q → Q1 be the projection, that is, p(x) = (x(2), x(3), . . . ). For each x, y ∈ ϕ(A), p(x) = p(y) implies x(1) = y(1). Indeed, for every i ∈ N, we have |x(1) − y(1)|  i −1 because x(i) = y(i). Then, p|ϕ(A) : ϕ(A) → Q1 is

184

2 Fundamental Results on Infinite-Dimensional Manifolds The graph of pr1 g (A) Q1 B

The graph of f

g (A) pr1

−1 − a 1

a1 1

−1

0

1

Fig. 2.23 The homeomorphism ψ

injective, hence it is an embedding. Let B = pϕ(A) and g = (p|ϕ(A))−1 : B → ϕ(A). Note that ϕ(A) is the graph of the map pr1 g : B → [−a1, a1 ]. By the Tietze Extension Theorem 1.3.1, pr1 g can be extended to a map f : Q1 → [−a1 , a1 ]. Then, there exists a homeomorphism ψ : Q → Q such that pψ = p, ψ|{±1} × Q1 = id, and ψ(f (x), x) = (0, x) for every x ∈ Q1 . The homeomorphism ψϕ is the desired one. Refer to Fig. 2.23. " ! Lemma 2.10.2 For every compactum A ⊂ I (Q) and ε > 0, there exists a homeomorphism h : Q → Q2 = Q × Q such that h(A) ⊂ I (Q) × {0}, h(I (Q)) = I (Q)2 = I (Q) × I (Q), and d(pr1 h, id) < ε (equivalently, d(h−1 , pr1 ) < ε), where pr1 : Q2 → Q is the projection onto the first factor of Q2 . In particular, A is Q-deficient in Q and it is also I (Q)-deficient in I (Q). Proof Choose n ∈ N so that 2−n+1 < ε. Let pn : Q → [−1, 1]n denote the projection onto the first n-coordinates. Let ϕ : Q → [−1, 1]n × QN be a homeomorphism such that ϕ(I (Q)) = (−1, 1)n × I (Q)N and q0 ϕ = pn , where q0 : [−1, 1]n × QN → [−1, 1]n is the projection. For each i ∈ N, let qi : [−1, 1]n × QN → Q be the projection onto the i-th factor of QN . Then, ϕ(A) ⊂

#

qi (ϕ(A)) ⊂ (−1, 1)n × I (Q)N .

i∈ω

By Lemma 2.10.1, we have homeomorphisms hi : Q → Q, i ∈ N, such that p1 hi (qi (ϕ(A))) = {0} and hi (I (Q)) = I (Q), where p1 = π1 (= pr1 ) is the projection onto the first factor of Q = [−1, 1]N. We define a homeomorphism h : QN → QN by h(x) = (hi (x(i)))i∈N. Let ψ :

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185

[−1, 1]n × QN → Q2 be a homeomorphism such that pn pr1 ψ = q0 , ψ((−1, 1)n × I (Q)) = I (Q)2 and pr2 ψ(x) = (p1 (qi (x)))i∈N for every x ∈ [−1, 1]n × QN , where pr1 , pr2 : Q2 → Q are the projections onto the first and the second factors of Q2 respectively. Then, ψ(id × h)ϕ : Q → Q2 is the desired homeomorphism. Indeed,    ψ(id × h)ϕ(A) ⊂ ψ q0 ϕ(A) × i∈N hi qi ϕ(A) ⊂ Q × {0}. Since pn pr1 ψ(id × h)ϕ = q0 (id × h)ϕ = q0 ϕ = pn , it follows that d(pr1 ψ(id × h)ϕ, id)  2−n+1 < ε.

" !

We will prove that every homeomorphism h : A → B between compacta in I (Q) can be extended to a homeomorphism h˜ : Q → Q. In the proof, we have to modify Klee’s Trick (Theorem 1.13.7). To this end, we need the following: Lemma 2.10.3 For each map f : Q → I (Q), there exists a homeomorphism ϕ : Q2 → Q2 such that pr1 ϕ = pr1 , ϕ(x, 0) = (x, f (x)) for each x ∈ Q, and ϕ(I (Q)2 ) = I (Q)2 . Proof We can define ϕ, ψ : Q2 → Q2 as follows: for each (x, y) ∈ Q2 , pr1 ϕ(x, y) = pr1 ψ(x, y) = x, pr2 ϕ(x, y)(n) =

 (1 + f (x)(n))y(n) + f (x)(n)

if − 1  y(n)  0,

(1 − f (x)(n))y(n) + f (x)(n) if 0  y(n)  1, ⎧ y(n) − f (x)(n) ⎪ ⎪ if − 1  y(n)  f (x)(n), ⎨ 1 + f (x)(n) pr2 ψ(x, y)(n) = ⎪ y(n) − f (x)(n) ⎪ ⎩ if f (x)(n)  y(n)  1. 1 − f (x)(n)

Then, ϕ and ψ are continuous, ψϕ = id and ϕψ = id. Therefore, ϕ is a homeomorphism with ϕ −1 = ψ. The other required conditions come from the definition. " ! Theorem 2.10.4 Let h : A → B be a homeomorphism between compacta in I (Q) with d(h, id) = supx∈A d(h(x), x) < ε. Then, h can be extended to a ˜ id) = supx∈Q d(h(x), ˜ homeomorphism h˜ : Q → Q such that d(h, x) < ε and ˜h(I (Q)) = I (Q).  Proof We may regard Q =  n∈N [−2−n , 2−n ] ⊂ 1 , and the pseudo-interior I (Q) of Q is identified with n∈N (−2−n , 2−n ). Let δ = ε − d(h, id) > 0. By

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2 Fundamental Results on Infinite-Dimensional Manifolds

Lemma 2.10.2, we have a homeomorphism ϕ : Q → Q2 such that d(pr1 ϕ, id) < δ/3 and ϕ(A ∪ B) ⊂

#

(−2−n , 2−n ) × {0}.

n∈N

Let A = pr1 (ϕ(A)) and B  = pr1 (ϕ(B)). Then, ϕ(A) = A × {0} and ϕ(B) = B  × {0}. We have the homeomorphism f = pr1 ϕhϕ −1 i|A : A → B  , where i : Q → Q × {0} ⊂ Q2 is the natural injection (i.e., i(x)  = (x, 0) for each x∈ Q). Choose 0 < an < 2−n , n ∈ N, so that A ∪ B   ⊂ n∈N [−an , an ] and −n − a ) < δ/3. We have the retraction r : 1 → n n∈N (2 n∈N [−an , an ] defined as follows:  min{x(n), an } if x(n)  0, r(x)(n) = max{x(n), −an } if x(n) < 0. Then, for each x, y ∈ 1 , r(x) − r(y)  x − y because |r(x)(n) − r(y)(n)|  |x(n) − y(n)| for every n ∈ N. Since n∈N (2−n − an ) < δ/3, it follows that d(r|Q, id) < δ/3. We can apply the Dugundji Extension Theorem 1.13.1 with Remark 1.11 to obtain a map g : Q → 1 such that g|A = f − id and g  = f − id = d(f, id), where  ·  is the sup-norm. Then, f˜ = r(g + id) : Q → n∈N [−an , an ] ⊂ Q is an extension of f with f˜ − id = d(f, id). Indeed, for each x ∈ Q, f˜(x) − x = r(g(x) + x) − r(x)  (g(x) + x) − x = g(x), hence f − id  f˜ − id  f − id, that is, f˜ − id = f − id = d(f, id), and if x ∈ A , then f˜(x) = r(g(x) + x) = rf (x) = f (x), that is, f˜|A = f . Moreover, note that d(f˜, r|Q)  d(f˜, id) + d(r|Q, id) < d(f, id) + δ/3. −1 can be On the other hand, by the Dugundji  Extension Theorem 1.13.1, f ∗ extended to a map f : Q → n∈N [−an , an ]. By Lemma 2.10.3, we have homeomorphisms ψi : Q2 → Q2 , i = 1, 2, 3 (Fig. 2.24), such that:

(1) pr1 ψ1 = pr1 , ψ1 (x, 0) = (x, r(x)) for each x ∈ Q, (2) pr2 ψ2 = pr2 , ψ2 (r(x), x) = (f˜(x), x) for each x ∈ Q, pr1 ψ2 − pr1  = f˜ − r < d(f, id) + δ/3, (3) pr1 ψ3 = pr1 , ψ3 (x, f ∗ (x)) = (x, 0) for each x ∈ Q,       −n −n 2 = −n −n 2 (4) ψi n∈N (−2 , 2 ) n∈N (−2 , 2 ) .

2.10 Z-Sets in the Hilbert Cube and Q-Manifolds

187 the graph of r

Q

2

A

id 1

f

3

B

the graph of f

A f

Q the graph of f˜ Fig. 2.24 The homeomorphisms ψ1 , ψ2 , and ψ3

Thus, we have a homeomorphism h˜ = ϕ −1 ψ3 ψ2 ψ1 ϕ : Q → Q. Observe that  #  −n −n (−2−n , 2−n ). h˜ n∈N (−2 , 2 ) = n∈N

For each x ∈ A , ψ3 ψ2 ψ1 (x, 0) = ψ3 ψ2 (x, r(x)) = ψ3 ψ2 (x, x) = ψ3 ψ2 (r(x), x) = ψ3 (f (x), x) = ψ3 (f (x), f −1 (f (x))) = (f (x), 0) = ϕhϕ −1 (x, 0). ˜ Hence, h|A = ϕ −1 ψ3 ψ2 ψ1 ϕ|A = h. Since d(ϕ −1 , pr1 ) = d(pr1 ϕ, id) < δ/3, it ˜ follows that h is δ/3-close to pr1 ψ3 ψ2 ψ1 ϕ = pr1 ψ2 ψ1 ϕ. Therefore, ˜ id)  d(h, ˜ pr1 ψ2 ψ1 ϕ) + d(pr1 ψ2 ψ1 ϕ, pr1 ψ1 ϕ) + d(pr1 ψ1 ϕ, id) d(h, < δ/3 + d(pr1 ψ2 , pr1 ) + d(pr1 ϕ, id) < d(f, id) + δ < ε. Thus, h˜ is the desired one.

" !

Let Homeo(Q) be the space of all homeomorphisms of Q onto itself with the sup-metric d(f, g) = supx∈Q d(f (x), g(x)), where the sup-metric d is not complete but Homeo(Q) is completely metrizable (as is shown in Sect. 2.1). Lemma 2.10.5 Let W = {1} × [−1, 1]N\{1} ⊂ Q. Then, there is a homeomorphism h : Q → Q such that h(W ) ⊂ I (Q).

188

2 Fundamental Results on Infinite-Dimensional Manifolds [−1 1]2

/2 Fig. 2.25 The homeomorphism ϕ

Proof First, we shall find a homeomorphism g1 : Q → Q such that g1 (W ) ⊂ {1} × (−1, 1)N\{1}. We consider the following closed subspace of Homeo(Q):    H = f ∈ Homeo(Q)  f (W ) ⊂ W . For each n > 1, the following set is open in H :    Hn = f ∈ H  prn f (W ) ⊂ (−1, 1) .  We shall show that each Hn is dense in H . Then, n∈N Hn would be dense in H by the Baire Category Theorem 1.3.15. Thus, the desired g1 could be found in  n∈N Hn . To see that Hn is dense in H , let f ∈ H and ε > 0. It suffices to find h ∈ Hn with d(h, id) < ε, because hf ∈ Hn and d(hf, f ) < ε. Take a homeomorphism ϕ : [−1, 1]2 → [−1, 1]2 as illustrated in Fig. 2.25 such that ϕ|[−1, 1 − 12 ε] × [−1, 1] = id, |pr2 ϕ(x) − pr2 (x)|  ε for each x ∈ [−1, 1]2 and ϕ({1} × [−1, 1]) = {1} × [−1 + 12 ε, 1 − 12 ε]. Then, the desired h ∈ Hn can be defined as follows: h(x)(i) = x(i), i ∈ N \ {1, n}, and (h(x)(x), h(x)(n)) = ϕ(x(1), x(n)) for each x ∈ Q. We have now obtained a homeomorphism g1 : Q → Q such that g1 (W ) ⊂ {1} × (−1, 1)N\{1}. Then, applying Lemma 2.10.2, we can obtain a homeomorphism g2 : Q → Q2 such that g2 g1 (W ) ⊂ {(1, 0, 0, . . . )} × I (Q). By the homogeneity of Q (Theorem 2.1.2), there is a homeomorphism g3 : Q → Q such that g3 (1, 0, 0, . . . ) = 0. Take a homeomorphism ψ : Q2 → Q with

2.10 Z-Sets in the Hilbert Cube and Q-Manifolds

189

ψ(I (Q)2 ) = I (Q) and define a homeomorphism h = ψ(g3 × id)g2 g1 : Q → Q. Then, h(W ) ⊂ I (Q).

" !

Theorem 2.10.6 For a closed set A in Q, the following are equivalent: (a) (b) (c) (d)

A is a Z-set in Q; A is Q-deficient in Q; There exists a homeomorphism h : Q → Q such that h(A) ⊂ I (Q); For each ε > 0, there exists a homeomorphism h : Q → Q such that h(A) ⊂ I (Q) and d(h, id) < ε.

Proof It is easy to see that {0} is a Z-set in Q, hence Q × {0} is a Z-set in Q × Q. Then, we have the implication (b) ⇒ (a). The implication (c) ⇒ (b) follows from Lemma 2.10.2 and the implication (d) ⇒ (c) is trivial. (a) ⇒ (d): For each n ∈ N, let −1 W2n−1 = pr−1 n (1) and W2n = prn (−1).

We consider the following subsets of Homeo(Q):    Hn = h ∈ Homeo(Q)  h(A) ∩ Wn = ∅ . Then, each Hn is open  in Homeo(Q). We shall show that each Hn is dense in Homeo(Q). Then, n∈N Hn is dense in Homeo(Q) by the Baire Category Theorem 1.3.15, which gives condition (d). To see that Hn is dense in Homeo(Q), let f ∈ Homeo(Q) and ε > 0. By Lemma 2.10.5, we have a homeomorphism ϕ ∈ Homeo(Q) such that ϕ(Wn ) ⊂ I (Q). By the uniform continuity of ϕ −1 , there is some δ > 0 such that d(x, y) < δ implies d(ϕ −1 (x), ϕ −1 (y)) < ε. Since ϕf (A) is a Z-set in Q and I (Q) is homotopy dense in Q, we have a map g  : ϕ(Wn ) → I (Q) \ ϕf (A) such that d(g  , id) < δ/2. Choose m ∈ N so that   2−m+1 < min δ/2, dist(g  ϕ(Wn ), ϕf (A)) and take an embedding k : ϕ(Wn ) → (−1, 1)N\{1,...,m} . Then, we can define an embedding g : ϕ(Wn ) → I (Q) \ ϕf (A) by g(x) = (pr1 g  (x), . . . , prm g  (x), k(x)).

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2 Fundamental Results on Infinite-Dimensional Manifolds

Since d(g, g  )  2−m+1 < δ/2, it follows that d(g, id) < δ. By Theorem 2.10.4, g can be extended to a homeomorphism g˜ : Q → Q such that d(g, ˜ id) < δ, which implies ˜ < ε. d(ϕ −1 g˜ −1 ϕf, f ) = d(ϕ −1 , ϕ −1 g) −1 −1 Since gϕ(W ˜ n ) ∩ ϕf (A) = ∅, we have Wn ∩ ϕ g˜ ϕf (A) = ∅, which means −1 −1 ϕ g˜ ϕf ∈ Hn . Hence, Hn is dense in Homeo(Q). This completes the proof. ! "

The following is a combination of Theorems 2.10.6 and 2.10.4: Corollary 2.10.7 Every homeomorphism h : A → B between Z-sets in Q can be extended to a homeomorphism h˜ : Q → Q. " ! Corollary 2.10.8 Every homeomorphism h : A → B between Z-sets in Q \ {0} can be extended to a homeomorphism h˜ : Q \ {0} → Q \ {0}. Proof Since A ∪{0} and B ∪{0} are closed in Q and {0} is a Z-set in Q, it is easy to see that A∪{0} and B ∪{0} are Z-sets in Q. As is easily observed, h can be extended to the homeomorphism h : A ∪ {0} → B ∪ {0} by h (0) = 0. By Corollary 2.10.7, we can extend h to a homeomorphism h¯ : Q → Q. The desired h˜ can be obtained ¯ as the restriction of h. " ! Recall that every Z-set in a Q-manifold is a strong Z-set by Proposition 2.8.1. The following is the Q-manifold version of Theorem 2.9.3: Theorem 2.10.9 For a closed set A in a Q-manifold M, the following are equivalent: (a) A is a Z-set in M; (b) A is Q-deficient in M; (c) There is a homeomorphism h : M → M × I such that h(x) = (x, 0) for each x ∈ A; (d) A is contained in some collared set in M. Proof The implications (c) ⇒ (d) ⇒ (a) are easy. We shall show the implications (a) ⇒ (b) ⇒ (c). (a) ⇒ (b): Each a ∈ A has an open neighborhood U in M with an open embedding ϕ : U → Q. Take another open neighborhood V of a in M so that cl V is compact and cl V ⊂ U . By Theorem 2.8.7(2), ϕ(A ∩ U ) is a Z-set in ϕ(U ), hence ϕ(A ∩ cl V ) is also a Z-set in ϕ(U ). Since ϕ(A ∩ cl V ) is compact, it is a Z-set in Q by Theorem 2.8.7(3). It follows from Theorem 2.10.6 that ϕ(A ∩ cl V ) is Q-deficient in Q. By Corollary 2.3.10(2), ϕ(A ∩ V ) = ϕ(A ∩ cl V ) ∩ ϕ(V ) is Q-deficient in ϕ(V ). Hence, A ∩ V is Q-deficient in V . This means that A is locally Q-deficient in M. It follows from Proposition 2.3.14 that A is Q-deficient in M. (b) ⇒ (c): Let ϕ : M → M×Q be a homeomorphism such that ϕ(x) = (x, 0) for each x ∈ A (Corollary 2.3.10(4)). On the other hand, Q × I ≈ Q is homogeneous (Theorem 2.1.2), hence we have a homeomorphism ψ : Q → Q × I such that

2.11 Complementary Basic Results on Q-Manifolds

191

ψ(0) = (0, 0). The following is the desired homeomorphism: h = (ϕ −1 × idI )(idM × ψ)ϕ : M → M × I.

" !

Using Theorem 2.10.9, the Strong Universality Theorem 2.3.16 and the Closed Embedding Approximation Theorem 2.3.17 can be rewritten as follows: Theorem 2.10.10 (STRONG UNIVERSALITY) Let M be a Q-manifold and A ∈ M0 with a closed set B ⊂ A and a map f : A → M such that f |B is a Zembedding. Then, for each U ∈ cov(M), there exists a Z-embedding h : A → M such that h|B = f |B and h is U-homotopic to f . " ! Theorem 2.10.11 (Z-EMBEDDING APPROXIMATION) Let M be a Q-manifold and A be a locally compact separable metrizable space with a closed set B ⊂ A and a proper map f : A → M such that f |B is a Z-embedding. Then, for each U ∈ cov(M), there exists a Z-embedding h : A → M such that h|B = f |B and p h U f (i.e., h is properly U-homotopic to f ). " !

2.11 Complementary Basic Results on Q-Manifolds In this section, we establish the Q-manifold versions of the Collaring Theorem 2.9.5 and the Unknotting Theorem 2.9.7. We also show the Classification and Open Embedding Theorems for R+ -stable Q-manifolds, where a Q-manifold M is said to be R+ -stable if M × R+ ≈ M. Moreover, we give a characterization of R+ stability for Q-manifolds. The Classification Theorem for all Q-manifolds will be shown in the next chapter. First of all, we shall prove the following Collaring Theorem: Theorem 2.11.1 (COLLARING) Let M and N be Q-manifolds such that N is closed in M. Then, N is a Z-submanifold of M if and only if N is collared in M. Proof The “if” part is trivial. We show the “only if” part. Since a locally collared set is collared (Theorem 2.5.7), it suffices to show that N is locally collared in M. Each x ∈ N has an open neighborhood U in M with an open embedding ϕ : U → Q. Let W = pr−1 1 (1) ⊂ Q, where pr1 : Q → [−1, 1] is the projection onto the first factor. We can easily find an open neighborhood V of x in N and an embedding ψ : cl V → W such that cl V is compact, cl V ⊂ U , and ψ(V ) is open in W . By Theorem 2.8.7(2), cl V is a Z-set in U , hence ϕ(cl V ) is also a Z-set in ϕ(U ). Since ϕ(cl V ) is compact, it is a Z-set in Q by Theorem 2.8.7(3). On the other hand, since W is a Z-set in Q, ψ(cl V ) is also a Z-set in Q. Applying Corollary 2.10.7, we can obtain a homeomorphism h : Q → Q such that hϕ| cl V = ψ. Then, hϕ(U ) is an open set in Q and ψ(V ) = hϕ(V ) is open in hϕ(U ) ∩ W . Hence, hϕ(V ) is collared in hϕ(U ) (cf. (G-1) in the proof of Theorem 2.5.7), which implies that V is collared in U . Thus, N is locally collared in M. " !

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2 Fundamental Results on Infinite-Dimensional Manifolds

Due to the above theorem, any Z-submanifold of each Q-manifold M can be regarded as a boundary of M, but it is not unique (cf. Remark 2.19, p. 182). Note that the Open Embedding Theorem does not hold for Q-manifolds, because any compact Q-manifold cannot be embedded into Q as an open set unless it is homeomorphic to Q. However, combining the Collaring Theorem 2.11.1 with Theorem 2.10.11, we can see that every compact Q-manifold M can be embedded into Q as a collared closed set. Since M × I ≈ M by Theorem 2.3.13, we can prove the following Q-manifold version of Theorem 2.5.11 in the same way: Theorem 2.11.2 Every compact Q-manifold M can be embedded in Q as a closed set having a Q-manifold bi-collared boundary. " ! For a non-compact Q-manifold M, let αM = M ∪ {∞} be the one-point compactification of M. Since αM is compact metrizable, we have a Z-embedding h : αM → Q. Then, h(M) is a Z-submanifold of Q \ {h(∞)}, hence it is collared in Q \ {h(∞)} by the Collaring Theorem 2.11.1. By the homogeneity of Q (Theorem 2.1.2), Q \ {h(∞)} ≈ Q \ {0}. The following can be obtained similarly: Theorem 2.11.3 Every Q-manifold M can be embedded in Q \ {0} as a closed set whose boundary is bi-collared and also a Q-manifold. " ! By the above argument, the following Q-manifold version of the Open Embedding Theorem 2.5.10 can also be obtained: Theorem 2.11.4 (OPEN EMBEDDING) For every Q-manifold M, M × [0, 1) can be embedded in Q as an open set. " ! In the Open Embedding Theorem 2.11.4, an open embedding of M × [0, 1) into Q is defined as the composition of the homeomorphism (h × id)|M × [0, 1) and a collar k : h(M) × [0, 1) → Q \ {0} of h(M), where h(∞) = 0. Consider the following: M ≈ M ×I

⊃

M × [0, 1)

h×id ≈

h(M) × [0, 1)

k

\ {0}.

For each Z-set A in M, h(A) is a Z-set in Q \ {0}. On the other hand, by Theorem 2.10.9, we have a homeomorphism f : M → M × I such that f (x) = (x, 0) for each x ∈ A. Then, U = f −1 (M × [0, 1)) is a homotopy dense open set in M with A ⊂ U and k(h × id)f |U : U → Q \ {0} is an open embedding. Thus, we have the following: Corollary 2.11.5 Let M be a Q-manifold and A a Z-set in M. Then, there is an open dense set U in M with an open embedding g : U → Q \ {0} such that A ⊂ U and g(A) is a Z-set in Q \ {0}. " ! Modifying the proof of the Unknotting Theorem 2.5.12, we can prove the following Z-Set Unknotting Theorem for Q-manifolds:

2.11 Complementary Basic Results on Q-Manifolds

193

Theorem 2.11.6 (Z-SET UNKNOTTING) Let M be a Q-manifold and h : A → B a homeomorphism between Z-sets in M. If h is properly homotopic to id in M, then h can be extended to a homeomorphism h˜ : M → M. Moreover, if f : A × I → M is a proper homotopy with f0 = id and f1 = h and U is an open cover of f (A × I) in M, then h˜ can be chosen so as to be ambiently invertibly U∗ -isotopic to id, where U∗ = {st(f ({x} × I), U) | x ∈ A}. Proof Take V ∈ cov(M) such that st3 V ≺ U, V is locally finite, and cl V is compact for each V ∈ V. Observe that V[cl V ] is finite for each V ∈ V. Then, it follows that sti V, i = 1, 2, 3, are also locally finite and the closures of these members are compact. First, we shall construct an ambient invertible st V-isotopy ψ : M × I → M such that A ∩ ψ1 (B) = ∅. Since A ∪ B is a Z-set in M, it is Q-deficient in M. By Corollary 2.3.10(4), we have a homeomorphism k : M × Q → M such that k(x, 0) = x for each x ∈ A ∪ B and k is V-close to prM . We define an ambient invertible isotopy ζ : Q × I → Q as follows: ζt (x) =

 ((1 − t/2)x(1) + t/2, x(2), x(3), . . . )

if x(1)  0,

((1 + t/2)x(1) + t/2, x(2), x(3), . . . )

if x(1)  0.

Then, ζ0 = id and ζ1 (0) = (1/2, 0, 0, . . . ). The desired isotopy ψ : M ×I → M can be defined by ψt = k(id×ζt )k −1 . In the same way as in the proof of Theorem 2.5.12, we have a proper homotopy f  : A × I → M from id to ψ1 h (cf. Proposition 1.3.9). Then, by Theorem 2.10.11, we have a Z-embedding f  : A × I → M such that f  |A × {0, 1} = f  |A × {0, 1} and f  is V-close to f  . On the other hand, we have a Z-embedding j : A → Q\{0} by Corollary 2.11.5. Then, j (A) × I is a Z-set in (Q \ {0}) × [−1, 2] and (Q \ {0}) × [−1, 2] ≈ Q \ {0} by the Stability Theorem 2.3.13. Using Corollary 2.11.5, we have an open neighborhood U of f  (A × I) in M with an open embedding g  : U → (Q \ {0}) × [−1, 2] such that g  f  (A × I) is a Z-set in (Q \ {0}) × [−1, 2]. Using Corollary 2.10.8, we can obtain a homeomorphism g  : (Q \ {0}) × [−1, 2] → (Q \ {0}) × [−1, 2] such that g  g  f  (x, t) = (j (x), t) for each (x, t) ∈ A × I. Then, note that {j (x)} × I = g  g  f  ({x} × I) ⊂ g  g  (st(f  ({x} × I), V)). Let W0 be an open neighborhood of g  g  f  (A × I) in (Q \ {0}) × [−1, 2] such that cl W0 ⊂ g  g  (U ). For each x ∈ A, define W (x) = W0 ∩ g  g  (st(f  ({x} × I), V))

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2 Fundamental Results on Infinite-Dimensional Manifolds

and let W = {W (x) | x ∈ A}. In the same way as the proof of Theorem 2.5.12, we can construct an ambient invertible W-isotopy ϕ  : (Q \ {0}) × [−1, 2] × I → (Q \ {0}) × [−1, 2] such that ϕ0 = id, ϕ  ({x} × I × I) ⊂ W (x), and ϕ1 (j (x), 0) = (j (x), 1) for every x ∈ A, where the last condition means that ϕ1 g  g  |A = g  g  ψ1 h. Since ϕt |(Q \ {0}) × [−1, 2] \ W0 = id, we can define an ambient invertible (g  g  )−1 (W)-isotopy ϕ : M × I → M by ϕt |U = (g  g  )−1 ϕt g  g  and ϕt |M \ U = id. Then, ϕ0 = id and ϕ1 |A = (g  g  )−1 ϕ1 g  g  |A = ψ1 h. Hence, h can be extended to a homeomorphism h˜ = ψ1−1 ϕ1 : M → M. In the same way as for Theorem 2.5.12, we can define an ambient invertible U∗ -isotopy η from id to h˜ = ψ1−1 ϕ1 . Thus, we have the result. " ! Remark 2.21 In the above, the properness of the homotopy is necessary. In fact, A = R × {0, 1} × Q is a Z-set in the Q-manifold M = R × I × Q. We define a homeomorphism h : A → A by h(t, i, x) = ((−1)i t, i, x) for each (t, i, x) ∈ A = R × {0, 1} × Q. Then, h  id in M = R × I × Q, but h cannot be extended to a homeomorphism of M onto itself. Indeed, assume that h is extended to a homeomorphism h˜ : M → M. Let U+ = (0, ∞) × I × Q, U− = (−∞, 0) × I × Q and L = {0} × I × Q. Choose β > 0 so that ˜ − ), (β, 0, 0) = h(β, 0, 0) ∈ h(U ˜ +) ˜ ˜ + )∪ h(U β ∈ prR h(L). Then, {β}×I×Q ⊂ h(U ˜ − ), which contradicts the connectedness of and (β, 1, 0) = h(−β, 1, 0) ∈ h(U {β} × I × Q. Suppose that f : M → N is a proper U-homotopy equivalence between Qmanifolds, where U ∈ cov(N). As observed in Sect. 2.6, Remark 2.9, we can apply the same construction as in the proof of Theorem 2.6.3 to Q-manifolds, where Theorem 2.4.9, the Strong Universality Theorem 2.3.15, the Collaring Theorem 2.5.9, and the Unknotting Theorem 2.5.12 are replaced by Theorem 2.10.9, the Closed Embedding Approximation 2.3.17, the Collaring Theorem 2.11.1, and the Z-setUnknotting Theorem 2.11.6, respectively. Thus, we can obtain the homeomorphism h = ψ −1 ϕ : M × R+ → N × R+ such that prN h is st(st2 U, V)-close to f prM , where V is an open star-refinement of U. Thus, we have the following: Proposition 2.11.7 Let M and N be Q-manifolds and U be an open cover of N. For each proper U-homotopy equivalence f : M → N, there exists a

2.11 Complementary Basic Results on Q-Manifolds

195

homeomorphism h : M × R+ → N × R+ such that prN h is st5 U-homotopic to f prM and h(M × {0}) ⊂ N × {0}. " ! By the way, we have the following: Proposition 2.11.8 Let X and Y be connected locally compact spaces. If f : X → Y is a homotopy equivalence, then there is a proper homotopy equivalence f∗ : X × R+ → Y × R+ such that prY f∗ = f prX . In particular, X  Y ⇒ X × R+  p Y × R+ . Proof Let g : Y → X be a homotopy inverse of f , that is, gf  id and fg  id. Let ϕ : X × I → X and ψ : Y × I → Y be homotopies such that ϕ0 = id, ϕ1 = gf , ψ0 =  id, and ψ1 = fg. Since  X and Y be connected locally compact, we can write X = n∈N Xn and Y = n∈N Yn , where each Xn and Yn are compact and contained in int Xn+1 and int Yn+1 , respectively. Using Urysohn maps, we can construct maps α : X → R+ and β : Y → R+ so that α(bd Xn ) = n, α(Xn \ int Xn−1 ) = [n − 1, n], β(bd Yn ) = n, β(Yn \ int Yn−1 ) = [n − 1, n]. We define maps f∗ : X × R+ → Y × R+ and g∗ : Y × R+ → X × R+ as follows: f∗ (x, t) = (f (x), max{t, α(x)}), g∗ (y, t) = (g(x), max{t, β(x)}). Then, prY f∗ = f prX . Each compact set B in Y × R+ is contained in some Yn × [0, n]. If x ∈ Xn+1 or t ∈ [0, n + 1], then max{t, α(x)}  n + 1, which implies that f∗ (x, t) ∈ Yn × [0, n]. Hence, f∗−1 (B) ⊂ f∗−1 (Yn × [0, n]) ⊂ Xn+1 × [0, n + 1], which implies that f∗−1 (B) is compact. Thus, f∗ is proper. Similarly, it can be shown that g∗ is proper. To show that g∗ f∗ p id and f∗ g∗ p id, we define a homotopies ϕ˜ : X × R+ × I → X × R+ and ψ˜ : Y × R+ × I → Y × R+ as follows: ϕ(x, ˜ t, s) = ˜ ψ(y, t, s) =

 (ϕ(x, 2s − 1), max{t, α(x), β(f (x))})

if 1/2  s  1,

(x, (1 − 2s)t + 2s max{t, α(x), β(f (x))}) if 0  s  1/2,  (ψ(y, 2s − 1), max{t, β(y), α(g(y))}) if 1/2  s  1, (y, (1 − 2s)t + 2s max{t, β(y), α(g(y))})

if 0  s  1/2.

Then, ϕ˜0 = id and ψ˜ 0 = id. For each (x, t) ∈ X × R+ , ϕ˜1 (x, t) = (ϕ1 (x), max{t, α(x), β(f (x))}) = g∗ f∗ (x, t).

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2 Fundamental Results on Infinite-Dimensional Manifolds

Hence, ϕ˜1 = g∗ f∗ . Similarly, ψ˜ 1 = f∗ g∗ . To see that ϕ˜ is proper, let A be a compact set in X ×R+ . Then, A ⊂ Xn ×[0, n] for some n ∈ N. If x ∈ Xn+1 or t ∈ [0, n+1], then max{t, α(x), β(f (x))}  max{t, α(x)}  n + 1, which implies that ϕ(x, ˜ t, s) ∈ Xn × [0, n] for each 1/2  s  1. Hence, ϕ˜ s−1 (Xn × [0, n]) ⊂ Xn+1 × [0, n + 1] for each 1/2  s  1. If x ∈ Xn or t ∈ [0, n], then for each 0  s  1/2, ϕ(x, ˜ t, s) = (x, (1 − 2s)t + 2s max{t, α(x), β(f (x))}) ∈ Xn × [0, n]. Hence, it follows that ϕ˜ s−1 (Xn × [0, n]) ⊂ Xn × [0, n] for each 0  s  1/2. Thus, we have ϕ˜ −1 (A) ⊂ Xn+1 × [0, n + 1] × I, which implies that ϕ˜ −1 (A) is compact. Similarly, we can show that ψ˜ is proper. Consequently, g∗ f∗ p id and f∗ g∗ p id. " ! For every Q-manifold M, M × R+ × R+ ≈ M × I × R+ ≈ M × R+ , where R+ ≈ [0, 1) and [0, 1)2 ≈ [0, 1) × I by Lemma 2.1.4. Combining Propositions 2.11.7 and 2.11.8, we have: Theorem 2.11.9 Let M and N be Q-manifolds. For each homotopy equivalence f : M → N, there exists a homeomorphism h : M × R+ → N × R+ such that prN h  f prM . " ! Recall that a space X is R+ -stable if X × R+ ≈ X. As we saw above, for every Q-manifold M, M × R+ is an R+ -stable Q-manifold. Lemma 2.11.10 For each R+ -stable Q-manifold M, there is a homeomorphism h : M × R+ → M that is homotopic to the projection. Proof In the following, we replace R+ with [0, 1). Let ϕ : M → M × [0, 1) be a homeomorphism. Since M × [0, 1) is a Q-manifold and Q × I ≈ Q, we apply the Stability Theorem 2.3.13 to obtain a homeomorphism f : M × [0, 1) × I → M × [0, 1) that is homotopic to the projection. On the other hand, as is observed above, we have a homeomorphism ψ : [0, 1)2 → [0, 1) × I by Lemma 2.1.4. Then, we can

2.11 Complementary Basic Results on Q-Manifolds

197

define a homeomorphism h : M × [0, 1) → M so as to make the following diagram commutative: M × [0, 1)

ϕ×id[0,1)

M × [0, 1) × [0, 1) idM ×ψ

M × [0, 1) × I

h

f

M

M × [0, 1)

ϕ

Since [0, 1) is contractible, it follows that pr[0,1) ψ  pr[0,1) , hence the following diagram is commutative up to homotopy: M × [0, 1) × [0, 1) idM ×ψ

M × [0, 1) × I

prM×[0,1)

prM×[0,1)

M × [0, 1)

Note that the bottom prM×[0,1) is homotopic to f . Thus, we have h = ϕ −1 ◦ f ◦ idM × ψ ◦ ϕ × id[0,1)  ϕ −1 ◦ prM×[0,1) ◦ idM × ψ ◦ ϕ × id[0,1)  ϕ −1 ◦ prM×[0,1) ◦ ϕ × id[0,1) = ϕ −1 ◦ ϕ ◦ prM = prM .

" !

Combining Theorem 2.11.9 and Lemma 2.11.10, we can easily obtain the following Classification Theorem for R+ -stable Q-manifolds: Theorem 2.11.11 (CLASSIFICATION) Let M and N be R+ -stable Q-manifolds. Then, every homotopy equivalence f : M → N is homotopic to a homeomorphism. Thus, two R+ -stable Q-manifolds are homeomorphic if they have the same homotopy type. " ! By the Open Embedding Theorem 2.11.4, we have the following: Theorem 2.11.12 (OPEN EMBEDDING) Every R+ -stable Q-manifold can be embedded in Q as an open set. ! " Due to the Classification Theorem 2.11.11 and the Open Embedding Theorem 2.11.12 above, R+ -stable Q-manifolds are tractable among Q-manifolds. In order to characterize such Q-manifolds, we introduce the notion of proper contractibility to infinity. A space X is said to be properly contractible to infinity

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2 Fundamental Results on Infinite-Dimensional Manifolds

provided, for each compact set C in X, there is a proper homotopy h : X × I → X with h0 = idX and h1 (X) ⊂ X \ C. Theorem 2.11.13 A connected Q-manifold M is R+ -stable if and only if M is properly contractible to infinity. Proof The “only if” part follows from the fact that M × R+ is properly contractible to infinity. This fact can be shown as follows: For each compact set C ⊂ M × R+ , choose a > 0 so that C ⊂ M × [0, a). The desired proper homotopy h : M × R+ × I → M × R+ can be defined by h(x, s, t) = (x, s + at) for every (x, s, t) ∈ M × R+ × I. To see the “if” part, it suffices to prove that M × I ≈ M × [0, 1), because M ≈ M × I by the Stability Theorem 2.3.13. Since M is locally compact and σ compact,  there are compact sets Xn , n ∈ N, in M such that Xn ⊂ int Xn+1 and M = n∈N Xn . Take 0 < t1 < t2 < · · · < 1 so that supn∈N tn = 1. We will inductively construct homeomorphisms hn : M × I → M × I, n ∈ N, so as to satisfy the following: (1) hn hn−1 · · · h1 (M × {1}) ⊂ (M \ Xn ) × {1}, (2) hn hn−1 · · · h1 |M × [0, tn ] = hn−1 · · · h1 |M × [0, tn ], (3) hn |Xn−1 × I = id. In the following, keep in mind that every proper map of M into itself can be approximated by closed embeddings (the Z-Embedding Approximation Theorem 2.10.11). Since M is properly contractible to infinity, we have a closed embedding f1 : M × {1} → (M \ X1 ) × {1} that is properly homotopic to id. By Theorem 2.10.9, M × {1} is a Z-set in M × [t1 , 1]. Applying the Z-Set Unknotting Theorem 2.11.6, we can extend f1 to a homeomorphism h1 : M × I → M × I such that h1 |M × [0, t1 ] = id. Then, h1 (M × {1}) ⊂ (M \ X1 ) × {1}. Suppose that homeomorphisms h1 , . . . , hn−1 have been defined so as to satisfy (1), (2), and (3). As is easily observed, (hn−1 · · · h1 )−1 (Xn−1 × I) ∩ (M × {1}) = ∅. Choose s ∈ (tn , 1) so that (hn−1 · · · h1 )−1 (Xn−1 × I) ⊂ M × [0, s]. Since M is properly contractible to infinity, we have a closed embedding fn : M × {1} → (M × {1}) \ (hn−1 · · · h1 )−1 (Xn × I) that is properly homotopic to id. Applying the Z-Set Unknotting Theorem 2.11.6, we can extend fn to a homeomorphism gn : M × I → M × I such that gn |M × [0, s] = id (cf. Fig. 2.26). Then, gn |(hn−1 · · · h1 )−1 (Xn−1 × I) = id.

(∗)

2.11 Complementary Basic Results on Q-Manifolds

199 h n −1 · · · h 1 (M × {1}) Xn

(h n −1 · · · h 1 )−1 (X n −1 × I) gn g n (M × {1}) 1

X n −1 1

s tn

h n −1 · · · h 1

t1

t1

0

0

M

M

(h n −1 · · · h 1 )−1 (X n −1 × I) Fig. 2.26 Homeomorphisms hn−1 · · · h1 and gn

Since gn is an extension of fn , we also have (hn−1 · · · h1 )gn (M × {1}) ⊂ (M \ Xn ) × {1}. The following homeomorphism satisfies (1), (2), and (3): hn = (hn−1 · · · h1 )gn (hn−1 · · · h1 )−1 : M × I → M × I. Indeed, this can be verified as follows: hn hn−1 · · · h1 (M × {1}) = (hn−1 · · · h1 )gn (M × {1}) ⊂ (M \ Xn ) × {1}. Since gn |M × [0, s] = id and tn < s, we have hn hn−1 · · · h1 |M × [0, tn ] = (hn−1 · · · h1 )gn |M × [0, tn ] = hn−1 · · · h1 |M × [0, tn ]. Moreover, it follows from (∗) that hn |Xn−1 × I = (hn−1 · · · h1 )gn (hn−1 · · · h1 )−1 |Xn−1 × I = id. Now, by induction, we have homeomorphisms hn , n ∈ N, satisfying (1), (2), and (3). Then, by virtue of (2), we can define h : M × [0, 1) → M × I by h(x) = lim hn hn−1 · · · h1 (x) for each (x, t) ∈ M × [0, 1). n∈N

Since M × [0, 1) =



n∈N M

× [0, tn ) and

h|M × [0, tn ) = hn−1 · · · h1 |M × [0, tn )

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2 Fundamental Results on Infinite-Dimensional Manifolds

is an open embedding into M ×I for each n ∈ N, itis easy to see that h is continuous, open, and injective. For each y ∈ M × I = n∈N Xn × I, choose n ∈ N so that y ∈ Xn × I. Then, we have x = (hn · · · h1 )−1 (y) ∈ M × [0, 1). Indeed, hn · · · h1 (x) = y ∈ (M \ Xn ) × {1}, which implies x ∈ M × {1} by (1). Since hn · · · h1 (x) ∈ Xn × I and hi |Xn × I = id for every i > n, it follows that h(x) = hn · · · h1 (x) = y. Hence, h is surjective. Consequently, h is a homeomorphism. ! "

Notes for Chapter 2 For topics related to this chapter, refer to the book [(1)] of Bessaga and Pełczy´nski. Concerning Q-manifolds, see the textbook [(3)] of J. van Mill and Chapman’s lecture notes [(2)]. In this chapter, we have treated almost all the contents of the book of van Mill except for the characterization of compact Q-manifolds. Without compactness, the characterization of Q-manifolds will be treated in the next chapter together with the one of 2 ()-manifolds. As a well-known application to Shape Theory, Chapman established in [42] the Complement Theorem, that is, two Z-sets X and Y in the Hilbert cube Q have the same shape type if and only if their complements Q \ X and Q \ Y are homeomorphic. As another remarkable application to Simple Homotopy Theory, Chapman proved in [45] the topological invariance of Whitehead torsion, which will be proved in Chapter 4. For Simple Homotopy Theory, refer to Cohen’s book [(20)]. The weight version of Lemma 2.1.1 is due to Cutler (cf. Lemma 1.1 in [80]). Theorem 2.1.2 was established by Keller [92]. Theorem 2.1.11 is due to Henderson [80] (cf. [85] and [81]). In this section, we need not assume that E is normable. The non-invertible isotopy in the proof of Proposition 2.1.12 is due to Anderson and Bing [12]; that of Proposition 2.1.13 is a modification of the one constructed by Crowell in [49]. Note that Crowell’s example is not locally connected. Lemma 2.1.16 is due to Anderon, Henderson, and West [17]. The results of Sect. 2.2 are due to Toru´nczyk [139] (the compact case was proved in [138]). In [140], the results have been generalized to metric linear spaces, where proofs are not elementary. The maps f and g in Sect. 2.2 were first defined on X × lim Rn and lim Rn by D.W. Henderson in − → − → [82] so as to prove that X × lim Rn ≈ lim Rn . − → − → The Stability Theorem 2.3.7 was first established for 2 -manifolds by Anderson and Schori [15] and generalized as Theorem 2.3.7 by Schori [131] (cf. [85]). For the Stability Theorem, we can assert more (cf. [69] and [114]). The results can be extended to local trivial bundles with Emanifold fibers (cf. [115]). Reformulating Lemma 2.3.1 by Yamashita, the proofs of Lemmas 2.3.1 and 2.3.2 were made slim. The proof of the Strong Universality Theorem 2.3.15 is due to [114]. In this section, only the assumption that E ≈ E N or ≈ EfN is used. Proposition 2.4.1 is due to Anderson and Bing [12], which is generalized to an infinitedimensional normed linear space E, that is, there exists an I-preserving homeomorphism ζ : E × I \ {(0, 0)} → E × I. This can be proved by using a non-complete norm. Refer to Bessaga and Pełczy´nski’s book [(1)], Ch. III, §5 (cf. [32]). The Negligibility Theorem was first established by Anderson, Henderson and West [17] and extended to Theorem 2.4.3 by Cutler [53]. In [4], Anderson proved that σ -compact sets are negligible in 2 and RN , that is a key in his proof of 2 ≈ RN . The proof of Proposition 2.4.14 is due to [115]. Theorem 2.5.7 was established by M. Brown [35]. Our proof of Theorem 2.5.7 is bases on Connelly’s proof [48]. The Open Embedding Theorem 2.5.10 and the Unknotting Theorem 2.5.12 were respectively established by Henderson [79] (cf. [81]) and Anderson and McCharen [14] for 2 -manifolds. They were extended in [84] (cf. [81]) and [40], respectively. The proof is due to [114]. These results can be extended to local trivial bundles with E-manifold fibers (cf. [116]).

Notes for Chapter 2

201

The Classification Theorem 2.6.1 was established by Henderson and Schori [84] (cf. [85] and [81]). Theorem 2.6.3 is due to Ferry [64]. The Triangulation Theorem for 2 -manifolds was first obtained by D.W. Henderson [79] (cf. [77, 78]), as a corollary of the Open Embedding Theorem by virtue of a triangulation result on open subsets of Hilbert space proved in [77]. Here, it is obtained as a combination of the Classification Theorem 2.6.1 and the Toru´nczyk Factor Theorem 2.2.14 (Corollary 2.2.15). The Bing Shrinking Criterion arose in [31] and has been generalized by several authors (cf. [103]). Theorem 2.7.1 here is due to Toru´nczyk [146]. The Generalized Schoenflies Theorem 2.7.7 was proved by B. Mazur [105] and M. Brown [34]. Although Lemmas 2.7.8, 2.7.9, 2.7.10 are due to Brown [34], our proof of Theorem 2.7.7 is different from Brown’s proof in [34]. The case n = 2 is established by A. Schoenflies [130] without the bi-collared condition. The bi-collared condition is necessary for n  3. In fact, J.W. Alexander [2] constructed an embedding of S2 in R3 such that the unbounded component of the complement is not simply connected. This example is called the Alexander horned sphere. The condition (z) in 2.14 on Theorem 2.8.6 is called Property Z in [6]. In [14], a closed set with Property Z is called a Z-set. T.A. Chapman [40] showed the equivalence of Property Z and E-deficiency (infinite deficiency) in E-manifolds (Theorem 2.9.3). The necessity of strong Z-sets was emphasized in the paper [30], where the example (p. 162) of a Z-set being not a strong Z-set was given (cf. [146]). Proposition 2.8.8 is due to Curtis, Dobrowolski, and Mogilski [51, Lemma 2.4]. The notation of “closed over a set” was introduced by Bestvina and Mogilski [29], who adopt Lemma 2.8.10 as the definition. Propositions 2.8.12, 2.8.13, and 2.8.14 were proved in [29]. Yamashita simplified the proof of Proposition 2.8.12, which is presented here. Proposition 2.8.17 was proved by Bestvina and Mogilski [29]. Proposition 2.9.1 is due to Toru´nczyk [142] (cf. [146]). The example in Remark 2.18 was given in [30]. In the case where E is locally convex, Z-sets in an E-manifold are strong Zsets, for which refer to the paper [83]. The non-locally convex case is not known. In [83], by using proper subdivisions of simplicial complexes, Henderson proved the implication (a) ⇒ (b) in Theorem 2.9.3. In our proof, we avoid proper subdivisions. Theorem 2.9.9 was established by Anderson [7]. In [5], R.D. Anderson showed that every compact set in the pseudo-interior I (Q) is Q-deficient (infinite deficient) in Q (Theorem 2.10.4), where Lemmas 2.10.1, 2.10.2, and 2.10.3 were also proved. In [6], he showed the equivalence of Property Z and Q-deficiency (infinite deficiency) in Q (Theorem 2.10.6), where Lemma 2.10.5 was also proved. In [41], using Theorem 2.9.3, Chapman proved the equivalence of Property Z and Q-deficiency (infinite deficiency) in Qmanifolds (Theorem 2.10.9). The Z-set Unknotting Theorem (Theorem 2.11.6) was established by Anderson and Chapman [13]. Theorem 2.11.9, the Classification Theorem for R+ -stable Q-manifolds (Theorem 2.11.11), and the Open Embedding Theorems (Theorems 2.11.4 and 2.11.12) are due to Chapman [43]. R.Y.T. Wong characterized in [160] the R+ -stability of Q-manifolds as in Theorem 2.11.13.

Chapter 3

Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Throughout this chapter, • let  be an infinite set (or an infinite discrete space) with card  = τ . In this chapter, we prove the Toru´nczyk characterizations of Hilbert manifolds (i.e., 2 ()-manifolds) and Hilbert cube manifolds (i.e., Q-manifolds). By using the characterization of Hilbert space, we show that every Fréchet space is homeomorphic to Hilbert space with the same weight. In particular, s = RN ≈ 2 . Additionally, it is shown that if X is a non-discrete compactum and Y is a completely metrizable separable ANR Y without isolated points, then the space C(X, Y ) of maps from X to Y is an 2 -manifold. To prove the characterization of 2 ()manifolds, we require the Toru´nczyk Factor Theorem 2.2.14, that is, for each completely metrizable ANR X with w(X)  τ , the product X × 2 () is an 2 ()manifold. For the characterization of Q-manifolds, we need to prove its Q-version, called the Edwards Factor Theorem 3.8.1, that is, for each locally compact ANR X, the product X × Q is a Q-manifold. By using the characterization of Q, we prove Keller’s Theorem, that is, every infinite-dimensional compact metrizable convex set in a locally convex topological linear space is homeomorphic to Q.

3.1 (Strong) Universality and U-Maps Given a class C of spaces, a space X is said to be C-universal or universal for C if any map f : C → X from C ∈ C can be approximated by closed embeddings, that is, for each open cover U of X, f is U-close to a closed embedding.1 It is said that

1 This

is a little stronger than a universal space in Dimension Theory, where a space X is called a universal space for C if X ∈ C and every space in C can be embedded into X, where it is not required to be embedded into X as a closed set (cf. Sect. 5.9 of [GAGT]). © Springer Nature Singapore Pte Ltd. 2020 K. Sakai, Topology of Infinite-Dimensional Manifolds, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-15-7575-4_3

203

204

3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

X is strongly C-universal or strongly universal for C if the following condition is satisfied: (SUC )

For each C ∈ C and each closed set D ⊂ C, if f : C → X is a map such that f |D is a strong Z-embedding, then, for each open cover U of X, there is a strong Z-embedding h : C → X such that h|D = f |D and h is U-close to f .

Obviously, the strong C-universality implies the C-universality. In this section and the next, we discuss and characterize the (strong) universality for the class M1 (τ ) of completely metrizable spaces with density  τ (equivalently, weight  τ ),2 and the class M0 of compacta (= compact metrizable spaces). Note that M0 = FQ and M1 (τ ) = F2 () (Theorem 1.13.9). For a completely metrizable ANR X, it will be proved that the M1 (τ )-universality implies the strong M1 (τ )universality, so they are equivalent (Theorem 3.1.19). For each f ∈ C(X, Y ) and U ∈ cov(Y ), we define    U(f ) = g ∈ C(X, Y )  g is U-close to f . Observe that, if V ∈ cov(Y ) is a star-refinement of U, then V(g) ⊂ U(f ) for each g ∈ V(f ). When Y is paracompact, C(X, Y ) has a topology such that {U(f ) | U ∈ cov(Y )} is a neighborhood basis of f . Such a topology is called the limitation topology. Let Emb(X, Y ) denote the subspace of C(X, Y ) consisting of all closed embeddings of X into Y . Then, f ∈ cl Emb(X, Y ) means that f can be approximated by closed embeddings. Thus, the definition of the C-universality can be rewritten as follows: • X is C-universal if Emb(Y, X) is dense in the space C(Y, X) with the limitation topology for every Y ∈ C. For the limitation topology, refer to Sect. 2.9 in [GAGT]. For each metrizable space Y , let MetrB (Y ) be the set of all admissible bounded metrics for Y . When Y is completely metrizable, let Metrc (Y ) be the set of all admissible bounded complete metric for Y . For each f ∈ C(X, Y ) and d ∈ MetrB (Y ), let    Ud (f ) = g ∈ C(X, Y )  supx∈X d(f (x), g(x)) < 1 . Proposition 3.1.1 For each metrizable space Y , {Ud (f ) | d ∈ MetrB (Y )} is a neighborhood basis of f ∈ C(X, Y ) in the space C(X, Y ) with the limitation topology. When Y is completely metrizable, {Ud (f ) | d ∈ Metrc (Y )} is a neighborhood basis of f ∈ C(X, Y ). 2.9.1

2 Recall

that M1 denotes the class of all completely metrizable spaces. The class of all separable completely metrizable spaces is denoted by M1 (ℵ0 ).

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When Y = (Y, d) is a metric space, for each f ∈ C(X, Y ) and α ∈ C(Y, (0, ∞)), let    Nα (f ) = g ∈ C(X, Y )  ∀x ∈ X, d(f (x), g(x)) < αf (x) . Proposition 3.1.2 When Y = (Y, d) is a metric space, {Nα (f ) | α ∈ C(Y, (0, ∞))} is a neighborhood basis of f ∈ C(X, Y ) in the space C(X, Y ) with the limitation topology. 2.9.3 In the above proposition, we can consider only Lipschitz maps with the Lipschitz constant  1 as α ∈ C(Y, (0, ∞)). In fact, this is derived from the following lemma: Lemma 3.1.3 Let U be a locally finite open cover of a metric space Y = (Y, d) such that diam st(y, U) < ∞ for each y ∈ Y . Then, the map α : Y → (0, ∞) defined by α(y) = supU ∈U dist(y, Y \ U ) is a Lipschitz map with the Lipschitz constant  1, that is,   α(y) − α(y  )  d(y, y  ) for every y, y  ∈ Y. Proof Let y, y  ∈ Y . Without loss of generality, it can be assumed that α(y) > α(y  ). We can take U ∈ U so that dist(y, Y \U ) > α(y  ). Since α(y  )  dist(y  , Y \ U ), we have dist(y, Y \ U ) − α(y  )  dist(y, Y \ U ) − dist(y  , Y \ U )  d(y, y  ). Thus, it follows that α(y) − α(y  ) = sup dist(y, Y \ U ) − α(y  )  d(y, y  ). U ∈U

" !

In the above lemma, as is easily observed,    α(y) = sup r > 0  ∃U ∈ U such that B(y, r) ⊂ U . From the definition of α, for each y, y  ∈ Y , d(y, y  ) < α(y) ⇒ ∃U ∈ U such that y, y  ∈ U. The limitation topology is favorable because of the following properties: Proposition 3.1.4 (Basic Properties of the Limitation Topology) Let X be an arbitrary space but Y and Z be paracompact. (1) The evaluation map ev : X × C(X, Y )  (x, f ) → f (x) ∈ Y is continuous with respect to the limitation topology.

2.9.9(1)

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(2) The composition C(X, Y ) × C(Y, Z)  (f, g) → gf ∈ C(X, Z) is continuous with respect to the limitation topology. (3) The inverse operation

2.9.9(2)

Homeo(Y )  h → h−1 ∈ Homeo(Y ) is continuous with respect to the limitation topology. Combining this with (2) above, the group Homeo(Y ) with the limitation topology is a topological group. 2.9.9(3)

Hereafter, we assume that the space C(X, Y ) and its subspace have the limitation topology. Theorem 3.1.5 For a completely metrizable space Y , the space C(X, Y ) is a Baire space, that is, the intersection of countably many dense open sets (or dense Gδ -sets) in C(X, Y ) is also dense. 2.9.4 The following strong version can be proved in the same way as above: Theorem 3.1.6 Let Y be a completely metrizable space, f ∈ C(X, Y ), and A ⊂ X be a closed set. Then, the following space is also a Baire space:    Cf |A (X, Y ) = g ∈ C(X, Y )  g|A = f |A . Sketch of Proof. Let Gn , n ∈ N, be open dense sets in Cf |A (X, Y ). For each g ∈ Cf |A (X, Y ) and a complete bounded metric d ∈ Metrc (Y ), we can inductively choose gn ∈ Cf |A (X, Y ) and dn ∈ Metr(Y ), n ∈ N, so that gn ∈ U2dn−1 (gn−1 ) ∩ Gn , Udn (gn ) ⊂ Gn , and dn  dn−1 , where g0 = g and d0 = d. Then, it is easy to see that (gn )n∈N is Cauchy with respect to the sup-metric induced by d. By the completeness, (gn )n∈N is uniformly convergent to (gn , g∞ ) < 1, which means g∞ ∈ Cf |A (X, Y ). It is easy to calculate d0 (g, g∞ ) < 1 and dn g∞ ∈ Ud (g) and g∞ ∈ Udn (gn ) ⊂ Gn . Hence, g∞ ∈ Ud (g) ∩ n∈N Gn .

Let CP (X, Y ) be the subspace of C(X, Y ) consisting of all proper maps. Due to Proposition 1.3.7, when X and Y are metrizable, a map f : X → Y is proper if and only if it is perfect.3 The following is the direct consequence of Proposition 1.3.8: Proposition 3.1.7 If Y is locally compact and paracompact, then CP (X, Y ) is clopen in C(X, Y ), where X is locally compact if CP (X, Y ) = ∅. 2.9.7

3 Even when Y is non-metrizable, if Y is locally compact, then this equivalence also holds [GAGT, Proposition 2.1.5].

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We also consider the class L of all locally compact metrizable spaces. Let L(τ ) be the subclass of L consisting of spaces with density  τ (equivalently, weight  τ ). Then, L(ℵ0 ) = FQ\{0} (Proposition 2.1.3). According to our definition, X is L(τ )-universal if Emb(Y, X) is dense in C(Y, X) for every Y ∈ L(τ ). However, the following fact should be remarked: Fact Any locally compact metrizable space X is not L(ℵ0 )-universal. Indeed, let f : R → X be a constant map and let U = {U, X \f (R)} ∈ cov(X), where cl U is compact. If f is U-close to g, then g(R) ⊂ U , which implies that cl g(R) is compact. Hence, g is not a closed embedding.

On this account, we define that X is properly L(τ )-universal (or properly universal for L(τ )) if Emb(Y, X) is dense in CP (Y, X) = ∅ for every Y ∈ L(τ ), where CP (Y, X) = ∅ if and only if Emb(Y, X) = ∅. If X is properly L(τ )-universal, then the discrete space  with card  = τ can be embedded in X as a closed set. Hence, we have the following: Lemma 3.1.8 If X is properly L(τ )-universal, then w(X)  τ .

" !

The following proposition can easily proved by the same method as the wellknown fact that every component of a locally connected space is clopen. Proposition 3.1.9 Every locally compact metrizable space X is a discrete union of locally compact separable subspaces, that is, a topological sum of locally compact separable metrizable spaces. Sketch of Proof. Take a locally finite open cover U ∈ cov(X) so that cl U is compact for every U ∈ U. Then, U is  star-finite (i.e., U[U ] is finite for every U ∈ U). For each U ∈ U, we define U ∗ = n∈N stn (U, U), where each stn (U, U) = st(stn−1 (U, U), U), st0 (U, U) = U (so st1 (U, U) = st(U, U)). Then, U ∗ is an equivalence class with respect to the equivalence relation ∼ on X defined as follows: x ∼ y ⇔ ∃U1 , . . . , Un ∈ U such that def

x ∈ U1 , y ∈ Un , Ui ∩ Ui+1 = ∅, i = 1, . . . , n − 1. Thus, for each U, V ∈ U, U ∗ = V ∗ ⇔ U ∗ ∩ V ∗ = ∅.Hence, U ∗ is clopen in X for each U ∈ U. As is easily observed, for every U ∈ U, U ∗ = n∈N cl stn (U, U) and cl stn (U, U), n ∈ N, are compact. Hence, U ∗ is σ -compact and separable.4 Since U ∗ is open in X, U ∗ is locally compact.

By virtue of Proposition 3.1.9 above, the following holds: Proposition 3.1.10 A locally compact metrizable space X is properly L(τ )universal if and only if X is properly L(ℵ0 )-universal and w(X)  τ . Proof The “only if” part follows from Lemma 3.1.8 and the fact L(ℵ0 ) ⊂ L(τ ). To see the “if” part, let f : Y → X be a proper map from Y ∈ L(τ ). By

4A

locally compact metrizable space is separable if and only if it is σ -compact.

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 Proposition 3.1.9, we can write X = λ∈  Xλ , where each Xλ is a locally compact separable metrizable space. Then, Y = λ∈ f −1 (Xλ ). Each Xλ has a countable compact cover {Xλ,n | n ∈ N}. Then, f −1 (Xλ ) = n∈N f −1 (Xλ,n ) and each f −1 (Xλ,n ) is compact because f is proper. Hence, f −1 (Xλ ) is a locally compact separable metrizable space and f |f −1 (Xλ ) is a proper map. For each U ∈ cov(X), we can choose an open refinement V ∈ cov(X) of U so that each V ∈ V meets only one of Xλ , λ ∈ , that is, st(Xλ , V) = Xλ for each λ ∈ . For each λ ∈ , we have a closed embedding hλ : f −1 (Xλ ) → X that is V-close to f |f −1 (Xλ ). Since hλ (f −1 (Xλ )) ⊂ st(Xλ , V) = Xλ for each λ ∈ , we can define a closed embedding h : Y → X by h|f −1 (Xλ ) = hλ for each λ ∈ . Since V ≺ U, h is U-close to f . " ! In order to discuss the proper L(τ )-universality of a locally compact metrizable space, we need the following theorem that can be obtained by combining Proposition 3.1.7 with Theorem 3.1.5: Theorem 3.1.11 For every locally compact metrizable spaces X and Y , the space 2.9.8 CP (X, Y ) is a Baire space. The following is a strong version corresponding to Theorem 3.1.6: Theorem 3.1.12 Let X and Y be locally compact metrizable spaces. For each f ∈ CP (X, Y ) and each closed set A ⊂ X, the following space is also a Baire space:    CPf |A (X, Y ) = g ∈ CP (X, Y )  g|A = f |A .

" !

Given U ∈ cov(X), a map f : X → Y is called a U-map if f −1 (V) ≺ U for some V ∈ cov(Y ). Let CU (X, Y ) denote the subspace of C(X, Y ) consisting of all U-maps. Then, Emb(X, Y ) ⊂ CU (X, Y ).5 Lemma 3.1.13 Let Y be paracompact and U ∈ cov(X). Then, CU (X, Y ) is open in the space C(X, Y ). 5.8.6 Sketch of Proof. Let f ∈ CU (X, Y ). Then, f −1 (V) ≺ U for some V ∈ cov(Y ). For a star-refinement W ∈ cov(Y ) of V, W(f ) ⊂ CU (X, Y ).

Lemma 3.1.14 Let X = (X, d) be a complete metric space  and Un ∈ cov(X), n ∈ N, with mesh Un < 2−n . Then, Emb(X, Y ) = n∈N CUn (X, Y ). Thus, when X is a completely metrizable space, Emb(X, Y ) is a Gδ -set in the space C(X, Y ). 5.8.7  Sketch of Proof. Each f ∈ n∈N CUn (X, Y ) is injective. For xn ∈ X, n ∈ N, if (f (xn ))n∈N is convergent, then (xn )n∈N is Cauchy, so convergent. Hence, f ∈ Emb(X, Y ).

By combining the above two lemmas with Theorem 3.1.5, the following can be obtained:

5 For

U-maps, refer to Sect. 5.8 of [GAGT].

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Proposition 3.1.15 Let X and Y be completely metrizable spaces. Suppose that, for each U ∈ cov(X), CU (X, Y ) is dense in the space C(X, Y ). Then, Emb(X, Y ) is also dense in C(X, Y ). 5.8.8 The following is easy to see: Lemma 3.1.16 Let U be an open cover of X consisting of open sets with compact " ! closure. Then, CU (X, Y ) ⊂ CP (X, Y ). We have the following locally compact version of Proposition 3.1.15: Proposition 3.1.17 Let X and Y be locally compact metrizable. Suppose that, for each open cover U of X consisting of open sets with the compact closure, CU (X, Y ) is dense in the space CP (X, Y ). Then, Emb(X, Y ) is also dense in CP (X, Y ). 5.8.9 When a map is a U-map on a closed set, it is a U-map on a neighborhood, that is: Proposition 3.1.18 Let X and Y be paracompact spaces, f : X → Y be a map, A ⊂ X be a closed set and U ∈ cov(X). If f |A is a (U|A)-map, then A has a neighborhood N in X such that f |N is a (U|N)-map. Proof Choose a locally finite open cover V of Y so that (f |A)−1 (Vcl ) = f −1 (Vcl )|A ≺ U|A ≺ U. Let V0 be an open star-refinement of V. By the continuity of f and the local finiteness of (f |A)−1 (Vcl ), we can obtain W ∈ cov(X) such that f (W) ≺ V0 and



  st(f −1 (cl V ) ∩ A, W)  V ∈ V ≺ U.

Then, N = st(A, W) is a neighborhood of A in X. We shall show that (f |N)−1 (V0 ) ≺ U, which means that f |N is a (U|N)-map. For each V0 ∈ V0 , choose V ∈ V so that st(V0 , V0 ) ⊂ V . Then, it suffices to see (f |N)−1 (V0 ) ⊂ st(f −1 (V ) ∩ A, W). For each x ∈ (f |N)−1 (V0 ) = f −1 (V0 ) ∩ N, we have a ∈ A such that x, a ∈ W for some W ∈ W, which implies that f (x), f (a) ∈ V1 for some V1 ∈ V0 . Then, it follows that f (a) ∈ st(V0 , V0 ) ⊂ V , so a ∈ f −1 (V ) ∩ A. Consequently, we have x ∈ st(f −1 (V ) ∩ A, W). " ! For an ANR X ∈ M1 (τ ), we can prove the M1 (τ )-universality implies the strong M1 (τ )-universality, so they are equivalent, that is: Theorem 3.1.19 A completely metrizable ANR X with weight τ is strongly M1 (τ )universal if and only if X is M1 (τ )-universal. To prove Theorem 3.1.19, we need the following lemma:

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Lemma 3.1.20 Let X ∈ M1 (τ ) be an ANR and A a strong Z-set in X. If X is M1 (τ )-universal, then the identity idX can be approximated by strong Zembeddings h : X → X such that h|A = id. Proof First, we will construct a map ϕ : X × (0, 1] → X × (0, 1] so that: (1) prX ϕ|X × [a, 1] is a closed embedding for each a ∈ (0, 1), (2) prX ϕ(X × (0, 1]) ∩ A = ∅, and (3) d(prX ϕ(x, t), x) < t for each (x, t) ∈ X × (0, 1]. For simplicity, we write M = C(X × (0, 1], X × (0, 1]), where we consider M as the space with the limitation topology. Then, M is a Baire space by Theorem 3.1.5. For each n ∈ N, let   Gn = f ∈ M  prX f |X × [2−n , 1] ∈ Emb(X × [2−n , 1], X),

 prX f (X × [2−n , 1]) ∩ A = ∅ ,

where it should be noted that prX f (X × [2−n , 1]) is closed in X. Recall that Emb(X×[2−n , 1], X) is a Gδ set in C(X×[2−n , 1], X) by Lemma 3.1.14. Moreover, the following is open in C(X × [2−n , 1], X):    On = g ∈ C(X × [2−n , 1], X)  cl g(X × [2−n , 1]) ∩ A = ∅ . Since the correspondence f → prX f |X × [2−n , 1] is continuous (3.1.4(2)) and Gn is the inverse image of a Gδ -set Emb(X ×[2−n , 1], X)∩On by this correspondence, it follows that Gn is a Gδ -set in M. To see that Gn is dense in M, let f ∈ M and U ∈ cov(X × (0, 1]). By the compactness of [2−n−1 , 1], we have W ∈ cov(X) such that 

  W × {t}  W ∈ W, t ∈ [2−n−1 , 1] ≺ U.

Since A is a strong Z-set in X and X is an M1 (τ )-universal ANR, we can easily obtain a closed embedding g : X × [2−n , 1] → X such that g(X × [2−n , 1]) ∩ A = ∅ and g W prX f |X × [2−n , 1]. By the Homotopy Extension Theorem 1.13.11, g can be extended to a map g˜ : X × (0, 1] → X such that g|X ˜ × (0, 2−n−1 ] = prX f |X × (0, 2−n−1 ] and g˜ W prX f rel. X × (0, 2−n−1 ].

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Then, f is U-close to the map f  ∈ Gn defined by prX f  = g˜ and pr(0,1]f  = pr(0,1] f , that is, ˜ t), pr(0,1] f (x, t)) for each (x, t) ∈ X × (0, 1]. f  (x, t) = (g(x, Thus, Gn is dense in M. Consequently, Gn is a dense Gδ set in M. Consider the following neighborhood of id ∈ M:    V = f ∈ M  ∀(x, t) ∈ X × (0, 1], d(prX f (x, t), x) < t .  Since M is a Baire space, we can obtain the desired ϕ ∈ V ∩ ( n∈N Gn ), which satisfies (1), (2), and (3). Here, note that prX ϕ is injective by (1). For any open cover U ∈ cov(X), using this ϕ, we will construct a strong Z-embedding h : X → X such that h|A = id and h is U-close to id. Due to Lemma 2.1.16, there exists V ∈ cov(X \ A) which is fitting in X and refines U. Take α ∈ C(X \ A, (0, 1]) so that Nα (idX\A ) ⊂ V(idX\A ) (Proposition 3.1.2), and define hα : X → X as follows: hα (x) =

 prX ϕ(x, α(x)) x

if x ∈ X \ A, if x ∈ A.

Then, hα (X \ A) ⊂ X \ A and d(hα (x), x) < α(x) for each x ∈ X \ A, which means hα |X \ A ∈ Nα (idX\A ). Thus, hα |X \ A is V-close to idX\A . Since hα |A = id and V is fitting in X, it follows that hα is continuous, where hα is a closed embedding if hα |X \ A is a closed embedding into X \ A (2.1.16(iii)). Because V ≺ U and hα |A = id, hα is U-close to idX . We have to show that hα is a strong Z-embedding, that is, hα is a closed embedding and hα (X) is a strong Z-set in X. We show that hα |X \ A is a closed embedding into X \ A. Then, as mentioned above, it follows that hα is a closed embedding. Since prX ϕ is injective, hα |X \ A is an injection into X \ A. To see that hα |X \ A is closed, let xi ∈ X \ A, i ∈ N, such that (hα (xi ))i∈N converges to y ∈ X \ A. If infi∈N α(xi ) = 0, then (xi )i∈N has a subsequence (xij )j ∈N such that limj →∞ α(xij ) = 0. In this case, limj →∞ xij = y because d(hα (xij ), xij ) < α(xij ) → 0 (j → ∞). Then, α(y) = limj →∞ α(xij ) = 0, which is a contradiction. Therefore, infn∈N α(xn ) > 0. Choose n ∈ N so that 2−n < infi∈N α(xi ). Because ϕ ∈ Gn , prX ϕ|X × [2−n , 1] is a closed embedding into X \ A. Since prX ϕ(xi , α(xi )) = hα (xi ) → y (i → ∞), it follows that (xi )i∈N is convergent in X \ A. Thus, hα |X \ A is a closed embedding. Finally, we show that hα (X) is a strong Z-set in X. For each W ∈ cov(X), let W ∈ cov(X) be a star-refinement of W. Since A is a strong Z-set in X, there is a map g : X → X such that g is W -close to idX and A ∩ cl g(X) = ∅. Choose an open set U in X so that A ∩ cl U = ∅ and cl g(X) ⊂ U . Let   W = W ∧ U, X \ cl U, X \ (A ∪ cl g(X)) ∈ cov(X).

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Take β ∈ C(X \ A, (0, 1]) so that β < α and Nβ (idX\A ) ⊂ W (idX\A ) (Proposition 3.1.2). Replacing α with this β, we can obtain a closed embedding hβ such that d(hβ (x), x) < β(x) for all x ∈ X. Thus, we have a map hβ g : X → X, which is W -close to g because d(hβ g(x), g(x)) < β(g(x)) for each x ∈ X. Since st(W , W ) ≺ W, it follows that hβ g is W-close to idX . Moreover, cl hβ g(X) ⊂ cl st(g(X), W ) ⊂ cl U ⊂ X \ A. Since hβ (X \ A) is closed in X \ A, prX ϕ is injective, and β(x) < α(x) for each x ∈ X, it follows that hα (X) ∩ cl hβ g(X) = (A ∪ hα (X \ A)) ∩ cl hβ g(X) ⊂ hα (X \ A) ∩ cl hβ (X \ A) = hα (X \ A) ∩ hβ (X \ A)    = prX ϕ(x, t)  α(x) = β(x) = t = ∅. Therefore, hα (X) is a strong Z-set in X.

" !

Remark 3.1 In the above, by (3), prX ϕ extends to the homotopy ϕ˜ : X × I → X by ϕ˜ 0 = id, that is a deformation. For any α ∈ C(X, (0, 1]), we can define a map hα : X → X hα (x) = ϕ(x, ˜ αA (x)), where αA (x) = min{α(x), d(x, A)}. Then, hαA ∈ Nα (idX ) and hαA |A = id. It can be proved that hαA is a strong Z-embedding, that is, hαA is a closed embedding and hαA (X) is a strong Z-set in X. The proof is a good exercise. With easy modification of the above proof, the following version of Lemma 3.1.20 can be shown: Lemma 3.1.21 Let X ∈ L(τ ) be an ANR and A a (strong) Z-set in X. If X is properly L(τ )-universal, then the identity idX can be approximated by strong Zembeddings h : X → X such that h|A = id. Modification in Proof of Lemma 3.1.20. Replace C(X × (0, 1], X × (0, 1]), C(X × [2−n , 1], X), Cf |D (C, X), and “M1 (τ )-universal” with CP (X × (0, 1], X × (0, 1]), CP (X × [2−n , 1], X), CPf |D (C, X), and “proper L(τ )-universal,” respectively. Then, Theorem 3.1.11 substitutes for Theorem 3.1.5 and Lemma 3.1.14 can be applied together with Lemma 3.1.16. To see that Gn is dense in C, note that prX f |X × [2−n , 1] is proper because f is proper. Choose U ∈ cov(X × (0, 1]) so that cl U is compact for each U ∈ U. Then, f  is also proper because it is U-close to f .

Now, we prove Theorem 3.1.19: Proof (Theorem 3.1.19) It suffices to show the “if” part. Let f : C → X be a map of C ∈ M1 (τ ) such that f |D is a strong X-embedding. By virtue of Lemma 3.1.20, it suffices to show that f can be approximated by closed embeddings g : Y → X

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such that f |D = g|D, that is, the following set is dense in Cf |D (C, X): Embf |D (C, X) = Cf |D (C, X) ∩ Emb(C, X). For each n ∈ N, take Un ∈ cov(C) so that mesh Un < 2−n for some d ∈ Metr(C), and let Fn = Cf |D (C, X) ∩ CUn (C, X). Then, each Fn is open in Cf |D (C, X) by Lemma 3.1.13 and Embf |D (C, X) =  n∈N Fn . Since Cf |D (C, X) is a Baire space by Theorem 3.1.6, it is enough to show that each Fn is dense in Cf |D (C, X). Let g ∈ Cf |D (C, X) and U ∈ cov(X). Take a star-refinement U ∈ cov(X) of U. We apply Proposition 2.8.12 to obtain a map g  : C → X such that g  |D = g|D = f |D, g  (C\D) ⊂ X\f (D), g  is closed over f (D), and g  is U -close to g. Because g  |D = f |D is a closed embedding, it is a (Un |D)-map. Due to Proposition 3.1.18, D has an open neighborhood N in C such that g  |N is a (Un |N)-map. Since g  is closed over f (D) and C \ N is closed in C, it follows that f (D) ∩ cl g  (C \ N) = f (D) ∩ g  (C \ N) ⊂ f (D) ∩ g  (C \ D) = ∅. Then, O = X \ cl g  (C \ N) is open set in X and D ⊂ g −1 (O) ⊂ N. Since CUn |N (N, X) is open in C(N, X) by Lemma 3.1.13, there is a locally finite open cover W of X such that W ≺ U , W(g  |N) ⊂ CUn |N (N, X), and cl st(cl g  (C \ N), st W) ∩ f (D) = ∅. Observe that st(W, W) ∩ f (D) = ∅ for each W ∈ W[X \ O] = W[cl g  (C \ N)]. Let D  = g −1 (cl st(cl g  (C \ N), st W)). Then, D  is closed in C and D ∩ D  = ∅. Because X is an M1 (τ )-universal ANR, there exists a closed embedding h : D  → X such that h W g  |D  . By using the Homotopy Extension Theorem 1.13.11, h can be extended to a map h : C → X such that h|D = g  |D = f |D and h W g  . Then, h is U -close to g  , so U-close to g. It remains to be shown that h ∈ Fn . To see h ∈ Fn , we may prove that h is a Un -map. Since the closed embedding h|D  = h is a (Un |D  )-map, we have V ∈ cov(X) such that V ≺ W and (h|D  )−1 (V ) ≺ Un |D  . Let V ∈ V such that V ∩ st(cl g  (C \ N), W) = ∅, i.e., st(V , W) ∩ cl g  (C \ N) = ∅.

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Since h is W-close to g  and V ≺ W, it follows that h−1 (V ) ⊂ g −1 (st(V , W)) ⊂ g −1 (st(cl g  (C \ N), st W)) = D  , which implies that h−1 (V ) = h−1 (V ) ∩ D  = (h|D  )−1 (V ). Therefore, h−1 (V [st(cl g  (Y \ N), W)]) ⊂ (h|D  )−1 (V ) ≺ Un . Because h|N ∈ W(g  |N), h|N is a (Un |N)-map. So, there is V ∈ cov(X) such that V ≺ W and (h|N)−1 (V ) ≺ Un |N. Let V ∈ V such that V ∩ st(cl g  (C \ N), W) = ∅, i.e., st(V , W) ∩ g  (C \ N) = ∅. Since h is W-close to g  , it follows that h−1 (V ) ⊂ g −1 (st(V , W)) ⊂ N. Then, h−1 (V ) = h−1 (V ) ∩ N = (h|N)−1 (V ). Hence, h−1 (V \ V [st(cl g  (C \ N), W)]) ≺ Un . For a common refinement V ∈ cov(X) of V and V , we have h−1 (V) ≺ Un . Consequently, h is a Un -map. ! " It is said that X is strongly properly L(τ )-universal or strongly properly universal for L(τ ) if X satisfies the following condition: p

(SUL(τ ) )

For each C ∈ L(τ ) and each closed set D ⊂ C, if f : C → X is a proper map and f |D is a strong Z-embedding, then f : C → X can be approximated by strong Z-embeddings g : C → X such that g|D = f |D. p

In the above condition (SUL(τ ) ), if X is locally compact, then “strong Zembedding” can be replaced with “Z-embedding” because every Z-set in X is a strong Z-set (Proposition 2.8.1). The following version of Theorem 3.1.19 can be proved by the same method with suitable modification: Theorem 3.1.22 A locally compact ANR X with weight τ is strongly properly L(τ )universal if and only if X is properly L(τ )-universal. Modification in Proof of Theorem 3.1.19. Let f : C → X be a proper map of C ∈ L(τ ) such that f |D is a (strong) Z-embedding. Lemma 3.1.21 substitutes for Lemma 3.1.20. Replace Cf |D (C, X) with CPf |D (C, X) and take Un ∈ cov(C) so as to satisfy the additional condition that cl U is compact for every U ∈ Un . Lemma 3.1.13 can used together with Lemma 3.1.16. To see that Fn is dense in CPf |D (C, X), take U ∈ cov(X) so that cl U is compact for each U ∈ U. Then, h is also proper because it is U-close to g.

Remark 3.2 In Theorem 3.1.22, “properly L(τ )-universal” can be replaced with “properly L(ℵ0 )-universal and w(X) = τ ” (Proposition 3.1.10).

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For completely metrizable ANRs, we characterize the M1 (τ )-universality as follows: Theorem 3.1.23 For a completely metrizable ANR X, in order to be M1 (τ )universal (resp. M1 (ℵ0 )-universal or M0 -universal), the following condition is necessary and sufficient: • For any locally finite-dimensional simplicial complex K with card K (0)  τ (resp. any countable locally finite simplicial complex K or any finite simplicial complex K), each map f : |K| → X can be approximated by OK -maps.6 Proof (Necessity) When X is M1 (ℵ0 )-universal or M0 -universal, let K be a countable locally finite simplicial complex or a finite simplicial complex, respectively. Then, |K| ∈ M1 (ℵ0 ) or |K| ∈ M0 , respectively. Since every closed embedding from |K| to X is an OK -map, each map f : |K| → X can be approximated by OK -maps. When X is M1 (τ )-universal, let K be a locally finite-dimensional simplicial complex with card K (0)  τ and let f : |K| → X be a map. Each U ∈ cov(X) has an open star-refinement V. We have a subdivision K   K such that K  ≺ f −1 (V) (Theorem 1.6.9). By Theorem 1.7.3, ϕ = id : |K| → |K  |m has a homotopy inverse ψ : |K  |m → |K| such that ψϕ K  id and ϕψ K  id. Then, f ψϕ is V-close to f and |K  |m ∈ M1 (τ ). By the M1 (τ )-universality, f ψ is V-close to a closed embedding h : |K  |m → X. Then, hϕ is V-close to f ψϕ, so U-close to f . Note that h(OK ) = W|h(|K|) for some W ∈ cov(X). Then, (hϕ)−1 (W) = OK , that is, hϕ : |K| → X is an OK -map. (Sufficiency) Let Y ∈ M1 (τ ) (resp. Y ∈ M1 (ℵ0 ) or Y ∈ M0 ). By Proposition 3.1.15, it suffices to show that CV (Y, X) is dense in C(Y, X) for any V ∈ cov(Y ), that is, each f ∈ C(Y, X) can be approximated by V-maps. For each U ∈ cov(X), take an open star-refinement U of U. Applying Theorem 1.13.21 with Remark 1.13, we can obtain a σ -discrete (resp. countable star-finite or finite) refinement W ≺ V (∈ cov(Y )) with the locally finite-dimensional nerve N(W) and a map ψ : |N(W)| → X such that ψϕ U f , where ϕ : Y → |N(W)| is a canonical map for W, that is, ϕ −1 (ON(W) (W )) ⊂ W for every W ∈ W. Since W is σ -discrete (resp. countable star-finite or finite), it follows that card N(W)(0) = card W  ℵ0 w(Y )  τ (resp. N(W) is countable locally finite or finite). Thus, we can apply the condition to obtain an ON(W) -map ψ  : |N(W)| → X that is U -close to ψ. Then, ψ  ϕ is a W-map, so it is a V-map. Since ψ  ϕ is U -close to ψϕ, it is U-close to f . " ! The following is the locally compact version of Theorem 3.1.23 above: Theorem 3.1.24 For a locally compact ANR X with density τ , in order to be properly L(ℵ0 )-universal (hence properly L(τ )-universal), the following condition is necessary and sufficient:

6 Recall

that OK is the open star cover of K defined in Sect. 1.5.

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• For any countable locally finite simplicial complex K, each proper map f : |K| → X can be approximated by OK -maps. Proof This can be proved by modifying the proof of Theorem 3.1.23. Just like the M1 (ℵ0 )-case, the necessity can be easily seen. To prove the sufficiency, we apply Proposition 3.1.17 instead of Proposition 3.1.15. Let Y be a locally compact separable metrizable space, f : Y → X be a proper map, and U ∈ cov(X) such that cl U is compact for every U ∈ U. Moreover, let U ∈ cov(X) be an open star-refinement of U. By Theorem 1.13.21 with Remark 1.13, every V ∈ cov(Y ) has a countable star-finite refinement W ∈ cov(Y ) with a map ψ : |N(W)| → X such that cl W is compact for every W ∈ W and ψϕ U f , where ϕ : Y → |N(W)| is a canonical map for W. Then, ϕ is proper because ϕ −1 (ON(W) (W )) ⊂ W for every W ∈ W. The star-finiteness of W means the local compactness of |N(W)|, hence ϕ is perfect (Proposition 1.3.7), so it is closed. Since ψϕ is U -close to the proper map f , it is proper. Then, it follows that ψ|ϕ(Y ) is proper. For each U ∈ U, ψ −1 (cl U ) ∩ ϕ(Y ) is compact. It is easy to find an open set U ∗ in |N(W)| U ∗ ∩ ϕ(Y ) = ψ −1 (U ) ∩ ϕ(Y ) and cl U ∗  such that ∗ is compact. Since ϕ(Y ) ⊂ U ∈U U , we havea subcomplex K of some small subdivision of N(W) such that ϕ(Y ) ⊂ |K| ⊂ U ∈U U ∗ . Then, ψ||K| is proper. Indeed, each compact set A ⊂ X is covered by finitely many U1 , . . . , Un ∈ U. Since ψ −1 (A) ∩ |K| ⊂

n  i=1

ψ −1 (Ui ) ∩ |K| ⊂

n 

cl Ui∗ ,

i=1

it follows that ψ −1 (A) ∩ |K| is compact. Thus, we can apply the condition to obtain an OK -map ψ  : |K| → X that is U -close to ψ. Then, ψ  ϕ is a W-map, so a V-map. Since ψ  ϕ is U -close to ψϕ, it is U-close to f . " !

3.2 The Discrete (or Disjoint) Cells Property In this section, improving Theorems 3.1.23 and 3.1.24, we give characterizations of the universalities for M0 and M1 (τ ). An indexed family (Aλ )λ∈ of subsets of a space X is said to be discrete (resp. locally finite) in X if each x ∈ X has a neighborhood U such that    card λ ∈   U ∩ Aλ = ∅  1 (resp. < ℵ0 ).7

is not the same as saying that the collection {Aλ | λ ∈ } is discrete (resp. locally finite) in X. Indeed, when Aλ = A = ∅ for every λ ∈ , {Aλ | λ ∈ } is discrete in X, but (Aλ )λ∈ is not discrete (resp. locally finite) in X if card  > 1 (resp.  ℵ0 ). Evidently, if (Aλ )λ∈ is discrete (resp. locally finite) in X, then {Aλ | λ ∈ } is discrete (resp. locally finite) in X.

7 This

3.2 The Discrete (or Disjoint) Cells Property

217

For each n ∈ ω, it is said that X has the τ -discrete (resp. τ -locally finite) ncells property if every map f : In ×  → X can be approximated by maps g : In ×  → X such that (gγ (In ))γ ∈ is discrete (resp. locally finite) in X, where  is a discrete space with card  = τ , gγ : In → X is defined by gγ (x) = g(x, γ ), and I0 = {0} when n = 0. The ℵ0 -discrete (resp. ℵ0 -locally finite) n-cells property is simply called the countable discrete (resp. countable locally finite) n-cells property. When X has the τ -discrete (resp. τ -locally finite) n-cells property for every n ∈ N, it is said that X has the τ -discrete (resp. τ -locally finite) cells property. The ℵ0 -discrete (resp. ℵ0 -locally finite) cells property is simply called the countable discrete (resp. countable locally finite) cells property. Evidently, if X has the τ -locally finite n-cells property, then w(X)  τ . The following are easily observed: Fact (1) The τ -discrete n-cells property implies the disjoint n-cells property and the τ  -discrete n -cells property for every ℵ0  τ   τ and 0  n  n. (2) A paracompact space X has the τ -discrete 0-cells property if and only if every neighborhood of each point of X contains a discrete set D with card D = τ . The “if” part of (2) can be shown as follows: When τ = ℵ0 , we regard  = (N, ), where  is the natural order. When τ > ℵ0 , we can assume that  = (, ) is a well-ordered set such that card (γ ) < card  = τ for every γ ∈ , where (γ ) = {γ  ∈  | γ  < γ }. Indeed, let  = (, ) be a well-ordered set with card  = τ . If card (γ ) = τ for some γ ∈ , take the minimum element γ0 ∈  such that card (γ0 ) = τ and replace  with (γ0 ). Each U ∈ cov(X) has a locally finite refinement V ∈ cov(X). Given a net (xγ )γ ∈ ∈ X  , choose Vγ ∈ V, γ ∈ , so that xγ ∈ Vγ . Each Vγ contains a discrete set Dγ with might occur that Dγ = Dγ  even if Vγ = Vγ  . Since {Dγ | γ ∈ } card Dγ = τ , where it  is locally finite, D = γ ∈ Dγ is discrete in X. By transfinite induction, we can choose yγ ∈ Dγ \ {yγ  | γ  < γ } ⊂ D, γ ∈ , because card (γ ) < τ = card Dγ . Then, (yγ )γ ∈ is discrete in X and {{xγ , yγ } | γ ∈ } ≺ U.

Obviously the τ -discrete n-cells property implies the τ -locally finite n-cells property. However, the converse is also true as in the following proposition. Therefore, these concepts are equivalent. Proposition 3.2.1 For each n ∈ ω, the τ -locally finite n-cells property implies the τ -discrete n-cells property. Proof As in the above proof of the “if” part of Fact (2), we can assume that  = (N, ) ( is the natural order) or  = (, ) is a well-ordered set such that card (γ ) < card  = τ for every γ ∈ , where (γ ) = {γ  ∈  | γ  < γ }. Let f : In ×  → X be a map and U ∈ cov(X). Since card   ℵ0 , we have card  2 = card . Applying the τ -locally finite n-cells property to the map f˜ : In ×  2 → X defined by f˜(z, γ , γ  ) = f (z, γ ), we can obtain a map g˜ such that g˜ is U-close to f˜ and (g(I ˜ n × {(γ , γ  )})(γ ,γ )∈ 2 is locally finite in X. By the compactness, each g(I ˜ n ×{(γ , γ  )}) meets only finitely many other g(I ˜ n ×{(δ, δ  )}).

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By transfinite induction (or induction), we can choose δ(γ ) ∈  for each γ ∈  so that     g˜ In × {(γ , δ(γ ))} ∩ g˜ In × {(γ  , δ(γ  ))} = ∅ for γ  < γ . Then, (g(I ˜ n × {(γ , δ(γ ))}))γ ∈ is discrete in X because it is locally finite and mutually disjoint. Hence, the desired map g : In ×  → X can be defined by g(z, γ ) = g(z, ˜ γ , δ(γ )). " ! A map ϕ : X → Y is said to be polyhedrally approximately soft provided that for any polyhedron Z, any map f : C → X of a subpolyhedron C of Z, and any open cover U ∈ cov(Y ), if ϕf extends to a map g : Z → Y , then f extends to f˜ : Z → X such that ϕ f˜ is U-close to g: X

ϕ

Y U

f

f˜

C

⊂

g

Z

In the above, when the condition dim Z  n is added, ϕ is said to be polyhedrally approximately n-soft. For soft maps, refer to Section 7.2 in [GAGT]. Proposition 3.2.2 Let ϕ : X → Y be a polyhedrally approximately n-soft proper map and n ∈ ω. If X has the τ -discrete n-cells property, then Y has also the τ discrete n-cells property. Proof By virtue of Proposition 3.2.1 above, the τ -discrete n-cells property can be replaced with the τ -locally finite n-cells property. Let f : In × → Y be a map. For any U ∈ cov(Y ), let V ∈ cov(Y ) be a star-refinement of U. Since ϕ is polyhedrally approximately n-soft, we have a map g : In ×  → X such that ϕg is V-close to f . By the τ -locally finite n-cells property of X, g is V-close to a map g  : In ×  → X such that (gγ (In ))γ ∈ is locally finite in X. Then, it follows that ϕgγ is U-close to f . Moreover, (ϕgγ (In ))γ ∈ is locally finite in Y . Indeed, if it is not locally finite at y ∈ Y , then we can find xi ∈ In and γi ∈  for each i ∈ N so that γi = γj if i = j , and (ϕgγ i (xi ))i∈N converges to y. Because ϕ is proper, it follows from Proposition 1.3.7 that (gγ i (xi ))i∈N has a convergent subsequence in X, which contradicts the local finiteness of (gγ (In ))γ ∈ . " ! For an ANR, the τ -discrete n-cells property is open hereditary, that is: Proposition 3.2.3 Let X be an ANR and n ∈ ω. If X has the τ -discrete (or τ -locally finite) n-cells property, then every open set W in X has the τ -discrete (or τ -locally finite) n-cells property. Proof By virtue of Proposition 3.2.1, it suffices to show that W has the τ -locally finite n-cells property.

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219

 Let f : In ×  → W be a map and U ∈ cov(W ). We can write W = i∈N Wi , where each Wi is open in X and cl Wi ⊂ Wi+1 . For each i ∈ N, let Ri = f −1 (cl Wi \ Wi−1 ), where W0 = ∅. Then, Ri is closed in In ×  and f (Ri ) ⊂ cl Wi \ Wi−1 . Take a star-refinement U0 ∈ cov(W ) of U and define U ∈ cov(W ) and Vi ∈ cov(X), i ∈ N, as follows:  U = U0 |Wi+1 \ cl Wi−1 and Vi = U |W2i ∪ {X \ cl W2i−1 }, i∈N

where W−1 = ∅. By the τ -discrete n-cells property of X, we have maps gi : In ×  → X, i ∈ N, such that gi Vi f and (gi (In × {γ })γ ∈ is discrete in X. Then, gi |R2i−1 U f |R2i−1 (in W ) for each i ∈ N. Using the Homotopy Extension Theorem 1.13.11, we can obtain a map g : In ×  → W such that g|R2i−1 = gi |R2i−1 for each i ∈ N and g U f . Then, it follows that g(Ri ) ⊂ Wi+1 \ cl Wi−2 for each i ∈ N. Let Aγ ,i = (In × {γ }) ∩ R2i−1 , γ ∈ , i ∈ N. As is easily observed, (g(Aγ ,i ))γ ∈ is discrete in W2i+1 \ cl W2i−2 , which implies that (g(Aγ ,i ))γ ∈,i∈N is locally finite in W . For each x ∈ W , let Vx be an open neighborhood of x in W such that Vx meets only finitely many g(Aγ ,i ), γ ∈ , i × N. Take V ∈ cov(W ) so that st V ≺ {Vx | x ∈ W } and V ≺ U . If h is V -close to g, then (h(Aγ ,i )γ ∈,i∈N is locally finite in W . Indeed, for each V ∈ V , st(V , V ) is contained in some Vx . If V ∩h(Aγ ,i ) = ∅, then st(V , V ) ∩ g(Aγ ,i ) = ∅, which implies that Vx ∩ g(Aγ ,i ) = ∅. Such Vx meets only finitely many g(Aγ ,i ), V meets only finitely many h(Aγ ,i ). Recall that g(Ri ) ⊂ Wi+1 \ cl Wi−2 for each i ∈ N. Let Bγ ,i = (In × {γ }) ∩ R2i , γ ∈ , i ∈ N. Then, g(Bγ ,i ) ⊂ W2i+1 \cl W2i−2 . In the same way as the above g, we can construct a map h : In ×  → W so that h V g and (h(Bγ ,i ))γ ∈ is discrete in W2i+2 \ cl W2i−3 , which implies that (h(Bγ ,i ))γ ∈,i∈N is locally finite in W . Since h V g, (h(Aγ ,i ))γ ∈,i∈N is also locally finite in W . Observe In × {γ } =

   (In × {γ }) ∩ Ri = (Ai ∪ Bi ). i∈N

i∈N

Therefore, (h(In × {γ }))γ ∈,i∈N is locally finite in W . " !  Remark 3.3 Let A = λ∈ Aλ be a topological sum of metrizable spaces. Suppose that an ANR X has the following property: (*) every map f : A → X can be approximated by maps g : A → X such that (g(Aλ ))λ∈ is locally finite in X.

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Then, by the same proof as Proposition 3.2.3, it can be proved that every open set in X has property (*). It is said that X has the disjoint n-cells property if every two maps f, g : In → X can be approximated by maps f  , g  : In → X such that f  (In ) ∩ g  (In ) = ∅. Namely, the disjoint n-cells property can be defined by replacing  with 2 = {0, 1}. If X has the disjoint n-cells property for every n ∈ N, it is said that X has the disjoint cells property. The open heritability of the disjoint n-cells property can be directly proved, which is simpler than Proposition 3.2.3.

Proposition 3.2.4 Let X be a paracompact space and n, n1 , . . . , nk ∈ N (k  2) with n1 , . . . , nk  n. If X has the disjoint n-cells property, then all maps fi : Ini → X, i = 1, . . . , k, can be approximated by maps fi : Ini → X such that fi (Ini ) ∩ fj (Inj ) = ∅ for i = j . Proof We shall prove the proposition by induction on k. To see the case k = 2, for m < n, let pm : In → Im and im : Im → In be the maps defined by pm (x) = (x(1), . . . , x(m)) and im (x) = (x(1), . . . , x(m), 0, . . . , 0). We write pn = in = idIn for convenience. Let fj : Inj → X, j = 1, 2, be maps and U ∈ cov(X). By the disjoint n-cells property, the maps fj pnj : In → X are U-close to maps gj : In → X such that g1 (In ) ∩ g2 (In ) = ∅. Then, the maps fj = gj inj : Inj → X are U-close to fj = fj pnj inj and f1 (In ) ∩ f2 (In ) = ∅. Now, assuming the case k − 1, we prove the case k. Let fi : Ini → X, i = 1, . . . , k, be maps. For each U ∈ cov(X), take its star-refinement V ∈ cov(X). By the inductive assumption, the maps f1 , . . . , fk−1 are V-close to maps fi : Ini → X, i = 1, . . . , k − 1, such that fi (Ini ) ∩ fj (Inj ) = ∅ for 1  i < j < k. Take V ∈ cov(X) so that     ni V ≺ V ∧ X \ f1 (In1 ), X \ k−1 i=2 fi (I ) .  , f are V -close to maps Again by the inductive assumption, the maps f2 , . . . , fk−1 k  n  n  n fi : I i → X, i = 2, . . . , k, such that fi (I i ) ∩ fj (I j ) = ∅ for 1 < i < j  k. Then, observe that f1 (In1 ) ∩ fi (In ) = ∅ for every i = 2, . . . , k − 1. Take V ∈ cov(X) so that

    ni V ≺ V ∧ X \ (f1 (In1 ) ∪ fk (Ink )), X \ k−1 i=2 fi (I ) . By the case k = 2, the maps f1 and fk are V -close to maps f1 : In1 → X and fk : Ink → X such that f1 (In1 ) ∩ fk (Ink ) = ∅. Then, observe that f1 (In1 ) ∩ fi (Ini ) = ∅ and fk (In1 ) ∩ fi (Ini ) = ∅ for every i = 2, . . . , k − 1. For each 1 < i < k, let fi = fi . Then, it follows that fi (In ) ∩ fj (In ) = ∅ for 1  i < j  k. Moreover, every fi is U-close to fi because st V ≺ U. " !

3.2 The Discrete (or Disjoint) Cells Property

221

Since In ×  ∈ M1 (τ ) and In × {0, 1} ∈ M0 , we have the following: Fact (1) The M1 (τ )-universality implies the τ -discrete cells property. (2) The M0 -universality implies the disjoint cells property. Remark 3.4 Let E be an infinite-dimensional normed linear space with dens E  τ . By Proposition 2.1.8, we can take  as a δ-discrete set in the sphere SE of E, where 0 < δ < 1. Taking an embedding h : In → E with h(In ) ⊂ (δ/3)BE , we can define a closed embedding f : In ×  → E by f (x, γ ) = h(x) + γ . Thus, each In ×  can be embedded in E as a closed set, that is, In ×  ∈ FE . Consequently, the FE -universality implies the τ -discrete cells property. For a simplex σ with dim σ > 0 and t ∈ I, the following notation is useful:    σ [t] = (1 − s)σˆ + sy  y ∈ ∂σ, 0  s  t . Then, σ [0] = {σˆ }, σ [1] = σ , and σ [t] ⊂ rint σ for every t < 1. Lemma 3.2.5 Let X be an ANR with the τ -discrete cells property and K be a finitedimensional simplicial complex with card K (0)  τ . Suppose that all maps from In , n ∈ N, to X can be approximated by maps with strong Z-set images. Then, each map f : |K| → X can be approximated by OK -maps. Proof By induction on dim K, we shall show the proposition. It is easy to see that the τ -discrete 0-cells property implies the case dim K = 0. Assume that the proposition is valid for any (n − 1)-dimensional simplicial complex. Let dim K = n and f : |K| → X be a map. For each U ∈ cov(X), let V ∈ cov(X) be a starrefinement of U. First, by the inductive assumption and the Homotopy Extension Theorem 1.13.11, we have a map f  : |K| → X such that f  ||K (n−1)| is an OK (n−1) -map and f  V f . Since OK (n−1) = OK |K (n−1) , we apply Proposition 3.1.18 to obtain an open neighborhood R of |K (n−1) | in |K| such that f  |R is an (OK |R)-map, that is, f  belongs to the following subset of C(|K|, X):    g ∈ C(|K|, X)  g|R is an (OK |R)-map . This set is open in C(|K|, X) because COK |R (R, X) is open in C(R, X) by Lemma 3.1.13 and the restriction operator of C(|K|, X) to C(R, X) is continuous by Proposition 3.1.4(2). Hence, we can find V ∈ cov(X) such that st V ≺ V and if a map g  : |K| → X is st V -close to f  , then g  |R is also an (OK |R)-map. Next, using the τ -discrete n-cells property, we can obtain maps fσ : σ → X, σ ∈ K \ K (n−1) , such that {fσ (σ ) | σ ∈ K \ K (n−1) } is discrete in X, fσ V f  |σ , and fσ (σ ) is a strong Z-set in X. For each σ ∈ K \ K (n−1) , let hσ : σ × I → X be a V -homotopy with hσ0 = f  |σ and hσ1 = fσ . For each σ ∈ K \ K (n−1) ,

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

choose tσ ∈ (0, 1) so that cl(σ \ σ [tσ ]) ⊂ R and define a map gσ : σ → X by gσ |σ [tσ ] = fσ |σ [tσ ] and   1−s gσ ((1 − s)σˆ + sy) = hσ (1 − s)σˆ + sy, for (y, s) ∈ ∂σ × [tσ , 1]. 1 − tσ Then, {gσ (σ [tσ ]) | σ ∈ K \ K (n−1) } is discrete in X, and each gσ (σ [tσ ]) is a strong Z-set in X. Observe that gσ |∂σ = f  |∂σ and gσ V f  |σ rel. ∂σ. g

Let g  = prX g˜ : |K| → X, that is, g  |σ = gσ for each σ ∈ K \ K (n−1) . Then, V f  and {g  (σ [tσ ]) | σ ∈ K \ K (n−1) } is discrete in X. Let 

Z=

g  (σ [tσ ]).

σ ∈K\K (n−1)

Since each g  (σ [tσ ]) = fσ (σ [tσ ]) is a strong Z-set in X, their discrete union Z is also a strong Z-set in X by Corollary 2.8.16. Due to Proposition 2.8.13, there is a homotopy ϕ : X × I → X such that ϕ0 = id, ϕ(X × (0, 1]) ∩ Z = ∅, and ϕ is closed over Z, that is, Z ∩ cl ϕ(A) = Z ∩ ϕ(A) for any closed set A in X × I. By Lemma 2.5.2, we have a map α : X → (0, 1] such that    {x} × [0, α(x)]  x ∈ X ≺ ϕ −1 (V ). Since {g  (σ [tσ ]) | σ ∈ K \K (n−1) } is discrete in X, X has a discrete open collection {Wσ | σ ∈ K \ K (n−1) } such that g  (σ [tσ ]) ⊂ Wσ for every σ ∈ K \ K (n−1) . Then, σ [tσ ] ⊂ g −1 (Wσ ). Since |K| is perfectly normal by Proposition 1.5.2, there is a map β : |K| → I such that 

β −1 (0) =

σ [tσ ] and

σ ∈K\K (n−1)



β −1 (1) = |K| \



 rint σ ∩ g −1 (Wσ ) ,

σ ∈K\K (n−1)

which implies 

β −1 ((0, 1]) = |K| \

σ ∈K\K (n−1)

=



σ ∈K\K (n−1)

σ [tσ ] =



(σ \ σ [tσ ])

σ ∈K\K (n−1)

kσ (∂σ × [0, 1)) ⊂ R and

3.2 The Discrete (or Disjoint) Cells Property

β −1 ([0, 1)) =





223

 rint σ ∩ g −1 (Wσ ) .

σ ∈K\K (n−1)

We now define a map g : |K| → X by g(x) = ϕ(g  (x), αg  (x)·β(x)) for each x ∈ |K|. Then, g|σ [tσ ] = g  |σ [tσ ] for each σ ∈ K \ K (n−1) , so we have Z=



g(σ [tσ ]).

σ ∈K\K (n−1)

Moreover, g is V-close to f because g is V -close to g  , g  is V -close to f  , f  is V-close to f , V ≺ V, and st V ≺ U. Since g is st V -close to f  , it follows that g|R is also an OK -map. Finally, we will show that g is an OK -map. Each x ∈ X has an open neighborhood W in X such that (g|R)−1 (W ) is contained in OK (v) for some v ∈ K (0) (the case (g|R)−1 (W ) = ∅ is possible). When x ∈ X \Z, W \Z = W ∩(X \Z) is an open neighborhood of x in X and g −1 (W \ Z) = g −1 (W ) \



g −1 (g(σ [tσ ]))

σ ∈K\K (n−1)

⊂ g −1 (W ) \



σ [tσ ]

σ ∈K\K (n−1)

=g

−1

 (W ) ∩ |K| \

= g −1 (W ) ∩





 σ [tσ ]

σ ∈K\K (n−1)

(σ \ σ [tσ ])

σ ∈K\K (n−1)

⊂ g −1 (W ) ∩ R = (g|R)−1 (W ) ⊂ OK (v). When x ∈ Z, x is contained in only one g(σ [tσ ]). Then, Wσ is an open neighborhood of x in X. Note that ϕ(x, 0) = x and x ∈ ϕ(X × (0, 1]) because x ∈ Z. Hence, ϕ −1 (x) = {(x, 0)}. Consider the following closed set in X × I: A = {(x  , t) ∈ Wσ × I | t  α(x  )}. Then, Z ∩ ϕ(A) = ∅. Since ϕ : X × I → X is closed over Z, it follows that Z ∩ cl ϕ(A) = ∅. Hence, x has an open neighborhood W0 in X such that W0 ⊂ Wσ and W0 ∩ ϕ(A) = ∅, so    ϕ −1 (W0 ) ⊂ (x  , t) ∈ Wσ × I  t < α(x  ) .

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

For each y ∈ g −1 (W0 ), (g  (y), αg  (y)β(y)) ∈ ϕ −1 (W0 ), which means β(y) < 1. Since g  (y) ∈ W0 ⊂ Wσ , it follows that y ∈ rint σ ∩ g −1 (Wσ ). Hence, g −1 (W0 ) ⊂ rint σ ⊂ OK (v) for v ∈ σ (0). Thus, g is an OK -map. " ! Remark 3.5 In Lemma 3.2.5, if the projection prX : X × I → X is a nearhomeomorphism, e.g., X is Q-stable (Corollary 2.3.11), then every map f : C → X from a compactum C can be approximated by maps with strong Z-set images. Indeed, for each U ∈ cov(X), take a homeomorphism ϕ : X × I → X that is U-close to prX . Then, f is U-close to the map f  : C → X defined as follows: f  (x) = ϕ(f (x), 0) for each x ∈ C. Then, f  (C) is a strong Z-set in X because it is collared (Lemma 2.8.2). Note that the disjoint cells property means the m-discrete cells property for every m ∈ N (cf. Proposition 3.2.4). Hence, the proof above is available, even if the τ -discrete cells property and the condition card K (0)  τ are replaced by the disjoint cells property and the condition card K (0) < ℵ0 (i.e., K is a finite simplicial complex). Thus, we have the following: Lemma 3.2.6 Let X be a separable completely metrizable ANR and K a finite simplicial complex. Suppose that all maps from In , n ∈ N, to X can be approximated by maps with strong Z-set images. Then, for any U ∈ cov(X), each map f : |K| → X is U-close to an OK -map. " ! In the above lemma, as is easily observed, the assumption implies that X has the disjoint cells property. In fact, we have this following characterization of the disjoint cells property for a separable completely metrizable ANR X: Proposition 3.2.7 For each separable completely metrizable ANR X, the following are equivalent: (a) X has the disjoint cells property; (b) Each pair of maps f, g : Q → X are arbitrarily close to maps f  , g  : Q → X with f  (Q) ∩ g  (Q) = ∅; (c) C(Q, X) has no isolated points and C(Q, X) contains a countable dense set {fi | i ∈ N} such that fi (Q) ∩ fj (Q) = ∅ if i = j ; (d) Every map f : Q → X can be approximated by maps g : Q → X such that g(Q) is a Z-set in X; (e) For each n ∈ N, every map f : In → X can be approximated by maps g : In → X such that g(In ) is a Z-set in X; (f) For each n ∈ N, every map f : In → X can be approximated by maps g : In → X such that g(In ) is a Zn -set in X. Proof The implications (e) ⇒ (f) ⇒ (a) are trivial. For each n ∈ N, let pn : Q → In and in : In → Q be the maps defined by pn (x) = (x(1), . . . , x(n)) and in (x) = (x(1), . . . , x(n), 0, 0, . . . ).

3.2 The Discrete (or Disjoint) Cells Property

225

Then, pn in = id, from which the implication (d) ⇒ (e) easily follows. On the other hand, in pn converges to idQ as n → ∞. We can use this fact to see the implication (a) ⇒ (b). In condition (c), {fi | i ∈ N} \ {fj } is dense in C(Q, X) for each j ∈ N because fj is not isolated in C(Q, X). Then, each fj (Q) satisfies condition (b) of Theorem 2.8.6, which means that fj (Q) is a Z-set in X. So, we have the implication (c) ⇒ (d). (b) ⇒ (c): We may assume that X = (X, d) is a separable complete metric space. The space C(Q, X) with the sup-metric is complete, where the same letter d stands for the sup-metric. It easily follows from (b) that C(Q, X) has no isolated points. Since C(Q, X) is separable metrizable,8 it has a countable dense set {gi | i ∈ N}. Then, {gi | i > k} is also dense in C(Q, X) for each k ∈ N. We define    M = (fi )i∈N ∈ C(Q, X)N  d(fi , gi ) < 2−i for each i ∈ N . Since M is a Gδ -set in the product space C(Q, X)N , M is also completely metrizable. For each j < k, we have the following open set:  Uj,k = {(fi )i∈N ∈ M | fj (Q) ∩ fk (Q) = ∅ ,  which is dense in M by (b). By the Baire Category Theorem 1.3.15, j 0, there is a map g : In → Mj for some j  i such that f |A = g|A and d(f, g) < ε; (ii) Given ε > 0, there is some δ > 0 such that any map f : In → Mi is εhomotopic to a map g : In → Mj for some j  i such that dist(f (In ), g(In )) > δ. Then, X has the discrete cell-tower property. Modification in Proof of Lemma 3.5.1. Because of the condition (i), replacing f with an arbitrarily close map, we can assume that each f (In ) is contained in some Mi(n) , where i(1) < i(2) < · · · Then, replacing the condition (USC) with (ii), we can perform the proof, that is, we can construct sequences gk : D → X and εk > 0, k ∈ N, so as to satisfy the conditions (1)–(4) and the following additional condition: (0) gk (In ) ⊂ Mj (n) , where j (n)  j (n − 1), i(n). In the construction of gk and εk , let δ > 0 be as the above (ii) instead of (USC). Moreover, we assume that gk (I1 ⊕ · · · ⊕ In−1 ) satisfies (0), hence gk (I1 ⊕ · · · ⊕ In−1 ) ⊂ Mj (n−1) . Then, applying (ii) with (i) instead of (USC),12 we have an ε-homotopy s : In ×I → Mj (n) such that s0 = gk−1 |In , j (n)  max{i(n), j (n−1)} and the condition (6) is satisfied. The map gk |In : I → X obtained by the same definition satisfies gk (In ) ⊂ Mj (n) , that is, the condition (0) is also satisfied.

12 We

use (i) to adjust the image of a homotopy so as to be contained in some Mj .

3.5 Fréchet Spaces and the Countable Product of ARs

253

We will apply Lemma 3.5.1 to topological groups. Let G be a topological group, where e ∈ G is the unit of G. A subset A of G is totally bounded provided, for each neighborhood U of e, there exists some finite set F ⊂ G such that A ⊂  U F = x∈F U x = {yx | x ∈ F, y ∈ U }, where F can be taken as a subset of A because there is a neighborhood V of e such that V V −1 ⊂ U (cf. p. 25). When G is metrizable, G has an admissible right invariant metric d, i.e., d(x, y) = d(xz, yz) for any x, y, z ∈ G (cf. Theorem 3.6.2 in [GAGT]). Then, the total boundedness defined above coincides with the one in a metric space (G, d), that is: • A is totally bounded in the sense of the above definition if and only if for each ε > 0, G is covered by finitely many subsets with diam  ε. In fact, since d is right invariant, we have Bd (e, ε)x = Bd (x, ε) for each x ∈ G and ε > 0, from which the above fact easily follows. — Refer to p. 26.

It is said that a subset X of G is multiplicative if X contains the unit e ∈ G and xy ∈ X for any x, y ∈ X. Proposition 3.5.3 Let X be a path-connected multiplicative subset of a topological group G with an admissible right invariant metric d. If X is locally path-connected at the unit e and any neighborhood of e in X is not totally bounded, then (X, d) has the uniformly separating cells property, hence every open set in X has the discrete cell-tower property. Proof For each ε > 0, choose η > 0 so that x ∈ X with d(x, e) < η is connected to e by a path in X with diam < ε. Also choose δ > 0 so that B(e, η) ∩ X has no finite cover consisting of open balls with radius 2δ. For any map f : In → X, the following A is compact:    A = f (x) · f (y)−1  x, y ∈ In ⊂ G. Then, there exists  some a ∈ B(e, η) ∩ X so that d(a, A) > δ. Otherwise, B(e, η) ∩ X ⊂ y∈A B(y, δ). Since A is compact, we have y1 , . . . , yk ∈ A such  that A ⊂ ki=1 B(yi , δ). Each y ∈ A is contained in some B(yi , δ), which implies  that B(y, δ) ⊂ B(yi , 2δ). Hence, B(e, η) ∩ X ⊂ ki=1 B(yi , 2δ), which contradicts the choice of δ. Let ρ : I → X be a path such that ρ(0) = e, ρ(1) = a, and diam ρ(I) < ε. Since X is multiplicative, we can define an ε-homotopy h : In × I → X by h(x, t) = ρ(t) · f (x) for each x ∈ In and t ∈ I. Then, h0 = f and, for each x, y ∈ In , d(h1 (x), f (y)) = d(a, f (y) · f (x)−1 )  d(a, A),

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

that is, dist(f (In ), h1 (In ))  d(a, A) > δ.

" !

When a topological group G is locally path-connected (equiv., locally connected at the unit e ∈ G because of homogeneity), each path-component of G is clopen in G and homeomorphic to the identity component G0 (= the component of the unit e ∈ G) that is a clopen subgroup of G. Applying Proposition 3.5.3 to the case X = G0 , we have the following: Corollary 3.5.4 Let G is a locally path-connected metrizable topological group. If any neighborhood of the unit e ∈ G is not totally bounded, then every open set in G has the discrete cell-tower property. " ! Due to Theorem 1.4.12, a topological linear space is finite-dimensional if and only if 0 has a totally bounded neighborhood. Combining the above corollary and the Toru´nczyk Characterization Theorem 3.4.7, we have the following: ´ ) A topological linear space is Theorem 3.5.5 (DOBROWOLSKI–TORU NCZYK 2 homeomorphic to  if and only if it is an infinite-dimensional separable completely metrizable AR. " !

Since a Fréchet space (= a locally convex completely metrizable topological linear space) is an AR, the following KADEC–ANDERSON THEOREM can be obtained as a special case. Theorem 3.5.6 (KADEC–ANDERSON) Every infinite-dimensional Fréchet space is homeomorphic to 2 . In particular, RN ≈ 2 .

separable " !

Now, applying Proposition 3.5.3 to convex sets, we will generalize Theorem 3.5.5 as follows: ´ ) A separable completely metrizTheorem 3.5.7 (DOBROWOLSKI–TORU NCZYK able convex set C in a metrizable topological linear space E is homeomorphic to 2 if C is an AR and some point of C has no totally bounded neighborhoods in C.

By the Toru´nczyk Characterization Theorem 3.4.7, this theorem can be reduced to the following lemma: Lemma 3.5.8 A separable convex set C in a metrizable topological linear space E has the discrete cell-tower property if some point of C has no totally bounded neighborhoods in C. Proof We may assume that 0 ∈ C has no totally bounded neighborhoods in C. First, we show that C × R has the discrete cell-tower property. To this end, consider the following subset of the product topological linear space E × R:     C ∗ = (tx, t)  x ∈ C, t ∈ R+ = rC × {r}. r0

In order to see that C × R ≈ C ∗ \ {(0, 0)} has the discrete cell-tower property, we can apply Proposition 3.5.3. So, we have to show that any neighborhood V

3.5 Fréchet Spaces and the Countable Product of ARs

255

of (0, 0) in C ∗ is not totally bounded. On the contrary, assume that V is totally bounded. Then, choose a circled neighborhood W of 0 in E and r ∈ (0, 1) so that (W × [0, r]) ∩ C ∗ ⊂ V . Since rW ⊂ W , it follows that r(W ∩ C) × {r} = (rW × {r}) ∩ (rC × {r}) ⊂ (W × [0, r]) ∩ C ∗ ⊂ V . Hence, r(W ∩ C) is totally bounded. For any neighborhood U of 0 in E, there is a finite set A in r(W ∩ C) such that r(W ∩ C) ⊂ A + rU . Then, W ∩ C ⊂ r −1 A + U , where r −1 A is a finite subset in W ∩ C. Thus, W ∩ C is totally bounded, which contradicts the assumption that 0 ∈ C has no totally bounded neighborhoods in C. Now, using the discrete cell-tower property of C × R, we show the locally finite cell-tower property of C, which implies the discrete cell-tower property. Let f :  n → C be a map. For each U ∈ cov(C), let I n∈N      U∗ = U × (−1, 1)  U ∈ U ∪ C × (R \ {0}) .  ∗ n  The map f : n∈N I → C × R defined by f (x) = (f (x), 0) is U -close to  n  n a map g : n∈N C × R such that (g (I ))n∈N is discrete in C × R. Then,  I → n → C is U-close to f . Since each g  (In ) is contained in g = prC g  : I n∈N C × [−1, 1] and prC |C × [−1, 1] is a proper map onto C, it easily follows that (g(In ))n∈N = (prC g  (In ))n∈N is locally finite in C. " ! of Theorem 3.5.7 above does not hold. In fact, the convex set  The converse −i , 2−i ) ⊂ 2 is homeomorphic to RN ≈ 2 . However, when E (−2 i∈N is completely metrizable and C is a closed convex set in E, the converse of Theorem 3.5.7 clearly holds, that is, we have the following characterization:13 Theorem 3.5.9 A closed convex set C in a completely metrizable topological linear space is homeomorphic to 2 if and only if C is a separable AR and not locally compact. " ! Next, we apply Lemma 3.5.1 to show that infinite products have the locally finite cell-tower property.  Proposition 3.5.10 Let X = i∈N Xi be the countable product of path-connected completely metrizable spaces. If infinitely many Xi ’s are non-compact then X has the uniform cells separation for some admissible metric, which implies that X has the discrete cell-tower property. Proof For each i ∈ N, let di ∈ Metr(Xi ). We assume that di is not totally bounded if Xi is non-compact. Namely, each non-compact Xi has no finite cover with mesh < εi for some εi > 0. Then, we can assume that εi  4. Because of this assumption,

13 A

metric space is compact if and only if it is totally bounded and complete.

256

3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

if Xi is non-compact and A ⊂ Xi is compact, then di (x, A)  1 for some x ∈ Xi . We define a metric d ∈ Metr(X) as follows: d(x, y) =



  min 2−i , di (x(i), y(i)) .

i∈N

We show that (X, d) has the uniform cells separation. For each ε > 0, choose k ∈ N so that 2−k < ε/2 and Xk is non-compact. For each map f : In → X = i∈N Xi and i ∈ N, let fi = pri f : In → Xi , where pri : X → Xi is the projection. Since Xk is non-compact, there is some a ∈ Xk with dk (a, fk (In ))  1. Since In is contractible and Xk is path-connected, we have a homotopy h : In × I →  Xk such that h0 = fk and h1 (In ) = a. We define a homotopy h∗ : In × I → X = i∈N Xi as follows: h∗ (x, t) = (f1 (x), . . . , fk−1 (x), h(x, t), fk+1 (x), . . . ). Then, h∗0 = f and g = h∗1 is the desired map. Indeed, for each x ∈ In and t ∈ I,   d(f (x), h∗ (x, t)) = min 2−k , dk (fk (x), h(x, t)) < ε/2, hence h∗ is an ε-homotopy. For each x, y ∈ In ,   d(f (x), g(y)) = d(f (x), h∗1 (y))  min 2−k , dk (fk (x), h1 (y))    min 2−k , dk (a, fk (In )) = 2−k , that is, dist(f (In ), g(In ))  2−k .

" !

As an application, we can easily prove the following theorem (the details are left to the reader):  Theorem 3.5.11 The countable product space X = i∈N Xi is homeomorphic to the separable Hilbert space 2 if and only if all Xi ’s are separable completely metrizable ARs and infinitely many Xi ’s are non-compact. From now, let us consider the non-separable case. We start with the following: Lemma 3.5.12 Assume that X has the discrete (or locally finite) self-copies  property. Let f : γ ∈ Yγ → X be a map such that (f (Yγ ))γ ∈ is σ -discrete (or σ -locally finite) in X. Then, for each U ∈ cov(X), f is U-close to a map g such that (g(Yγ ))γ ∈ is discrete (or locally finite) in X. Proof By the discrete (or locally finite) self-copies property, we have closed idX and embeddings hi : X → X, i ∈ N, such that every hi is U-close to  (hi (X))i∈N is discrete (or locally finite) in X. We can write  = i∈N i such that each (f (Yγ ))γ ∈i is discrete (or locally finite) in X. The desired map

3.5 Fréchet Spaces and the Countable Product of ARs

257

 g : γ ∈ Yγ → X can be defined by g(y) = hi (f (y), i) for y ∈ Yγ , γ ∈ i , i ∈ N. " ! Lemma 3.5.13 Let X be a completely metrizable ANR with w(X) = τ . Then, X is an 2 ()-manifold if X has the locally finite self-copies property and satisfies the following condition with respect to some d ∈ Metr(X): (∗) For any n ∈ N and ε > 0, each f : In ×  → X is ε-close to a map g such that (g(In × {γ }))γ ∈ is σ -locally finite in X. Proof It is easy to see that the locally finite self-copies property implies the locally finite f.d. τ -polyhedra property (or the locally finite τ -skeletal tower property). We will show that X has the τ -locally finite cells property. Then, we can apply Theorem 3.4.2 (or 3.4.3) to have the result. To see the τ -locally finite cells property, let f : In ×  → X and α : X → (0, 1) be maps. By Lemma 3.5.12, it suffices to find a map g : In ×  → X such that g ∈ Nα (f ) and (g(In × {γ }))γ ∈ is σ -locally finite in X. For each i ∈ N, let    i = γ ∈   2−i  inf{αf (z, γ ) | z ∈ In } < 2−i+1 .  Then,  = i∈N i . By condition (∗), we have maps gi : In × i → X, i ∈ N, such that d(gi , f |In × i ) < 2−i and (gi (In × {γ }))γ ∈i is σ -locally finite in X. Let g : In ×  → X be the map defined by g|In × i = gi . Then, g is clearly the desired one. " ! Remark 3.11 In Remark 3.10, when τ = ℵ0 (i.e.,  = N), condition (i) implies (∗). Then, the space X of Remark 3.10 has no locally finite self-copies property because X ≈ 2 . Hence, X is not 2 -stable, i.e., X × 2 ≈ X. The following lemma will be used in the proof of the non-separable version of Theorem 3.5.11.  Lemma 3.5.14 Let X = i∈N Xi be the countable product of completely metrizable connected ANRs (or ARs) such that w(X) = τ . If each Xi is an AR containing a closed copy of 2 () except for finitely many i ∈ N, then X is an 2 ()-manifold (or X ≈ 2 ()). Proof By virtue of Theorem 3.4.2, it suffices to verify M1 (τ )-universality. We may assume that Xi is an AR containing a closed copy of 2 () for every i > 1. Let f : Y → X be a map of Y ∈ M1 (τ ) and α : X → (0, 1) be a map. We use the same metric d for X as in the proof of Proposition 3.5.10, that is, d(x, y) =



  min 2−i , di (x(i), y(i)) ,

i∈N

where di ∈ Metr(Xi ) for each i ∈ N. Since Xi contains a closed copy of 2 (), we have a closed embedding hi : Y → Xi for each i > 1. For each i > 1, let λi : Xi2 × I → Xi be an equi-connecting map for Xi , that is, λi (x, y, 0) = x,

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λi (x, y, 1) = y, and λi (x, x, t) = x for each x, y ∈ X and t ∈ I. We define a map g : Y → X as follows: g(y) = (pr1 f (y), . . . , pri f (y), λi+1 (pri+1 f (y), hi+1 (y), 2i α(f (y)) − 1), hi+2 (y), hi+3 (y), . . . ) if 2−i < α(f (y))  2−i+1 . Then, d(f (y), g(y)) < α(f (y)) for each y ∈ Y . We show that g is a closed embedding. Assume that g(yn ) → x (n → ∞). Let s = infn∈N α(f (yn )). If s = 0 then α(f (yni )) → 0 for some n1 < n2 < · · · , whence f (yni ) → x, which means α(x) = 0. This is a contradiction. Then, we can choose k ∈ N so that 2−k < s. For each n ∈ N, α(f (yn )) > 2−k , hence prk+2 g(yn ) = hk+2 (yn ). Since hk+2 is a closed embedding, (yn )n∈N is convergent in Y . Thus, g is a closed embedding. " ! Now, we prove the non-separable version of Theorem 3.5.11:  Theorem 3.5.15 Let X = i∈N Xi be the countable product of metrizable spaces such that w(X) = τ = supin w(Xi ) for each n ∈ N. Then, X ≈ 2 () if and only if all Xi ’s are completely metrizable ARs and infinitely many Xi ’s are non-compact. Proof The “only if” part is trivial. To see the “if” part, it suffices to show that X contains a closed copy of 2 (). In fact, considering products of infinitely many Xi instead of Xn , we can assume that each Xn contains a closed copy of 2 (), hence the result follows from the lemma above. As is well known, every path-connected space is arcwise connected (cf. [GAGT, Corollary 5.14.7]). Then, it is easy to see that the product Y1 × Y2 of non-compact path-connected completely metrizable spaces contains a closed copy of the half real 2 line R+ = [0, ∞). By Theorem 3.5.11, RN + ≈  . Without loss of generality, we may  assume that w(X) = limi∈N w(X2i ), where every X2i is an AR. Then, Y = i∈N X2i is a completely metrizable AR with w(Y ) = w(X) and X contains a closed copy of Y × 2 . We show that Y × 2 ≈ 2 (). To this end, it suffices to verify condition (∗) in Lemma 3.5.13. We use the following metric for Y × 2 : d((y, z), (y  , z )) =



  min 2−i , di (y(i), y  (i)) + z − z .

i∈N

Let n ∈ N and f : In ×  → Y × 2 be a map. For each ε > 0, choose k ∈ N −k so that 2 < ε. Let Z = i>k X2i . Since w(Z) = w(X) = card , Z has a σ discrete open basis B = {Bγ | γ ∈ }. Then, we have a map ϕ :  → Z such that ϕ(γ ) ∈ Bγ for each γ ∈ . We define a map g : In ×  → Y × 2 as follows: g(x, γ ) = (pk (f (x, γ )), ϕ(γ ), q(f (x, γ ))) ∈

k # i=1

X2i × Z × 2 = Y × 2 ,

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259

 where pk : Y × 2 → ki=1 X2i and q : Y × 2 → 2 are the projections. Since B is σ -discrete in Z, it follows that ϕ() is σ -discrete in Z, which implies that (g(In × {γ }))γ ∈ is σ -discrete in Y × 2 . Thus, Y × 2 satisfies condition (∗) in Lemma 3.5.13, hence Y × 2 ≈ 2 (). " ! As a special case of Theorems 3.5.11 and 3.5.15, we have the following: Corollary 3.5.16 For every infinite set , J ()N ≈ 2 ().

" !

The Kadec–Anderson Theorem 3.5.6 can be extended to the non-separable case as follows: ´ ) Every infinite-dimensional Theorem 3.5.17 (KADEC–ANDERSON–TORU NCZYK Fréchet space is homeomorphic to a Hilbert space with the same weight.

Proof Since the separable case has been proved, we prove the non-separable case. Let X be an infinite-dimensional non-separable Fréchet space with w(X) = τ > ℵ0 . We may assume that X has an admissible F -norm  ·  (Theorem 1.4.15). First, observe that X contains an infinite-dimensional separable closed linear subspace X0 . Indeed, such an X0 can be obtained as the closure of the linear span of a linearly independent countable subset A ⊂ X. Note that X0 is a separable Fréchet space, which is homeomorphic to 2 by the Kadec–Anderson Theorem 3.5.6. By the Bartle–Graves–Michael Theorem 1.4.16, X ≈ X1 × 2 for some X1 , hence X × 2 ≈ X1 × 2 × 2 ≈ X1 × 2 ≈ X. Then, as mentioned after Corollary 3.3.14, it follows that X has the discrete selfcopies property. By virtue of Corollary 3.4.5, it suffices to see that any closed set A in X with w(A) < τ is a Z-set. This will be proved in more general setting as the following proposition. " ! Proposition 3.5.18 Let C be a non-separable convex set in a metrizable topological linear space E. Then, every closed set A in C with dens A < dens C is a Z-set in C. Proof Due to Theorem 1.4.15, E has an admissible F -norm  · . Without loss of generality, we may assume that 0 ∈ C. Let n ∈ N and ε > 0. It suffices to show that each map f : In → C is ε-close to a map g : In → C \ A. Observe that dens fl(A ∪ f (In )) = dens(A ∪ f (In )) = dens A < dens C. Hence, there exists v ∈ C \ cl fl(A ∪ f (In )). Because f (In ) − v ≈ f (In ) is compact, we can find δ > 0 such that δ(f (In ) − v) ⊂ B(0, ε). We define a map g : In → C \ fl A as follows: g(x) = (1 − δ)f (x) + δv for every x ∈ In .

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Then, g is ε-close to f . Indeed, for each x ∈ In , f (x) − g(x) = δ(f (x) − v) ∈ B(0, ε), which means that f (x) − g(x) < ε. " ! Remark 3.12 As Theorem 3.5.6 was generalized to Theorems 3.5.7 and 3.5.17 can be generalized to convex sets in Fréchet spaces. By virtue of Proposition 3.5.18, it suffices to show that every non-separable convex Gδ set in a Fréchet space has the locally finite self-copies property. T. Banakh and I. Zarichnyy [24] showed that the locally finite self-copies property can be weakened to the locally finite displacing property (cf. Remark 3.9, p. 249). Using the result of [20], T. Banakh and R. Cauty [21] showed that every non-separable convex Gδ set in a Fréchet space has the locally finite displacing property (cf. Remark 3.8, p. 239). Thus, every non-separable convex Gδ set in a Fréchet space is homeomorphic to a Hilbert space. For details, refer to the papers [20, 21], and [24].

3.6 The Function Space C(X, Y ) In this section, it is proved that the space C(X, Y ) of maps from a non-discrete compactum X to a completely metrizable ANR Y without isolated points is an 2 manifold. Recall that the compact-open topology on C(X, Y ) is generated by the sets    K; U  = f ∈ C(X, Y )  f (K) ⊂ U , where K is any compact set in X and U is any open set in Y . On the other hand, the pointwise convergence topology on C(X, Y ) is generated by the sets    x; U  = f ∈ C(X, Y )  f (x) ∈ U , where x is any point of X and U is any open set in Y . For a subset F ⊂ C(X, Y ), let clp F denote the closure of F with respect to the pointwise convergence topology. Then, it follows that cl F ⊂ clp F. When Y = (Y, d) is a metric space, a subset F ⊂ C(X, Y ) is said to be equicontinuous at x0 ∈ X provided that for each ε > 0, there is a neighborhood U of x0 in X such that f (U ) ⊂ Bd (f (x0 ), ε) for all f ∈ F. It is said that F is equicontinuous on X if it is equi-continuous at each point of X. Proposition 3.6.1 Let X be a locally compact space and Y = (Y, d) a metric space. Every compact subset F ⊂ C(X, Y ) is equi-continuous on X. Proof To show that F is equi-continuous at x0 ∈ X, let ε > 0. For each f ∈ F, since X is locally compact and f is continuous, x0 has a compact neighborhood Nf in X such that f (Nf ) ⊂ B(f (x0 ), ε/2). Note that Nf ; B(f (x0 ), ε/2) is an open

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261

neighborhood of f in C(X, Y ). Since F is compact, we have f1 , . . . , fn ∈ F such that F⊂

n 

Nfi ; B(fi (x0 ), ε/2).

i=1

 Then, N = ni=1 Nfi is a neighborhood of x0 in X. Each f ∈ F is contained in some Nfi ; B(fi (x0 ), ε/2), that is, f (Nfi ) ⊂ B(fi (x0 ), ε/2). For each x ∈ N, since f (x), f (x0 ) ∈ B(fi (x0 ), ε/2), it follows that d(f (x), f (x0 ))  d(f (x), fi (x0 )) + d(fi (x0 ), f (x0 )) < ε/2 + ε/2 = ε. Hence, f (N) ⊂ B(f (x0 ), ε). This means that F is equi-continuous at x0 .

" !

Proposition 3.6.2 Let X be an arbitrary space and Y = (Y, d) a metric space. If F ⊂ C(X, Y ) is equi-continuous on X, then clp F is also equi-continuous on X. Proof We show the equi-continuity of clp F at each x0 ∈ X. For each ε > 0, since F is equi-continuous at x0 , there is a neighborhood U of x0 in X such that f (U ) ⊂ Bd (f (x0 ), ε/3) for all f ∈ F. Then, it suffices to show that g(U ) ⊂ Bd (g(x0 ), ε) for all g ∈ clp F. On the contrary, assume that g(U ) ⊂ Bd (g(x0 ), ε) for some g ∈ clp F. Then, we have x1 ∈ U such that d(g(x0 ), g(x1 ))  ε. Because g ∈ clp F, x0 , Bd (g(x0 ), ε/3) ∩ x1 , Bd (g(x1 ), ε/3) ∩ F = ∅, that is, there is some f ∈ F such that d(f (x0 ), g(x0 )), d(f (x1 ), g(x1 )) < ε/3. Since x1 ∈ U , it follows that d(f (x0 ), f (x1 )) < ε/3. Then, d(g(x0 ), g(x1 ))  d(g(x0 ), f (x0 )) + d(f (x0 ), f (x1 )) + d(f (x1 , g(x1 )) < ε/3 + ε/3 + ε/3 = ε, which is a contradiction because d(g(x0 ), g(x1 ))  ε. Therefore, g(U ) ⊂ Bd (g(x0 ), ε) for all g ∈ clp F. " ! Using the above proposition, we can show the following well-known theorem: Theorem 3.6.3 (ARZELÀ–ASCOLI) Let X be an arbitrary space and Y = (Y, d) a metric space. Suppose that F ⊂ C(X, Y ) satisfies the following conditions: (i) F is equi-continuous on X, (ii) clY F(x) = clY {f (x) | f ∈ F} is compact for each x ∈ X. Then, cl F is compact, that is, F has the compact closure with respect to the compact-open topology.

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

 Proof The product space x∈X clY F(x) is compact by the Tychonoff Theorem [GAGT, Theorem 2.1.1]. Since evx : Cp (X, Y ) → Y is continuous, we have (clp F)(x) = evx (clp F) ⊂ clY evx (F) = clY F(x),  which implies clp F ⊂ x∈X clY F(x). Hence, the subspace clp F of Cp (X, Y ) is compact. Thus, it suffices to show that cl F = clp F as spaces. Since id : C(X, Y ) → Cp (X, Y ) is continuous, we have cl F ⊂ clp F. Then, it suffices to show the continuity of the inclusion of Cp (X, Y ) with the pointwise convergent topology to C(X, Y ) with the compact-open topology. By Proposition 3.6.2, (i) implies that clp F is equi-continuous on X. The evaluation ev : Cp (X, Y )×X → Y is not continuous in general (cf. Remark 1.1). However, the restriction ev| clp F×X is continuous, where clp F has the pointwise convergent topology. Indeed, let (f, x) ∈ clp F × X. For each ε > 0, x; Bd (f (x), ε/2) is a neighborhood of f in Cp (X, Y ). By the equi-continuity of clp F, there is a neighborhood U of x in X such that f  (U ) ⊂ Bd (f  (x), ε/2) for all f  ∈ clp F. Then, for each (f  , x  ) ∈ x; Bd (f (x), ε/2) × U , d(f  (x  ), f (x))  d(f  (x  ), f  (x)) + d(f  (x), f (x)) < ε/2 + ε/2 = ε, hence ev(f  , x  ) = f  (x  ) ∈ Bd (f (x), ε) = Bd (ev(f, x), ε). By Proposition 1.1.2(1), ev| clp F × X induces the map of clp F to C(X, Y ), which is the inclusion. Hence, clp F ⊂ cl F and the compact-open topology is not finer than the pointwise convergence topology on clp F. Consequently, clp F = cl F as spaces. " ! Theorem 3.6.4 Let X be a compactum and Y a metrizable space. (1) If Y is completely metrizable, then so is C(X, Y ). (2) If Y is separable, then so is C(X, Y ). (3) If Y is an ANR, then so is C(X, Y ). Proof (1): Let d be an admissible complete metric on Y . By Proposition 1.1.2(5), the sup-metric induced by d is admissible for C(X, Y ). Let (fn )n∈N be a Cauchy sequence in C(X, Y ) with respect to the sup-metric. For each x ∈ X, (fn (x))n∈N is a Cauchy sequence of Y = (Y, d). Then, we can define f (x) = limn→∞ fn (x) for each x ∈ X. We will show that f ∈ C(X, Y ) and limn→∞ fn = f . To see that f ∈ C(X, Y ), let x ∈ X and ε > 0. First, choose n0 ∈ N so that d(fi , fj ) < ε/3 for each i, j  n0 . By the continuity of fn0 , x has a neighborhood U in X such that fn0 (U ) ⊂ Bd (fn0 (x), ε/9). Since f (x) = limn→∞ fn (x), we have nx ∈ N such that d(fi (x), f (x)) < ε/9 for each i  nx . Similarly, for each

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263

x  ∈ U , we have nx  ∈ N such that d(fi (x  ), f (x  )) < ε/9 for each i  nx  . Let i = max{n0 , nx , nx  }. Then, it follows that d(f (x  ), f (x))  d(f  (x  ), fi (x  )) + d(fi (x  ), fn0 (x  )) + d(fn0 (x  ), fn0 (x)) + d(fn0 (x), fi (x)) + d(fi (x), f (x)) < ε/9 + ε/3 + ε/9 + ε/3 + ε/9 = ε. Hence, f is continuous. Finally, we show that limn→∞ fn = f . For each ε > 0, take the same n0 ∈ N as above. Then, we show that d(fn , f ) < ε for each n  n0 . For each x ∈ X, take the same nx ∈ N as above. Let i = max{nx , n}, Then, we have d(fn (x), f (x))  d(fn (x), fi (x)) + d(fi (x), f (x)) < ε/3 + ε/9 = 4ε/9. Hence, d(fn , f )  4ε/9 < ε. (2): From Proposition 1.1.2(5), it follows that C(X, Y ) is second countable, so it is separable. (3): Let Z be a metrizable space, A a closed set in Z, and let f : A → C(X, Y ) be a map. Then, f induces a map f˜ : A × X → Y defined by f˜(z, x) = f (z)(x) for each (z, x) ∈ A × X. Since Y is an ANR, A × X has a neighborhood W in Z × X and f˜ extends to a map f  : W → Y . Because X is compact, there is a neighborhood U of A in Z such that U × X ⊂ W . Then, f  |U × X induces a map f¯ : U → C(X, Y ) defined by f¯(z)(x) = f  (z, x) for each (z, x) ∈ U × X. Observe that f˜ is an extension of f . " ! As another application of the Toru´nczyk Characterization of 2 -manifolds, we prove the following: Theorem 3.6.5 Let X be a non-discrete compactum and Y be a separable completely metrizable ANR without isolated points. Then, the space C(X, Y ) with the compact-open topology is an 2 -manifold. Proof By Theorem 3.6.4, C(X, Y ) is a completely metrizable separable ANR. Hence, it suffices that C(X, Y ) has the discrete cell-tower property. To this end, we will apply Lemma 3.5.2. Since X is non-discrete, X has a cluster points x∞ . Take a sequence (xi )i∈N of distinct points of X converging to x∞ , and define a tower M1 ⊂ M2 ⊂ · · · ⊂

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

C(X, Y ) as follows:    Mi = f ∈ C(X, Y )  f (xj ) = f (x∞ ) for all j  i . Since Y has no isolated points, Y has an admissible complete metric d such that every component of Y has the diameter > 1 (cf. Proposition 1.3.22(1)). By Theorem 1.13.37, we can assume that Y = (Y, d) is a uniform ANR. Adopting the sup-metric induced by d, we verify the conditions (i) and (ii) of Lemma 3.5.2. (i): Let A ⊂ In be a compactum, f : In → C(X, Y ) a map with f (A) ⊂ Mi0 , and ε > 0. Then, f induces a map f˜ : In × X → Y defined by f˜(z, x) = f (z)(x). Since Y is a uniform ANR, we have δ > 0 such that any two δ-close maps to Y are ε-homotopic (Theorem 1.13.32). Since f (In ) is a compact subset of C(X, Y ), it is equi-continuous by Proposition 3.6.1, that is, there exists a neighborhood U of x∞ in X such that f (z)(U ) ⊂ B(f (z)(x∞ ), δ), that is, f˜({z} × U ) ⊂ B(f˜(z, x∞ ), δ). Then, we have i1  i0 such that xi ∈ U for all i  i1 . Note that f˜(z, xi ) = f˜(z, x∞ ) for every z ∈ A and i  i0 , and d(f˜(z, xi ), f˜(z, x∞ )) < δ for every z ∈ In and i  i1 . Using the Homotopy Extension Theorem 1.13.11, we can easily obtain a map g  : In × X → Y such that g  ε f˜, g  |A × X = f˜|A × X, and g  (z, xi ) = g  (z, x∞ ) = f˜(z, x∞ ) for every z ∈ In and i  i1 . Then, g  induces a map g : In → C(X, Y ) defined by g(z)(x) = g  (z, x), where g is ε-close to f and g|A = f |A. Moreover, g(z)(xi ) = g  (z, xi ) = g  (z, x∞ ) = g(z)(x∞ ) for every z ∈ In and i  i1 , which means that g(In ) ⊂ Mi1 . (ii): For each ε > 0, we take δ > 0 as in (i), where we may assume that δ < 1. Each map f : In → Mi0 induces a map f˜ : In × X → Y defined by f˜(z, x) = f (z)(x), where f˜(z, xi ) = f˜(z, x∞ ) for every z ∈ In and i  i0 . By the compactness of In × X, there exist y1 , . . . , ym ∈ Y such that f˜(In × X) ⊂  m j =1 B(yj , δ/3). Since every component of Y has the diameter > δ, we can choose vj ∈ B(yj , δ) \ B(yj , 2δ/3) for each j = 1, . . . , m. Using the Homotopy Extension Theorem 1.13.11, we can obtain a map gj : Y → Y such that gj ε id, gj |Y \ B(yj , ε) = id, and gj (B(yj , δ/3) = {vj }. Then, the following holds: (*) max{d(y, gj (y)) | j = 1, . . . , m} > δ/3 for every y ∈ f˜(In × X). Again using the Homotopy Extension Theorem 1.13.11, we can easily obtain a map g  : In × X → Y such that g  ε f˜, g  (z, x∞ ) = f˜(z, x∞ ), g  (z, xi ) = f˜(z, xi ) for i > i0 + m, and g  (z, xi0 +j ) = gj f˜(z, xi0 +j ) for j = 1, . . . , m.

3.6 The Function Space C(X, Y )

265

Then, g  induces a map g : In → C(X, Y ) defined by g(z)(x) = g  (z, x), where the homotopy g  ε f˜ induces a homotopy g ε f . Moreover, g(In ) ⊂ Mi0 +m because g(z)(xi ) = g  (z, xi ) = f˜(z, xi ) = f (z)(x∞ ) = g(z)(x∞ ) for i > i0 + m. It remains to show that dist(f (In ), g(In ))  δ/6. Conversely, assume that dist(f (In ), g(In )) < δ/6. Then, we have z, z ∈ In such that d(f (z), g(z )) < δ/6. Observe that d(f˜(z, x∞ ), f˜(z , x∞ )) = d(f˜(z, x∞ ), g  (z , x∞ )) = d(f (z)(x∞ ), g(z )(x∞ ))  d(f (z), g(z )) < δ/6. Moreover, for each j = 1, . . . , m, d(f˜(z, x∞ ), gj f˜(z , x∞ )) = d(f˜(z, xi0 +j ), gj f˜(z , xi0 +j )) = d(f˜(z, xi0 +j ), g  (z , xi0 +j ))  d(f (z), g(z )) < δ/6. Consequently, d(f˜(z , x∞ ), gj f˜(z , x∞ ))  d(f˜(z , x∞ ), f˜(z, x∞ )) + d(f˜(z, x∞ ), gj f˜(z , x∞ )) < δ/6 + δ/6 = δ/3, which contradicts (*). Thus, we have dist(f (In ), g(In ))  δ/6.

" !

Remark 3.13 In Theorem 3.6.5, if X is discrete, then X has only finitely many points. In this case, C(X, Y ) = Y X ≈ Y n , where n = card X < ∞. If Y has an isolated point y0 , then the constant map cy0 : X → Y with cy0 (X) = {y0 } is an isolated point of C(X, Y ). Remark 3.14 When X is non-compact and Y = (Y, d) is a separable complete metric space, it was proved by A. Yamashita [162] that (1) if Y is an ANRU (= ANR in the category of metric spaces and uniform continuous mappings) and the diameters of components of Y are bounded away from zero, then the space C(X, Y ) with the sup-metric is an 2 (2ℵ0 )-manifold; (2) if Y is a connected complete Riemannian manifold, then the subspace CB (X, Y ) of bounded maps is an 2 (2ℵ0 )manifold. At the end of this section, we will introduce one of the famous longstanding problems, called the HOMEOMORPHISM GROUP PROBLEM. For a compactum X,

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

let Homeo(X) be the homeomorphism group of X, that is, the subspace of C(X, X) consisting of all homeomorphisms of X onto itself, which is a topological group with function composition as the group operation (cf. Proposition 1.1.2(3)). For a closed set A ⊂ X, let HomeoA (X) = {h ∈ Homeo(X) | h|A = id}, where Homeo∅ (X) = Homeo(X). For a metric d ∈ Metr(X), we can define a metric d ∗ on Homeo(X) as follows: d ∗ (f, g) = d(f, g) + d(f −1 , g −1 ) = sup d(f (x), g(x)) + sup d(f −1 (x), g −1 (x)). x∈X

x∈X

Then, d ∗ is an admissible complete metric. For an n-manifold M with the boundary ∂M, we write Homeo∂M (M) = Homeo∂ (M). The HOMEOMORPHISM GROUP PROBLEM is the following fascinating problem:

•

? HOMEOMORPHISM GROUP PROBLEM

For a compact n-manifold M (n ∈ N), is Homeo∂ (M) an 2 -manifold? When n = 1, 2, this problem is affirmative. Indeed, R.D. Anderson proved in [11] that Homeo∂ (X) is an 2 -manifold for every non-discrete finite graph X (i.e., X = |K| for some one-dimensional finite simplicial complex K). The proof of Homeo∂ (I) ≈ 2 can be found in the paper [91]. The two-dimensional case was proved in [100] by R. Luke and W.K. Mason, that is, Homeo∂ (M) is an 2 manifold for every compact surface (two-dimensional manifold) M. Generalizing this, W. Jakobsche [88] showed that Homeo(X) is an 2 -manifold for a compact two-dimensional polyhedron X. The Homeomorphism Group Problem is still open for n > 2. The 2 -stability of Homeo∂ (X) was independently proved by R. Geoghegan [68] and J. Keesling [91] under some assumption, that is, X has an open set (called a cone path) admitting some kind of structure in [68] or a nontrivial flow in [91], where the former implies the latter. Thus, by virtue of the Toru´nczyk Factor Theorem 2.2.14, the Homeomorphism Group Problem is reduced to the problem of whether Homeo∂ (M) ˇ ˘i [39] is an ANR or not. R.D. Edwards and R.C. Kirby [60] and A.V. Cernavski independently showed that Homeo(M) is locally contractible for every compact nmanifold. Furthermore, S. Ferry [64] and H. Toru´nczyk [141] independently proved the following: Theorem 3.6.6 For every compact Q-dimensional manifold M, Homeo(M) is an ANR (hence an 2 -manifold). By using the result of Edwards and Kirby in [60], Haver [70] reduced the Homeomorphism Group Problem to the following problem:

3.7 Cell-Like Images of Q-Manifolds

267

•

? REDUCED HOMEOMORPHISM GROUP PROBLEM

For n > 2, is Homeo∂ (Bn ) an ANR?

The following result (or the method for proving) is called the ALEXANDER TRICK, which was demonstrated by J.W. Alexander in [1]: Theorem 3.6.7 (ALEXANDER TRICK) For every n ∈ N, Homeo∂ (Bn ) is contractible. Namely, a homeomorphism of the n-ball Bn fixing the boundary sphere Sn−1 is canonically isotopic to the identity. Proof A contraction ϕ : Homeo∂ (Bn ) × I → Homeo∂ (Bn ) can be defined as follows:  x if x  1 − t, ϕ(h, t)(x) = (1 − t)h((1 − t)−1 x) if x  t < 1. Then, ϕ1 (h) = id and ϕ1 (h) = h for every h ∈ Homeo∂ (Bn ).

" !

In the above proof, ϕt (Bd (id, ε)) ⊂ Bd (id, ε) for each t ∈ I, where d is the sup-metric induced by the usual norm. Indeed, d(ϕt (h), id)  (1 − t)d(h, id) for each h ∈ Homeo∂ (Bn ). Since Homeo∂ (Bn ) is homogeneous as a topological group, it follows that Homeo∂ (Bn ) is locally contractible.

3.7 Cell-Like Images of Q-Manifolds We want to characterize Q-manifolds in the same way as the Characterization of 2 -Manifolds 3.4.7. Since Q ≈ C(Q) by Theorem 2.7.5 and we have obtained Q-manifold versions of all results used in the proof of Proposition 2.9.1, the Qmanifold version of Proposition 2.9.1 is also valid. However, it cannot be applied because we have not proved that X × Q is a Q-manifold for a locally compact ANR X. This will be proved in the next section. To prove this and the characterization of Q-manifolds, we need a more general result. Lemma 3.7.1 Let X be a separable locally compact ANR with the disjoint cells property. Then, the space C(Q, X) has a countable dense set {αi | i ∈ N} consisting of Z-embeddings such that αi (Q) ∩ αj (Q) = ∅ if i = j . Proof Due to Proposition 3.2.8, X is M0 -universal. Then, condition (b) of Proposition 3.2.7 can be strengthened as follows: (b’) All maps f, g : Q → X are arbitrarily close to Z-embeddings f  , g  : Q → X with f  (Q) ∩ g  (Q) = ∅.

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Because of the separability, C(Q, X) has a countable dense set {fi | i ∈ N}. As for the implication (b) ⇒ (c) of Proposition 3.2.7, we can apply the Baire Category Theorem 1.3.15 (using the above (b’) instead of (b)) to find (αi )i∈N ∈ C(Q, X)N such that each αi is an embedding with d(αi , fi ) < 2−i and αi (Q) ∩ αj (Q) = ∅ for i = j , where d is the sup-metric with respect to an admissible complete metric for X. Then, each αi is a Z-embedding and {αi | i ∈ N} is dense in C(Q, X). " ! For each map f : X → Y , we define    N(f ) = x ∈ X  f −1 (x) is non-degenerate . Proposition 3.7.2 Let f : M → X be a cell-like map from a Q-manifold M onto an ANR X. If f −1 (N(f )) is contained in a Z-set A in M, then f is a nearhomeomorphism and f (A) is a Z-set in X (hence so is cl N(f )). Proof First, note that f is a fine homotopy equivalence by Theorem 1.15.9 and f (M \ A) = X \ f (A) because f −1 (N(f )) ⊂ A. Since A is a Z-set in M and f (A) is closed in X, it is easy to see that f (A) is a Z-set in X. As the proper image of a locally compact space, X is locally compact, hence it is completely metrizable. To show that f is a near-homeomorphism, we apply the Bing Shrinking Criterion 2.7.1. Since f is a closed map, for each U ∈ cov(M) and V ∈ cov(X), it suffices to construct a homeomorphism ϕ : M → M such that {ϕf −1 (x) | x ∈ X} ≺ U and f ϕ is V-close to f (cf. Remark 2.12): {f −1 (x) | x ∈ X}

f

M

ϕ shrinks

ϕ

U

M

V

X

f g

Choose locally finite open covers U ∈ cov(M) and V ∈ cov(X) so that the closures of each U ∈ U and V ∈ V are compact, st2 V ≺ V and st U ≺ U ∧ f −1 (V ). Let g : X → M be a V -homotopy inverse of f and h : M × I → M an f −1 (V )homotopy such that h0 = idM and h1 = gf , where h is proper by Proposition 1.3.8. By Theorem 2.10.11 and Proposition 1.3.8, h1 |A is properly U -homotopic to a Z-embedding, which extends to a homeomorphism ϕ : M → M such that ϕ is st2 f −1 (V )-isotopic to id by Theorem 2.11.6. Then, f ϕ is V-close to f . If f −1 (x) is non-degenerate, then h1 (f −1 (x)) = gf (f −1 (x)) = g(x) and f −1 (x) ⊂ A, hence ϕf −1 (x) ⊂ st(g(x), U ). Then, {ϕf −1 (x) | x ∈ X} ≺ U. This completes the proof. " ! For each perfect map f : X → Y between ANRs X and Y , the mapping cylinder Mf (= M(f )) is an ANR by Corollary 1.13.17. If f is cell-like, then so is the

3.7 Cell-Like Images of Q-Manifolds

269

quotient map q : X × I → Mf and every non-degenerate q −1 (y) is contained in the Z-set X × {0} in X × I. By the above proposition, we have the following MAPPING CYLINDER THEOREM that is a generalization of Theorem 2.7.5: Theorem 3.7.3 (MAPPING CYLINDER) Let f : M → X be a cell-like map from a Q-manifold M onto an ANR X. Then, the mapping cylinder M(f ) (= Mf ) is a Q-manifold with X a Z-set in M(f ) and the natural (quotient) map qf : M × I → M(f ) is a near-homeomorphism. " ! Strengthening Proposition 3.7.2 a little, we have the following generalization of the Q-manifold version of Proposition 2.9.1: Proposition 3.7.4 Let f : M → X be a cell-like map from a Q-manifold M onto an ANR X. If cl N(f ) is a Z-set in X, then f is a near-homeomorphism. Proof For each U ∈ cov(X), X has an admissible complete metric d such that {Bd (x, 1) | x ∈ X} ≺ U and each Bd (x, 1) is compact (cf. 1.3.22(1)). We shall construct a cell-like map g : M → X such that d(f, g)  1/2, N(g) ⊂ cl N(f ), and cl g −1 (N(g)) is a Z-set in M. Then, we can apply Proposition 3.7.2 to obtain a homeomorphism that is 1/2-close to g, hence it is U-close to f . Thus, f is a near-homeomorphism. Let A = cl N(f ). Note that f |M \ f −1 (A) : M \ f −1 (A) → X \ A is a homeomorphism, hence X \ A is a Q-manifold. By Lemma 3.7.1, C(Q, M) has a countable dense set {αi | i ∈ N} consisting of Z-embeddings. Note that a closed set in M is a Z-set if it misses all αi (Q). Then, it suffices to construct a cell-like map g : M → X so that N(g) ⊂ A and g −1 (A) ∩ αi (Q) = ∅, i.e., gαi (Q) ∩ A = ∅ for all i ∈ N. Let ε0 = 2−1 , g0 = f , and take an locally finite open cover V of X with mesh V < ε0 /10. Since A is a Z-set in X and X \ A is a Q-manifold, we apply the Strong Universality Theorem 2.10.10 (or 2.10.11) to obtain a V-homotopy h : Q × I → X such that h0 = g0 α1 , h1 : Q → X \ A is a Z-embedding and ht |Q \ α1−1 g0−1 (Nd (A, ε0 )) = g0 α1 |Q \ α1−1 g0−1 (Nd (A, ε0 )) for every t ∈ I. Then, β1 = g0−1 h1 : Q → M \ g0−1 (A) ⊂ M is a Z-embedding and β1 |Q \ α1−1 g0−1 (Nd (A, ε0 )) = α1 |Q \ α1−1 g0−1 (Nd (A, ε0 )). By the approximate softness of the cell-like map g0 = f (Theorems 1.15.9 and 1.15.1), we have a homotopy h˜ : Q × I → M such that h˜ 0 = α1 , h˜ 1 = β1 , h˜ t |Q \ α1−1 g0−1 (Nd (A, ε0 )) = α1 |Q \ α1−1 g0−1 (Nd (A, ε0 )) for every t ∈ I, and g0 h˜ is V-close to h. The last condition implies that h˜ is a g0−1 (st V)-homotopy. We can apply the Z-Set Unknotting Theorem 2.11.6 to obtain a homeomorphism ϕ1 : M → M such that ϕ1 α1 = β1 , ϕ1 is f −1 (st2 V)-close to id, and ϕ1 |M \

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

g0−1 (Nd (A, ε0 )) = id. Thus, we have a cell-like map g1 = g0 ϕ1 : M → X such that d(g1 , g0 ) < ε0 /2 and g1 |M \ g0−1 (Nd (A, ε0 )) = g0 |M \ g0−1 (Nd (A, ε0 )). Since g1−1 (x) = ϕ1−1 g0−1 (x) for every x ∈ X and ϕ is a homeomorphism, we have N(g1 ) = N(g0 ), so cl N(g1 ) = A. Since g1 α1 = h1 , it follows that g1 α1 (Q) ∩ A = ∅. We define ε1 =

1 2

  min ε0 , dist(g1 α1 (Q), A) > 0.

By the same construction as above, we can inductively obtain cell-like maps gi : M → X and εi > 0, i ∈ N, such that cl N(gi ) = A, d(gi , gi−1 ) < εi−1 /2, −1 −1 gi |M \ gi−1 (Nd (A, 2−i )) = gi−1 |M \ gi−1 (Nd (A, 2−i )),

εi  εi−1 /2, and εi < dist(gi αj (Q), A) for each j  i. Then, (gi )i∈N is uniformly convergent to a map g : M → X, which is a fine homotopy equivalence by Theorem 1.15.11. Moreover, d(g, gi ) 

∞  j =1

εi+j /2 

∞ 

2−j εi = εi ,

j =1

which implies that gαj (Q) ∩ A = ∅ for each j  i. Then, d(g, f ) = d(g, g0 )  ε0 = 1/2 and g −1 (A) misses all αi (Q). Since g is proper by Proposition 1.3.8, g is cell-like by Theorem 1.15.9. To see N(g) ⊂ A, let x ∈ X \ A and choose i ∈ N so that 2−i < d(x, A), that is, x ∈ X \ Nd (A, 2−i ). Let y ∈ g −1 (x). If y ∈ gi−1 (Nd (A, 2−(i+1) )), then d(g(y), gi (y)) < εi  2−(i+1) , hence x = g(y) ∈ Nd (A, 2−i ). This is a contradiction. Therefore, y ∈ M \ gi−1 (Nd (A, 2−(i+1) )). Since gj (y) = gi (y) for every j  i, it follows that x = g(y) = gi (y), that is, y ∈ gi−1 (x). Thus, we have g −1 (x) ⊂ gi−1 (x). Since N(gi ) ⊂ A, gi−1 (x) is a singleton, hence so is g −1 (x). Consequently, N(g) ⊂ A. " ! As mentioned at the beginning of this section, we need a substitution of the Qmanifold version of Proposition 2.9.1. For a map f : M → X and a closed set A ⊂ X, we define the mapping cylinder reduced over A to be the space     M(f )A = X ∪ f −1 (X \ A) × (0, 1] = X ∪ (M \ f −1 (A)) × (0, 1] that admits the topology generated by open sets in   (X \ A) ∪ (M \ f −1 (A)) × (0, 1] = M(f |M \ f −1 (A)) (⊂ M(f ))

3.7 Cell-Like Images of Q-Manifolds

271

and sets U ∪ (f −1 (U \ A) × (0, 1]), where U is open in X. Then, the natural map πA : M(f ) → M(f )A can be defined as follows: πA |X ∪ (f −1 (X \ A) × (0, 1]) = id and πA |f −1 (A) × (0, 1] = f prf −1 (A) = cf . Let τA : M(f )A → X be the map induced by the collapsing, that is, τA πA = cf . Then, τA  id rel. X in M(f )A by the homotopy h : M(f )A × I → M(f )A defined by h0 = τA , ht |X = id for each t ∈ I and ht (x, s) = (x, st) for each t ∈ (0, 1] and (x, s) ∈ (M \ f −1 (A)) × (0, 1]. Thus, X is a strong deformation retract of M(f )A . In the case where f = idX , M(idX )A coincides with the reduced product (X × I)A . The product space X × C(E) can be regarded as the mapping cylinder M(prX ) of the projection prX : X × E → X, and the reduced product (X × C(E))A as corresponding to the reduced mapping cylinder M(prX )A . The following REDUCED MAPPING CYLINDER THEOREM is a substitution of the Q-manifold version of Proposition 2.9.1: Theorem 3.7.5 (REDUCED MAPPING CYLINDER) Let f : M → X be a celllike map from a Q-manifold M onto an ANR X and A be a Z-set in X. Then, the reduced mapping cylinder M(f )A is a Q-manifold with A a Z-set in M(f )A and the natural map πA : M(f ) → M(f )A is a near-homeomorphism. Proof First of all, note that the mapping cylinder M(f ) is a Q-manifold by the Mapping Cylinder Theorem 3.7.3 above, and that the map πA is cell-like and A = N(πA ) = cl N(πA ). As observed above, X is a strong deformation retract of M(f )A . Since M(f )A \ X = (M \ f −1 (A)) × (0, 1] and X are ANRs, M(f )A is an ANR by Theorem 1.13.16. Then, Proposition 3.7.4 can be applied, that is, it suffices to show that A is a Z-set in M(f )A . For each open over U ∈ cov(M(f )A ), there is an open cover V of A in X such that    V ∪ (f −1 (V \ A) × (0, 1])  V ∈ V ≺ U. Take open sets W0 , W1 in X so that A ⊂ W1 ⊂ cl W1 ⊂ W0 ⊂ cl W0 ⊂



V

 and let ϕ, ψ : X → I be Urysohn maps with ϕ(cl W0 ) = 0, ϕ(X \ V) = 1, ψ(X \ W0 ) = 0, and ψ(cl W1 ) = 1. Since A is a Z-set in X, X \ A is homotopy dense in X by Theorem 2.8.6, hence there is a homotopy h : X × I → X such that h0 = id and ht (X) ⊂ X \ A for each t ∈ (0, 1]. By Lemma 2.5.2, it can be

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

assumed that h is a V-homotopy, where ht |X \ g : M(f )A → M(f )A \ A as follows:



V = id. We can define a map

g(x) = h(x, ψ(x)) for each x ∈ X and ⎧ ⎪ if 0 < t  ϕf (x), ⎪ ⎨(x, t) g(x, t) = (x, ϕf (x)) if t > ϕf (x) > 0, ⎪ ⎪ ⎩h(f (x), ψf (x)) if t  ϕf (x) = 0. Then, g is U-close to id.

" !

Proposition 3.7.6 Let f : M → X be a cell-like map from a Q-manifold M onto an ANR X with the disjoint cells property. Then, the collapsing cf : M(f ) → X of the mapping cylinder M(f ) (= Mf ) of f is a near-homeomorphism, and hence X is also a Q-manifold. Proof First of all, notice that the disjoint cells property of X implies that every proper map from a locally compact separable metrizable space can be approximated by Z-embeddings (Proposition 3.2.9). Moreover, the mapping cylinder M(f ) is a Q-manifold by the Mapping Cylinder Theorem 3.7.3. Since f is a proper (= perfect) map, the collapsing cf : M(f ) → X is also a cell-like map. To prove that cf is a near-homeomorphism, we can apply the Bing Shrinking Criterion 2.7.1 with Remark 2.11. For each U ∈ cov(M(f )) and V ∈ cov(X), it suffices to find a near-homeomorphism ϕ : M(f ) → M(f ) such that cf ϕ is Vclose to cf and {ϕcf−1 (x) | x ∈ X} ≺ U. Our strategy is the same as the proof of (b) ⇒ (a) in Theorem 3.4.2. We introduce a similar notation as in the proof of Proposition 2.9.1 and Theorem 3.4.2. For each W ⊂ X and α ∈ C(X, (0, 1]), we write    W [0, α] = W ∪ (x, t) ∈ cf−1 (W ) × (0, 1]  t  αf (x) . For each N ⊂ M and β  γ ∈ C(M, (0, 1]),    N[β, γ ] = (x, t) ∈ N × (0, 1]  β(x)  t  γ (x) . For simplicity, let M[1] = M × {1}. We can easily find α ∈ C(X, (0, 1)) and V0 ∈ cov(X) such that V0 ≺ V and {V [0, α] | V ∈ V0 } ≺ U. Moreover, we have α0 < α1 < α2 < · · · ∈ C(M, (0, 1)) and U0 ∈ cov(M) such that α1 < αf , limn→∞ αn (x) = 1, and 

  U [αn−1 , αn+1 ]  U ∈ U0 , n ∈ N ≺ U.

3.7 Cell-Like Images of Q-Manifolds

273

Indeed, for each x ∈ M, let   γ  (x) = sup s ∈ (0, 1)  ∀t ∈ [α(x)/2, 1], ∃U ∈ U

 such that {x} × ([t, t + s] ∩ I) ⊂ U .

Then, as in the proof of Theorem 3.4.2, it can be seen that γ  is lower semicontinuous. By Theorem 1.3.14, we have a map γ : M → (0, 1) with 2γ (x) < γ  (x) and γ (x) < αf (x). The maps α0 < α1 < α2 < · · · can be defined as in the proof of Theorem 3.4.2, that is, αn (x) =

αf (x) + (n − 1)γ (x) . 1 + αf (x) + (n − 1)γ (x)

Then, α1 (x) < αf (x), limn→∞ αn (x) = 1, and 0 < αn (x) − αn−1 (x) < γ (x) < γ  (x)/2 for each n ∈ N. For each x ∈ M, we have n ∈ N such that 1 − αn (x) < γ (x). Then, we can easily obtain U0 ∈ cov(M) with the desired property. Let V1 , V2 ∈ cov(X) be locally finite open refinements of V0 such that each member of both of V1 has the compact closure and ∗

st3 V2 ≺ V1 ≺ V0 . By Proposition 3.2.9, we have a Z-embedding v : M[1] → X such that v V2 cf . Let A = v(M[1]), and let π : M(f ) → M(f )A be the natural map and τ : M(f )A → X be the map induced by the collapsing, i.e., τA π = cf . Due to the Reduced Mapping Cylinder Theorem 3.7.5, A is a Z-set in M(f )A and π is a near-homeomorphism. Using the Z-set Unknotting Theorem 2.11.6, we have a homeomorphism h : M(f ) → M(f )A such that h τ −1 (V2 ) π and h|A = id. Due to the Mapping Cylinder Theorem 3.7.3, X is a Z-set in M(f ), hence so is h−1 (A) = A. Then, h−1 v : M[1] → M(f ) is a Z-embedding and cf h−1 v = τ πh−1 v V2 τ v = v V2 cf |M[1], where the above two V2 -homotopies are proper by Proposition 1.3.8. Since cf  id by the homotopy hf with cf hf = cf prM(f ) , it follows that h−1 v is properly cf−1 (st V2 )-homotopic to id. By the Z-set Unknotting Theorem 2.11.6, we have a

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

homeomorphism g : M(f ) → M(f ) such that g|M[1] = h−1 v and g c−1 (st2 V2 ) id. f

 0 = {U × (0, 1] | U ∈ U0 } is an Then, hg(M[1]) = A = v(M[1]). Note that U  0 ) is an open cover of A in M(f )A . Let open cover of M[1] in M(f ), hence hg(U W0 be an open cover of A in X such that     0 ). π(cf−1 (W0 )) = τ −1 (W0 ) = W [0, 1]A  W ∈ W0 ≺ hg(U Thus, we have a near-homeomorphism η = g −1 h−1 π : M(f ) → M(f ) such that  0 (Fig. 3.4). Since η(cf−1 (W0 )) ≺ U cf η = cf g −1 h−1 π st2 V2 cf h−1 π = τ πh−1 π V2 τ π = cf , it follows that c f η is V1 -close to cf . Let M0 = W0 and choose an open neighborhood N0 of A in X so that cl N0 ⊂ M0 . Then, η(cf−1 (N0 )) = g −1 h−1 τ −1 (N0 ) is an open neighborhood of

if (M) = η(cf−1 (A)) in M(f ). As in the proof of the Reduced Mapping Cylinder

Theorem 3.7.5, we have β0 ∈ C(M, (0, 1)) such that M[β0 , 1] ⊂ η(cf−1 (N0 )). c −1 f (N 0 )

c −1 f (N 1 )

−1 (N

−1 (N

0)

1)

M (f )A

M (f ) c −1 f (A) A id

= g −1 h −1 (c −1 f (N 1 ))

h

−1 (

X

2)

i f (M )

cf

0

M (f )

(c −1 f (N 0 ))

g X

Fig. 3.4 The homeomorphism η

c −1 f (st

2)

id

X

A

3.7 Cell-Like Images of Q-Manifolds

275

Then, η(M(f ) \ cf−1 (N0 )) = M(f ) \ η(cf−1 (N0 )) ⊂ M[0, β0 ]. Since hg(M[β0 , 1]) is a neighborhood of A in M(f )A , A has open neighborhoods N1 and M1 in X such that cl N1 ⊂ M1 ⊂ cl M1 ⊂ N0 and τ −1 (cl M1 ) ⊂ hg(M[β0 , 1]). Then, η(cf−1 (N1 )) = g −1 h−1 τ −1 (N1 ) is an open neighborhood of if (M) = η(cf−1 (A)) in M(f ) and η(cf−1 (cl M1 )) ⊂ M[β0 , 1]. Hence, we have

β1 ∈ C(M, (0, 1)) such that M[β1 , 1] ⊂ η(cf−1 (N1 )) and β1 > (1 + β0 )/2 (> β0 ), where it should be noted that η(cf−1 (X \ N1 )) = M(f ) \ η(cf−1 (N1 )) ⊂ M[0, β1 ]. Thus, we can inductively choose open neighborhoods Ni , Mi of A in X and βi ∈ C(M, (0, 1)), i ∈ ω, such that cl Ni ⊂ Mi ⊂ cl Mi ⊂ Ni−1 ,



Ni = A, βi >

i∈ω

M[βi , 1] ⊂

η(cf−1 (Ni ))

1 + βi−1 and 2

⊂ η(cf−1 (cl Mi )) ⊂ M[βi−1 , 1].

Then, it follows that η(cf−1 (X \ cl N1 )) ⊂ M[0, β1 ] and η(cf−1 (Mi \ cl Ni+1 )) ⊂ M[βi−1 , βi+1 ] for each i ∈ N. Sliding along the (0, 1]-coordinates as illustrating in Fig. 3.5, we can obtain a homeomorphism ξ : M(f ) → M(f ) such that cf ξ = cf , ξ |X ∪ if (M) = id, and ψ(x, βi (x)) = (x, αi (x)) for x ∈ M and i ∈ ω. i f (M )

M (f )

M (f )

2 1 2 0 1

X Fig. 3.5 The homeomorphism ξ

X

0

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Then, ξ η : M(f ) → M(f ) is a near-homeomorphism such that cf ξ η is V1 -close (hence V-close) to cf . We verify that each x ∈ X has a neighborhood Wx in X such that ξ η(cf−1 (Wx )) = ξ η(cf (Wx )) is contained in some member of U. If x ∈ X \ cl N1 , choose Vx ∈ V1 so that x ∈ Vx and let Wx = Vx \ cl N1 . Since cf ξ η = cf η is V1 -close to cf , we have cf ξ η(Wx [0, 1]) ⊂ st(Vx , V1 ), which is contained in some V ∈ V0 . Then, it follows that ξ η(cf−1 (Wx )) ⊂ ξ(V [0, β1 ]) = V [0, α1 ] ⊂ V [0, α], which is contained in some member of U. When x ∈ M1 \ A, x ∈ Mi \ cl Ni+1 for some i ∈ N. Choose Wx ∈ W0 so that x ∈ Wx and let Wx = Wx ∩ (Mi \ cl Ni+1 ). Then, η(cf−1 (Wx )) ⊂ cf−1 (U ) for some U ∈ U0 . It follows that ξ η(cf−1 (Wx )) ⊂ ξ(U [βi−1 , βi+1 ]) = U [αi−1 , αi+1 ], which is contained in some member of U. When x ∈ A, ξ(g −1 h−1 (τ −1 (x))) = g −1 h−1 (τ −1 (x)) is a singleton, hence it is contained in some U ∈ U. Then, x has an open neighborhood Wx in X such that ξ η(cf−1 (Wx )) = ξ(g −1 h−1 (τ −1 (Wx ))) ⊂ U. " !

Thus, ξ η is the desired near-homeomorphism.

By the Z-Embedding Approximation Theorem 2.10.11, the following is a direct consequence of Proposition 3.7.6: Corollary 3.7.7 Let f : M → N be a cell-like map between Q-manifolds. Then, the collapsing cf : M(f ) → N is a near-homeomorphism. " ! Now, we can prove the following: Theorem 3.7.8 Every cell-like map f : M → N between Q-manifolds is a nearhomeomorphism. Proof Consider the following commutative diagram: M ×I

qf

M(f ) cf

prM

M

f

N,

3.8 Characterization and Classification of Q-Manifolds

277

Y

X Mf

f (X ) −1

0

1

Fig. 3.6 The mapping cylinder Mf ∗

where the projection prM , the natural (quotient) map qf , and the collapsing cf are near-homeomorphisms by the Stability Theorem 2.3.13 with Corollary 2.3.11, the Mapping Cylinder Theorem 3.7.3, and Corollary 3.7.7, respectively. Thus, it follows that f is a near-homeomorphism. " ! By the above theorem, the following can be easily obtained: Theorem 3.7.9 Every compact contractible Q-manifold M is homeomorphic to the Hilbert cube Q. Proof The projection pr2 : M × Q → Q is cell-like, hence it is a nearhomeomorphism by Theorem 1.15.9. On the other hand, M × Q ≈ M by the Stability Theorem 2.3.13. Consequently, we have M ≈ Q. ! "

3.8 Characterization and Classification of Q-Manifolds In this section, we establish the Toru´nczyki characterization of Q-manifolds. To this end, we first prove the EDWARDS FACTOR THEOREM, which is the Q-version of the Toru´nczyk Factor Theorem 2.2.14, that is: Theorem 3.8.1 (EDWARDS) For every locally compact ANR X, X × Q is a Qmanifold. Since a compact contractible Q-manifold M is homeomorphic to Q (Theorem 3.7.9), we have the following: Corollary 3.8.2 For every compact AR X, X × Q is homeomorphic to the Hilbert cube Q. " ! To prove the Edwards Factor Theorem 3.8.1 above, we use the fat mapping cylinder. For a map f : X → Y , let f ∗ : X → Y × [−1, 0] be the map defined by f ∗ (x) = (f (x), 0) for each x ∈ X. The mapping cylinder Mf ∗ can be regarded as the space (Y × [−1, 0]) ∪ Mf , where Y × {0} is identified with Y ⊂ Mf .14 The collapsing cf ∗ is the unique extension of cf (Fig. 3.6).

14 When

p. 145).

f is a proper map, this is the same space M(f ) ∪Y =Y ×{0} Y × I defined in Sect. 2.6 (see

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Lemma 3.8.3 Let r : Q → X be a retraction of Q onto X and d be an admissible metric for Q. For each ε > 0, there is a cell-like map β : Q × [−1, 1] → Mr ∗ such that β −1 (x, 1) = {(x, 1)} for each x ∈ X, β|Q × {−1} = id, and d(prX r¯ β, prX (r × id)) < ε, where r¯ : Mr ∗ → X × [−1, 1] is the map defined by r¯ (x, t) = (r(x), t) for each (x, t) ∈ (Q × [−1, 0]) ∪ (Q × (0, 1]). Proof By Corollary 2.3.10(4) and Stability Theorem 2.3.13, for any small δ1 > 0, we have a homeomorphism h : Q × [−1, 1] × Q → Q × [−1, 1] such that h is δ1 -close to the projection and h(x, ±1, 0) = (x, ±1) for every x ∈ Q. Let k : Q → Q be a map such that k −1 (0) = X and k|Q \ X is injective. We define ζ : Q → Q × [−1, 0] × Q by ζ(x) = (r(x), 0, k(x)) for each x ∈ Q. Since ζ is injective and Q × {1} × Q is a Z-set in Q × I × Q, it follows that ζ is a Z-embedding. Note that ζ(x) = (x, 0, 0) for every x ∈ X and pr1 ζ = r, where pr1 : Q × [−1, 0] × Q → Q is the projection onto the first factor. We define a homeomorphism ξ : Q × [−1, 1] × Q → Q × [−1, 0] × Q ⊂ Mζ by ξ(x, t, y) = (x, (t − 1)/2, y) (Fig. 3.7). Using the Z-Set Unknotting Theorem 2.11.6, we can easily construct a homeomorphism ϕ : Q × [−1, 1] → Mζ such that ϕ(Q × {−1}) = Q × {−1} × Q, ϕ(x, 0) = ζ (x) for each x ∈ Q, and ϕ|Q × (0, 1] = id. For any small δ2 > 0, we can obtain a homeomorphism η : Q × [−1, 1] → Q × [−1, 0] such that η|Q × [−1, −δ2 ] = id and η({x} × [−δ2 , 1]) = {x} × [−δ2 , 0] for each x ∈ Q. We have a cell-like map p : Mζ → Mr∗ such that p|Q × (0, 1] = id and p|Q × [−1, 0] × Q is the projection onto Q × [−1, 0]. Thus, we have a cell-like map β = pϕη−1 ϕ −1 ξ h−1 : Q × [−1, 1] → Mr∗ . Observe that β|Q × {−1} = id and β −1 (x, 1) = {(x, 1)} for each x ∈ X. We can choose δ2 > 0 so small that prX r¯ pϕηϕ −1 is ε/2-close to prX r¯ pcζ = prX r¯ p. Then, it follows that d(prX r¯ β, rprQ pξ h−1 ) = d(prX r¯ β, prX r¯ pξ h−1 ) = d(prX r¯ pϕη−1 ϕ −1 ξ h−1 , prX r¯ pϕηϕ −1 ϕη−1 ϕ −1 ξ h−1 ) < ε/2. Observe that prQ pξ is the projection of Q × [−1, 1] × Q onto the first factor. We can choose δ1 > 0 so small that prX r¯ h = rprQ h is ε/2-close to prX r¯ pξ = rprQ pξ , hence d(prX r¯ pξ h−1 , prX (r ×id)) < ε/2. Consequently, d(prX r¯ β, prX (r ×id)) < ε. This completes the proof. " !

3.8 Characterization and Classification of Q-Manifolds Q × [− 1, 1]

Q h ≈

−1

279

1

0

Q

−1

1

0

M

Q Q p

Q

Q

Mr −1

X 1

1

0

−1

Mr

(Q) = (Q × {0})

0 −1

Q

−1

−1

−

2

0

1

Fig. 3.7 Mζ and constructing β

Lemma 3.8.4 For a retraction r : Q → X, there is a cell-like map ϕ : Mr → X×I such that ϕ(x) = (x, 0) for each x ∈ X. Proof Take any metric d ∈ Metr(X) and define the admissible metric ρ for X × I as follows:   ρ((x, t), (y, s)) = max d(x, y), |t − s| . For each n ∈ ω, we define the space M n = the mapping cylinder M defined by

2n

n i=1 Mi ,

where each Min is a copy of

 $ %  & ' i−1 i−1 i , Min = X × ∪ Q × . r×id{(i−1)/2n } 2n 2n 2n Then, X × I is contained in the space M n as a retract with the retraction rn : M n → X × I defined by rn (x, t) = (r(x), t) for each (x, t) ∈ M n (Fig. 3.8). The mapping cylinder Mr can be regarded as M 0 = M10 .

280

3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Q ······ X 0

1 2n

2 n −1 2n

2 2n

1

Fig. 3.8 The space M n

We shall construct cell-like maps fn : M n → M n+1 , n ∈ ω. Observe that '  & 4i − 3 i n+2 , ∪ Q× , Min = M4i−3 2n+2 2n and the following set is a copy of Mr ∗ = (Q × [−1, 0]) ∪ Mr in Lemma 3.8.3:  & ' 4i − 3 2i − 1 n+1 Q× ∪ M2i , , n ∈ N, i = 1, . . . , 2n . 2n+2 2n+1 We can apply Lemma 3.8.3 to obtain a cell-like map & fn,i : Q ×

'  & ' 4i − 3 i 4i − 3 2i − 1 n+1 , , → Q × ∪ M2i 2n+2 2n 2n+2 2n+1

−1 such that fn,i |Q×{(4i−3)/2n+2 } = id, fn,i (x, i/2n ) = {(x, i/2n )} for each x ∈ X, n+2 = id. and d(prX rn+1 fn,i , prX (r×id)) < 2−n , which extends over Min by fn,i |M4i−3 n Now, we have a cell-like map fn : M n → M n+1 defined by fn |Mi = fn,i for each i = 1, . . . , 2n (Fig. 3.9). Since d(prX rn+1 fn , prX rn ) < 2−n , it follows that ρ(rn+1 fn , rn ) < 2−n .

M r (n + 2, 4i − 3)

M r (n + 1, 2i − 1)

M r (n + 1, 2i)

M r (n, i) Q

Q fn, i

=

X i −1 2n

X

i 2n

2i −1 2 n +1 4i −3 2 n +2

Fig. 3.9 Construction of the cell-like map fn

2i −1 2 n +1

2i 2 n +1 4i −3 2 n +2

2i 2 n +1

3.8 Characterization and Classification of Q-Manifolds

281

For each n ∈ ω, since C(M n , X × I) is complete with respect to the sup-metric, we can obtain a map ϕn : M n → X × I as the limit of the Cauchy sequence  Then, ρ(rn , ϕn )
δ > 0 such that Bρ ((x, t), ε) ⊂ U and Bd (x, δ) is contractible in Bd (x, ε). Choose n ∈ N so that 2−n+3 < δ and n  3. Claim A = ϕn−1 (x, t) is contractible in ϕn−1 (U ). This Claim implies that ϕ −1 (x, t) is contractible in ϕ −1 (U ). In fact, ϕ = ϕn fn−1 · · · f0 , where fn−1 · · · f0 is a fine homotopy equivalence by Corollary 1.15.10 and Theorem 1.15.9. Then, by Corollary 1.15.3, ϕ −1 (x, t) = (fn−1 · · · f0 )−1 (A) is contractible in ϕ −1 (U ) = (fn−1 · · · f0 )−1 (ϕn−1 (U )). Thus, it follows that ϕ is cell-like. Now, we shall prove the Claim. For each a, a  ∈ A, since ϕn (a) = ϕn (a  ) = x, it follows that ρ(rn (a), rn (a  ))  ρ(rn (a), ϕn (a)) + ρ(ϕn (x  ), rn (x  )) < 2−n+1 + 2−n+1 = 4 · 2−n . This means that A is contained in the union of some five successive Min , that is, j +4 A ⊂ i=j Min for some j = 1, . . . , 2n − 4. Since each Min is a copy of the j +4 mapping cylinder Mr , we can obtain a strong deformation retraction of i=j Min onto X × {(j − 1)/2n } as the composition of collapsings. Thus, there is a homotopy j +4 j +4 h : i=j Min × I → i=j Min such that h0 = id, h1 is a retraction onto X × {(j − 1)/2n }, and, (z, s) ∈ Min (j  i  j + 4) implies  & '   ' j −1 i−1 i−1 i h({(z, s)} × I) ⊂ {r(z)} × ∪ {z} × . , , 2n 2n 2n 2n When (z, s) ∈ A ∩Min , since ρ(ϕn (z, s), (r(z), s))  ρ(ϕn , rn ) < 2−n+1 , it follows that diamρ ϕn h({(z, s)} × I) 

5 1 2 8 + n + n = n = 2−n+3 < ε. 2n 2 2 2

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Since ϕn (z, s) = (x, t), we have ϕn h({(z, s)} × I) ⊂ Bρ ((x, t), ε) ⊂ U, hence h({(z, s)} × I) ⊂ ϕn−1 (U ). Therefore, h(A × I) ⊂ ϕn−1 (U ). Moreover, since d(x, r(z)) < 2−n+1 < δ in the above, it follows that h1 (A) ⊂ Bd (x, δ) × {(j − 1)/2n } ⊂ Bd (x, ε) × {(j − 1)/2n }, where the second set is contractible in the third one. Here, recall that ϕn |X × {(j − 1)/2n } = id. Then, Bd (x, ε) × {(j − 1)/2n } ⊂ ϕn−1 (Bd (x, ε) × {(j − 1)/2n }) ⊂ ϕn−1 (U ). Consequently, h1 (A) is contractible in ϕn−1 (U ), hence so is A. This completes the proof. " ! Lemma 3.8.5 Let X be a non-compact separable locally compact ANR. Then, the one-point compactification α(X × [0, 1)) = (X × [0, 1)) ∪ {∞} of X × [0, 1) is a compact AR. When X is an AR, {∞} is a Z-set in α(X × [0, 1)). Proof Since X is separable and locally compact, we can apply the Urysohn Metrization Theorem 1.3.4 to see that α(X × [0, 1)) is metrizable. We can regard α(X × [0, 1)) as the following quotient space: (αX × I)/(({∗} × I) ∪ (αX × I)), where αX = X ∪ {∗} is the one-point compactification of X. Let q : αX × I → α(X × [0, 1)) be the quotient map. Then, there exists the deformation h : (αX × I) × I → αX × I defined by ht (x, s) = (x, 1 − (1 − t)(1 − s)), which induces the contraction of α(X×[0, 1)) onto {∞} fixing {∞}. Thus, {∞} is a strong deformation retract of α(X × [0, 1)). By the Kruse–Liebnitz Theorem 1.13.16, α(X × [0, 1)) is an ANR, hence it is an AR because it is contractible. When X is an AR, we shall show that {∞} is a Z-set in α(X × [0, 1)). For each open cover U of α(X × [0, 1)), we take U ∈ U such that ∞ ∈ U . Choose open neighborhoods V and W of ∗ in αX so that V × I ⊂ q −1 (U ) and cl W ⊂ V . Let k : αX → I be an Urysohn map with k((αX) \ V ) = 0 and k(cl W ) = 1. Define a map r : αX × I → αX × I as follows:  r(x, t) =

(x, t)

if t  k(x),

(x, k(x)) if t < k(x).

Choose 0 < t0 < t1 < 1 so that αX × [t0 , 1] ⊂ q −1 (U ). Let ϕ : X × I → X be a contraction with ϕ1 (X) = {x0 }. Define a map f : r(αX × I) → X × [0, 1)

3.8 Characterization and Classification of Q-Manifolds

283

as follows: ⎧ (x , t ) ⎪ if t1  t  1, ⎪ ⎨ 0 1 t − t

0 , t if t0  t  t1 , ϕ x, f (x, t) = ⎪ t1 − t0 ⎪ ⎩ (x, t) if 0  t  t0 . Then, the map f r : αX × I → X × [0, 1) induces the map g : α(X × [0, 1)) → X × [0, 1), where g|(X \ V ) × [0, t0 ] = id and g(α(X × [0, 1)) \ ((X \ V ) × [0, t0 ])) ⊂ U, which implies that g is U-close to id. Thus, it follows that {∞} is a Z-set in α(X × [0, 1)). " ! We can now prove the Edwards Factor Theorem 3.8.1. Here, we also use Proposition 3.7.6. Proof (Edwards Factor Theorem 3.8.1) First, note that if X × [0, 1) × Q (≈ X × (0, 1] × Q) is a Q-manifold, then so is X × Q because X × I × Q = (X × [0, 1) × Q) ∪ (X × (0, 1] × Q). When X is a compact AR, we may assume that X is a closed set in Q. Then, there is a retraction r : Q → X whose mapping cylinder Mr is an AR (Corollaries 1.13.17 and 1.13.3). We define S = Mr ∗ = (Q × [−1, 0]) ∪ Mr , where X × {0} ⊂ Q × [−1, 0] is identified with X = X × {1} ⊂ Mr . Then, S is an AR (1.13.6(5)). On the other hand, we have a cell-like map f : Q × [−1, 1] → S by Lemma 3.8.3. Let T = (Q × [−1, 0]) ∪ (X × [0, 1]), where T can be regarded as a subspace of S. By Lemma 3.8.4, we have a cell-like map g : Mr → X × [0, 1] such that g(x) = (x, 0) for each x ∈ X, which can be extended to the cell-like map g˜ : S → T by g|Q ˜ × [−1, 0] = id. Then, we have the cell-like map h = gf ˜ × idQ : Q × [−1, 1] × Q → T × Q. Since T × Q is a compact AR with the disjoint-cells property, we can apply Proposition 3.7.6 to obtain T × Q ≈ Q × [−1, 1] × Q ≈ Q.

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Hence, X × (0, 1] × Q is a Q-manifold as an open set in T × Q. In the general case, it suffices to show the theorem for each component of X. Then, we may assume that X is connected, and hence separable. Due to Lemma 3.8.5, the one-point compactification α(X × [0, 1)) is a compact AR. Since α(X ×[0, 1))×Q is a Q-manifold, its open set X ×[0, 1)×Q is also a Q-manifold. " ! By combining the Edwards Factor Theorem 3.8.1 and Proposition 3.7.6, we can establish the following characterization of Q-manifolds: Theorem 3.8.6 (CHARACTERIZATION OF Q-MANIFOLDS) For a separable locally compact ANR X, the following are equivalent: (a) X is a Q-manifold; (b) X has the disjoint cells property; (c) X is M0 -universal. Proof Since the equivalence (b) ⇔ (c) was proved in Theorem 3.2.9 and the implication (a) ⇒ (c) follows from the Strong Universality of Q-manifolds (Theorem 2.10.10 or 2.3.16), it suffices to prove the implication (b) ⇒ (a). By the Edwards Factor Theorem 3.8.1 above, X × Q is a Q-manifold. Since the projection " prX : X × Q → X is a cell-like map, the result follows from Proposition 3.7.6. ! Since every compact contractible Q-manifold M is homeomorphic to Q (Theorem 3.7.9), we have the following characterization of the Hilbert cube: Theorem 3.8.7 (CHARACTERIZATION OF Q) A space X is homeomorphic to the Hilbert cube if and only if X is a compact AR with the disjoint cells property. ! " To establish the Classification Theorem for Q-manifolds, we introduce the notion of CE equivalence, which is a generalization of simple homotopy equivalence. It is said that ANRs X and Y are CE equivalent if there exist cell-like maps f : Z → X and g : Z → Y of an ANR Z. Theorem 3.8.8 (CLASSIFICATION) Two Q-manifolds M and N are homeomorphic if and only if they are CE equivalent. Proof The “only if” part is trivial. To see the “if” part, let f : Z → M and g : Z → N be cell-like maps of an ANR Z. Since f (or g) is proper, Z is locally compact. By the Edwards Factor Theorem 3.8.1, Z × Q is a Q-manifold. Then, f prZ : Z × Q → M is a cell-like map, hence it is a near-homeomorphism by Theorem 3.7.8. Similarly, gprZ : Z × Q → N is also a near-homeomorphism. Consequently, M ≈ Z × Q ≈ N. " ! The Triangulation Theorem for Q-manifolds will be proved in Chap. 4, where not only some more results on Q-manifolds but also some results on Algebraic Topology are required. However, the Triangulation Theorem for R+ -stable Qmanifolds can be easily obtained:

3.9 Keller’s Theorem and the Countable Product of ARs

285

Theorem 3.8.9 (TRIANGULATION) For each R+ -stable Q-manifold M, there exists a locally finite simplicial complex K such that M ≈ |K| × Q. Proof Due to Theorem 1.13.24, M has the proper homotopy type of a locally finite simplicial complex K. Then, |K| × Q is a Q-manifold by the Edwards Factor Theorem 3.8.1 and it is also R+ -stable by Theorem 2.11.13. Hence, M ≈ |K| × Q by the Classification Theorem 2.11.11 for R+ -stable Q-manifolds: " ! As a corollary, we have the following: Corollary 3.8.10 For each Q-manifold M, there exists a locally finite simplicial " ! complex K such that M × R+ ≈ |K| × Q.

3.9 Keller’s Theorem and the Countable Product of ARs In this section, we apply characterization of Q to prove Keller’s Theorem on infinite-dimensional compact convex sets and the compact version of Theorem 3.5.11. We start to show the latter:  Theorem 3.9.1 The countable product X = i∈N Xi of non-degenerate spaces is homeomorphic to the Hilbert cube if and only if all Xi , i ∈ N, are compact ARs. Proof The “only if” part is trivial. To prove the “if” part, since X is a compact AR, it suffices to show that X has the disjoint cells property. For each i ∈ N, let di ∈ Metr(Xi ). We define a metric d for X as follows: d(x, y) =



  min 2−i , di (x(i), y(i)) .

i∈N

Let f, g : In → X be maps. For each ε > 0, choose i0 ∈ N so that 2−i0 < ε. Because Xi0 is non-degenerate, there are two distinct points a, b ∈ Xi0 . We can define maps f  , g  : In → X as follows: pri f  (x) = pri g  (x) =

 a

if i = i0 ,

pri f (x) otherwise,  b if i = i0 , pri g(x)

otherwise.

Then, f  (In )∩g  (In ) = ∅, and f  and g  are ε-close to f and g, respectively. Hence, X has the disjoint cells property. " ! Next, we prove the following GENERALIZED KELLER’S THEOREM:

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

´ Theorem 3.9.2 (KELLER–DOBROWOLSKI–TORU NCZYK ) A convex set C in a metrizable topological linear space E is homeomorphic to Q if and only if it is an infinite-dimensional compact AR.

When E = 2 , the above theorem is called KELLER’S THEOREM. To prove the above theorem, we use the following lemma: Lemma 3.9.3 Let E = (E,  · ) be an F -normed linear space, C a convex set in E and P a compact polyhedron. For any ε > 0, each map f : P → C is ε-close to a map g : P → C such that g(P ) is contained in a (convex) linear cell in C. Proof Let n = dim P and take a triangulation K of P so that diam f (σ ) < ε/(n + 1) for every σ ∈ K. Since C is convex, we can define g : P = |K| → C on each σ = v1 , . . . , vk  ∈ K as follows: g(x) =



ti f (vi ) if x =

k 

ti vi , t1 , . . . , tk ∈ I with

i=1

v∈K (0)

k 

ti = 1.

i=1

Then, it follows that g(P ) ⊂ f (K (0) ) and f (x) − g(x) 

k 

ti (f (x) − f (vi )) 

i=1

k 

f (x) − f (vi )

i=1

 (n + 1) diam f (σ ) < ε, that is, f is ε-close to g.

" !

Proof (Theorem 3.9.2) Due to the Characterization of Q (Theorem 3.8.7), it suffices to show that C has the disjoint cells property. By Theorem 1.4.15, E has an admissible F -norm  · . Let f, g : In → C be maps and U ∈ cov(C) and ε > 0. By Lemma 3.9.3, we may assume that f (In ) ∪ g(In ) is contained in a cell D in C. Since C is infinite-dimensional and D is finite-dimensional, we have v ∈ C \ fl D. Because C − C is compact, we can find δ > 0 so that δ(C − C) ⊂ B(0, ε). We define a map f  : In → C as follows: f  (x) = (1 − δ)f (x) + δv for each x ∈ In . Then, f (x)−f  (x) = δ(f (x)−v) ∈ B(0, ε), which means that f (x)−f  (x) < ε. Namely, f  is ε-close to f . Let g  = g : In → C. Since g  (In ) ⊂ D, it follows that f (In ) ∩ g  (In ) = ∅. " ! In the above proof, the proof of the disjoint cells property of C can be performed without the compactness of C. Then, we have also the following: Theorem 3.9.4 A convex set in a metrizable topological linear space is a Qmanifold if and only if it is an infinite-dimensional locally compact ANR (which implies an AR). " !

3.9 Keller’s Theorem and the Countable Product of ARs

287

The following classification of infinite-dimensional locally compact closed convex sets was proved by V. Klee [95]: Theorem 3.9.5 (KLEE) An infinite-dimensional locally compact closed convex set in a normed linear space is homeomorphic to one of Q, Q × R+ or Q × Rn , n ∈ N. Here, we do not give a proof of this theorem. For details, one can refer to the book of Bessaga and Pełczy´nski [(1)], Chap. III, §7. The Toru´nczyk characterization of Q can also be applied to Hyperspace Theory. For a metric space X = (X, d), the hyperspace of all non-empty compacta in X is denoted by Comp(X), which has the Hausdorff metric dH defined as follows:15   dH (A, B) = max supx∈A d(x, B), supy∈B d(y, A)    = inf r > 0  A ⊂ Nd (B, r), B ⊂ Nd (A, r) .16 Let Cont(X) be the subspace of Comp(X) consisting of all continua in X. It is well known that Cont(I) ≈ B2 . In fact, Cont(I) consists of all closed subintervals of I, that is,    Cont(I) = [a, b]  0  a  b  1 . We can define ϕ : Cont(I) → R2 as follows: ϕ([a, b]) = ((a + b)/2, (b − a)/2) ∈ R2 each [a, b] ∈ Cont(I). As is easily observed, ϕ is a continuous injection. Because of the compactness of Cont(I), ϕ is an embedding. Since ϕ(Cont(I)) is a triangle with vertices 0 = (0, 0), (1/2, 1/2), e1 = (1, 0) in the plane R2 , it follows that Cont(I) ≈ B2 .

An arc A in X (i.e., A ≈ I) is called a free arc in X if int A = A \ ∂A, where (A, ∂A) ≈ (I, ∂I) (= (I, {0, 1})). If X has a free arc, then Cont(X) contains a two-dimensional open set. The following CURTIS–SCHORI–WEST HYPERSPACE THEOREM is a celebrated application, which was conjectured by M. Wojdysławski [158] in 1938. In [159], Wojdysławski proved that Comp(X) and Cont(X) are ARs for every Peano continuum X, whose shorter proof was given by J.L. Kelley [93]. Theorem 3.9.6 (CURTIS–SCHORI–WEST) For every non-degenerate metric Peano continuum X, Comp(X) is homeomorphic to Q. Additionally, if X has no free arcs, then Cont(X) is also homeomorphic to Q. Here, we do not give the proof of the above theorem (Appendix of [143]). For the hyperspace Comp(X), one can refer to the book of Jan van Mill [(3)].

d is bounded (i.e., diamd X < ∞), this definition can be acceptable for the space Cld(X) of all non-empty closed sets in X. 16 Recall that d(x, B) = inf y∈B d(x, y) and Nd (B, r) = {x ∈ X | d(x, B) < r}. 15 When

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3 Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Notes for Chapter 3 In this chapter, we concentrated our attention on Toru´nczyk’s characterizations of Q-manifolds and 2 ()-manifolds. The main results were established in [143], [145], and [146]. For survey, refer to [144], [102, p. 250]. The disjoint n-cells property was introduced by Toru´nczyk to characterize Q-manifolds in [143]. In [37], J.W. Cannon independently introduced the disjoint disk property to distinguish topological manifolds from homological manifolds, which was generalized by R.D. Edwards. The discrete n-cells property was introduced by Toru´nczyk to characterize 2 ()-manifolds in [145]. Proposition 3.2.3 was pointed out by Koshino during the writing of the next chapter. Proposition 3.2.12 was proved by Bestvina and Mogilski [29] and its non-separable version, Proposition 3.2.11, is due to Sakai and Yaguchi [129]. The characterizing conditions of Hilbert manifolds given in [145] (cf. [146]) are little modified in Theorem 3.4.2. Theorem 3.4.7 was given in [145] but, as its non-separable version, Theorem 3.4.3 is formulated and proved in the present book. In [30], an alternative proof for the characterization separable Hilbert manifolds was given. Lemma 3.5.1 is due to Bessaga, Dobrowolski, and Toru´nczyk [55], which is modified to Lemma 3.5.2 by Sakai [126]. M.I. Kadec [89, 90] proved that all separable infinite-dimensional Banach spaces are homeomorphic. R.D. Anderson [4, 5] proved that RN is homeomorphic to Hilbert space 2 . As a combination of these two results and earlier results obtained by Bessaga and Pełczy´nski in [26] and [27], the Kadec–Anderson Theorem 3.5.6 was established. T. Dobrowolski and H. Toru´nczyk generalized this result to Theorem 3.5.5 in [54] and later to Theorem 3.5.7 in [55]. The non-separable version of the Kadec–Anderson Theorems 3.5.6 and 3.5.17, was obtained by Toru´nczyk in [145]. Theorem 3.6.5 was proved in a more general formulation in [126]. In [142], H. Toru´nczyk proved the same result when X has an open set U such that bd U is compact and collared in cl U (cf. Geoghegan’s result in [68]). The characterization of compact Q-manifolds is also treated in the book [(3)] of Jan van Mill, where its alternative proof is given. However, we do not assume the compactness in this chapter. It is not trivial to derive the non-compact version from the characterization of compact Q-manifolds. In [148], J.J. Walsh also gave another alternative proof, where Miller’s theorem was used. Our approach is a combination of these two. In [59], one more proof due to R.D. Edwards can be found. The Mapping Cylinder Theorem 3.7.3 was proved by West [151]. Fathi, Marin, and Visetti [62] established Lemma 3.7.2, which produces the Mapping Cylinder Theorem 3.7.3. Theorem 3.8.1 is established by R.D. Edwards but his own proof was not published; the first published proof is in Chapman’s Lecture Notes [(2)]. The proof of its compact case is in the book of van Mill [(3)], where Lemmas 3.8.3 and 3.8.4 are key lemmas. The first part of Lemma 3.8.5 was first proved by Fathi and Visetti [61] in the case where X is an AR, but the proof presented here is due to Sakai and Wong [128]. In the proof of Theorem 3.8.1, the reduction of the locally compact case to the compact case is due to Fathi and Visetti [61]. Theorem 3.9.1 was established by J.E. West [149] when each Xi is a non-degenerate Q-factor, i.e., Xi × Q ≈ Q (cf. Theorem 3.8.1). Theorem 3.9.2 was originally proved by O.H. Keller [92] when a topological linear space is completely metrizable and locally convex (i.e., a Fréchet space) and generalized to the present form by T. Dobrowolski and H. Toru´nczyk [54]. The proof presented here is due to [54] but there is a gap in the proof of Lemma 3 in [54]. Theorem 3.9.6 was finally proved by D.W. Curtis and R.M. Schori [52] but in advance, R.M. Schori and J.E. West [132] had first proved that Comp(I) ≈ Q.

Chapter 4

Triangulation of Hilbert Cube Manifolds and Related Topics

In this chapter, we focus on proving the Triangulation Theorem for Q-manifolds. In particular, for each compact Q-manifold M, we have to find a finite simplicial complex K such that |K| × Q ≈ M. This is bound up with the Borsuk conjecture asserting that every compact ANR has the homotopy type of a finite simplicial complex. It is related to the homotopy type problem of whether a space has the homotopy type of a finite simplicial complex if it is homotopy dominated by a finite simplicial complex, where the first simplicial complex might be different from the second (cf. the Whitehead–Milnor Theorem 1.10.4 and Remark 1.9 on p. 57). To prove the Triangulation Theorem for Q-manifolds, we will use Wall’s work on the homotopy type of a finite simplicial complex (Theorem 4.1.6) and a result on simple homotopy equivalences (Theorem 4.1.5). Combining the Triangulation Theorem for compact Q-manifolds with the Edwards Factor Theorem 3.8.1, we know that the Borsuk conjecture is true. We will also prove the topological invariance of Whitehead torsion, which had been a longstanding problem. As is mentioned in Preface, this chapter requires further results other than results contained in [GAGT] and added in Chap. 1. In Sect. 4.1, we give a brief introduction of the simple homotopy type and write down some results from Simple Homotopy Theory without proofs, which will be used in Sects. 4.4 and 4.6. Concerning this theory, Cohen’s excellent textbook [(20)] is our source of information. At the end of Sect. 4.1, we write down a result from Wall’s Obstruction Theory without proof, which will be used in Sect. 4.6. Since the covering space arguments will be applied to Q-manifolds in Sect. 4.6, we add Sect. 4.2 concerning covering spaces, which is not contained in [GAGT]. Additionally, we give in Sect. 4.2 some elementary results in Algebraic Topology (the homotopy and homology groups), which will be used in Sect. 4.3. The contents in Sect. 4.2 are contained in any standard textbook on Algebraic Topology.

© Springer Nature Singapore Pte Ltd. 2020 K. Sakai, Topology of Infinite-Dimensional Manifolds, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-15-7575-4_4

289

290

4 Triangulation of Hilbert Cube Manifolds and Related Topics X (I n ) Y

e

Y

Fig. 4.1 An elementary collapse

4.1 Simple Homotopy Equivalences Let X and Y be compact polyhedra.1 It is said that X collapses to Y by an elementary collapse, written X &e Y , if Y is a subpolyhedron of X and there is a PL embedding ϕ : In → X for some n ∈ N such that X = Y ∪ ϕ(In ) and ϕ(In ) ∩ Y = ϕ(In−1 × {0}) (see Fig. 4.1), where it should be noted that bdX ϕ(In ) = ϕ(In−1 × {0}). Then, a strong deformation retraction rϕ : X → Y can be defined as follows: rϕ |Y = id and rϕ (ϕ(x)) = ϕ(x(1), . . . , x(n − 1), 0) for each x ∈ In . Note that rϕ−1 (y) = {y} or ≈ I for each y ∈ Y . This PL map rϕ is called an elementary collapse. When X &e Y , we also say that Y expands to X by an elementary expansion and write Y e' X, where the inclusion Y ⊂ X is a homotopy equivalence and called an elementary expansion. We say that X collapses to Y and write X & Y if there is a finite sequence of elementary collapses X = X0 &e X1 &e · · · &e Xn = Y, which also means that Y expands to X, written Y ' X. When Y is a singleton, X is said to be collapsible, written X & 0. Note that if X is collapsible, then X is contractible. It is said that X and Y have the same simple homotopy type or X has the simple homotopy type of Y (X is simple homotopy equivalent to Y ) if there is a finite sequence of compact polyhedra Xi , i = 0, 1, . . . , n, such that X0 = K, Xn = Y , and Xn−1 & Xn or Xn−1 ' Xn . A map f : X → Y is called a simple homotopy

1 More generally, X and Y might be CW complexes. In fact, the Simple Homotopy Theory has been developed for CW complexes. Refer to Cohen’s book [(20)].

4.1 Simple Homotopy Equivalences

291

equivalence if f is homotopic to a finite composition f1

f2

fn

X = X0 −→ X1 −→ · · · −→ Xn−1 −→ Xn = Y, where each Xi is a compact polyhedron and each fi is an elementary collapse or an elementary expansion. Evidently, if X and Y have the same simple homotopy type, then they must have the same homotopy type, and every simple homotopy equivalence is a homotopy equivalence. But, the converse is a problem. Obviously, a compact polyhedron has the same simple homotopy type of a singleton if it is collapsible. However, the converse is wrong. To give such an example and to show that simple homotopy type is a useful concept, consider the two-dimensional polyhedron    H = ∂I3 ∪ [0, 1/2] × ∂([0, 1/3] × [0, 1/2])     ∪ {1/2} × I2 ∪ [1/2, 1] × ∂([2/3, 1] × [0, 1/2])   \ ({0, 1/2} × (0, 1/3) × (0, 1/2)) ∪ ({1/2, 1} × (2/3, 1) × (0, 1/2)) , that is a house with two rooms and two entrances illustrated in Fig. 4.2, where the entrance to the right-hand room is on the left and the entrance to the left-hand room is on the right. Then, H is not collapsible because bdH ϕ(I2 ) = ϕ(I × {0}) for any PL embedding ϕ : I2 → H . However, I3 is collapsible and I3 & H (a rigorous proof is messy), equivalently H ' I3 , so H has the simple homotopy type of a singleton {0}. In particular, H  {0}, that is, H is contractible. This example also cancels the apparent restrictiveness of the definition of simple homotopy type. The following is evident from the definition: Proposition 4.1.1 Let f : X → Y , g : Y → Z, and h : X → Z be maps of compact polyhedra such that h  gf . If any two of the maps f , g, and h are simple homotopy equivalences, then so is the third one. " !

Fig. 4.2 A house with two rooms

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4 Triangulation of Hilbert Cube Manifolds and Related Topics

Combining the Edwards Factor Theorem 3.8.1 with 3.7.8, we have the following: Theorem 4.1.2 Let f : X → Y be a simple homotopy equivalence between compact polyhedra. Then, f × id : X × Q → Y × Q is homotopic to a homeomorphism. Proof By the Edwards Factor Theorem 3.8.1, X × Q and Y × Q are Q-manifolds. When f is an elementary collapse, since f ×id : X×Q → Y ×Q is cell-like, f ×id is a near-homeomorphism by Theorem 3.7.8. If f is an elementary expansion, then f is the inclusion and there is an elementary collapse g : Y → X, where f is a homotopy inverse of g because g is a strong deformation retraction. Since g × id is homotopic to a homeomorphism h : Y × Q → X × Q, it follows that f × id is homotopic to h−1 . Indeed, f × id = h−1 h(f × id)  h−1 (g × id)(f × id) = h−1 . The result easily follows from the definition of a simple homotopy.

" !

The converse of the above theorem is also valid, which will be proved in Sect. 4.6 (Theorem 4.6.5). For each compact polyhedron X, an Abelian group Wh(X) called the Whitehead group of X can be defined, and for each homotopy equivalence f : Y → X from another compact polyhedron Y to X, an element τ (f ) ∈ Wh(X) named the Whitehead torsion of f can be defined so that • f is a simple homotopy equivalence if and only if τ (f ) = 0. Then, τ (f ) = τ (f  ) for any homotopy equivalence f  : Y → X homotopic to f . It is known that the Whitehead group Wh(X) is determined by the fundamental group π1 (X). — Refer to §6 and §22 of Cohen’s book [(20)]. The following TOPOLOGICAL INVARIANCE OF WHITEHEAD TORSION had been a longstanding problem raised by J.H.C. Whitehead [156] and was proved by T.A. Chapman [45] by using Q-manifolds: Theorem 4.1.3 (TOPOLOGICAL INVARIANCE OF WHITEHEAD TORSION) Every homeomorphism f : X → Y between compact polyhedra is a simple homotopy equivalence, that is, τ (f ) = 0. In Theorem 4.6.5, using Q-manifolds, we will characterize simple homotopy equivalences. Theorem 4.1.3 above can be obtained as a particular case. To prove Theorem 4.6.5, we will establish the Handle Straightening Theorem 4.6.2 and apply the following result on Whitehead torsion: Theorem 4.1.4 Let f : X → Y be a map between compact polyhedra. Suppose that X = X1 ∪ X2 , Y = Y1 ∪ Y2 , each Xi and Yi are subpolyhedra of X and Y , respectively, such that f (Xi ) ⊂ Yi , and let X0 = X1 ∩ X2 and Y0 = Y1 ∩ Y2 . If the restrictions f |Xi : Xi → Yi , i = 0, 1, 2, are simple homotopy equivalences (i.e.,

4.1 Simple Homotopy Equivalences

293

τ (f |Xi ) = 0 for each i = 0, 1, 2), then f : X → Y is also a simple homotopy equivalence (i.e., τ (f ) = 0).2 The Handle Straightening Theorem 4.6.2 also will play a very important role in our proof of the Triangulation Theorem 4.7.3 for compact Q-manifolds. To prove the Handle Straightening Theorem 4.6.2, we will establish the Splitting Theorem 4.4.1. In order to prove these theorems, we need the following two theorems, whose proofs cannot be given in this book because they are heavily reliant on algebra. The first is the following theorem concerning the Whitehead group: Theorem 4.1.5 For a compact connected polyhedron X, if π1 (X) is free or free Abelian, then Wh(X) = 0, hence every homotopy equivalence f : Y → X of another compact connected polyhedron Y to X is a simple homotopy equivalence.3 The second is related to Wall’s finiteness obstruction: Theorem 4.1.6 Let X be a compact connected space that is homotopy dominated by a finite simplicial complex. If π1 (X) is free or free Abelian, then X has the homotopy type of a finite simplicial complex.4 The following theorem will play an important role in our proof of the Triangulation Theorem for Q-manifolds, which is a combination of Theorems 4.1.2 and 4.1.5: Theorem 4.1.7 Let X and Y be compact connected polyhedra with the same homotopy type such that the fundamental group of X (and Y ) is free or free Abelian. Then, every homotopy equivalence f : X × Q → Y × Q is homotopic to a homeomorphism. Proof We can easily obtain a homotopy equivalence f0 : X → Y such that f  f0 × idQ . Due to Theorem 4.1.5, f0 is a simple homotopy equivalence. Then, we have the result by Theorem 4.1.2. " ! In Remark 1.9 of Sect. 1.10, p. 57, we mentioned the question of whether a compact space homotopy dominated by a simplicial complex has the homotopy type of a finite simplicial complex. Note that if a compact space is homotopy dominated by a simplicial complex K, then it is homotopy dominated by a finite subcomplex of K (cf. Remark 1.12, p. 70). In [147], C.T.C. Wall answered a more general question of whether a space homotopy dominated by a finite simplicial complex has

2 This is a direct consequence of the Sum Theorem (23.1) in Cohen’s book [(20)]. As mentioned in Chapman’s lecture notes [(2)], p. 56, there is a geometric proof which is implicit in Section 6 in Cohen’s book [(20)]. 3 In Cohen’s book [(20)], it is shown in (21.1) that Wh(X) is equal to the Whitehead group Wh(π1 (X)) for the group π1 (X), which reduces the free Abelian case to (11.3) and the free case to a combination of (11.2) and (11.6), but the proofs of (11.3) and (11.6) are not given because of difficulty, where two papers [25] and [134] are cited for (11.3) and (11.6), respectively. The proofs of Wh({0}) = 0 (11.1) and Wh(Z) = 0 (11.2) are given in the book [(20)]. 4 This theorem will be used in the proof of the Splitting Theorem 4.4.1, which is the key to proving the Handle Straightening Theorem 4.6.2.

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4 Triangulation of Hilbert Cube Manifolds and Related Topics

the homotopy type of a finite simplicial complex.5 In fact, let X be a connected space homotopy dominated by a finite simplicial complex. Then, he defined an invariant [X] in an Abelian group K˜ 0 (Zπ1 (X)) determined by the fundamental group π1 (X) such that [X] = 0 if and only if X has the homotopy type of a finite simplicial complex. Thus, Theorem 4.1.6 follows from the algebraic result that K˜ 0 (Zπ1 (X)) = 0 if π1 (X) is free or free Abelian. As is the case with the topological invariance of Whitehead torsion, the proof of the Borsuk conjecture also depends on the Triangulation Theorem for compact Q-manifolds (Theorem 4.7.3). Thus, to prove both the Borsuk conjecture and the topological invariance of Whitehead torsion, we need the above algebraic results. So, the author is awaiting geometric proofs of Theorems 4.1.5 and 4.1.6. By the way, S. Ferry gave in [65] a geometrical approach to Wall’s finiteness obstruction as an analogy of Cohen’s approach to the Whitehead group, where an invariant [X] can be realized as an element of Wh(X × S1 ). Note For finite simplicial complexes K and L, it is said that K collapses to L by an elementary simplicial collapse, written K &e L, if L is a subcomplex of K and K \ L = {σ, vσ }, where vσ is the join of a vertex v ∈ L(0) and a simplex σ ∈ K \ L. When K &e L, it is also said that L expands to L by an elementary simplicial expansion, written L e' K. Obviously, K &e L implies |K| &e |L| (L e' K implies |L| e' |K|), where the converse does not hold. Thus, K &e L and |K| &e |L| (L e' K and |L| e' |K|) have different meanings even if the same notation &e ( e' ) is used. It is said that K simplicially collapses to L (or L simplicially expands to K), written K & L (or L ' K), if there is a finite sequence of elementary simplicial collapses K = K0 &e K1 &e · · · &e Kn = L. It is said that K and L have the same simple-homotopy type or K has the simplehomotopy type of L (K is simple-homotopy equivalent to L) if there is a finite sequence of simplicial complexes Ki , i = 0, 1, . . . , n, such that K0 = K, Kn = L, and Kn−1 & Kn or Kn−1 ' Kn . It can be easily seen (but a proof is bothersome) that, if X &e Y , then there is a simplicial complex K with a subcomplex L ⊂ K such that X = |K|, Y = |L| and K & L. Thus, it follows that |K| & |L| if and only if there are simplicial subdivisions K   K and L  L such that K  & L . Consequently, we have the following: •

5 Refer

For compact polyhedra X and Y , X & Y if and only if K & L for some triangulations K and L of X and Y , respectively.

to Sect. 1 of [111]: “Historical background.”

4.2 Covering Spaces and Algebraic Preliminaries

295

4.2 Covering Spaces and Algebraic Preliminaries As observed in the previous section, the fundamental group is important in Simple Homotopy Theory and Wall Obstruction Theory. In calculation of this group, the covering space is indispensable. Let p : E → B be a surjective map. An open set U in B is said to be evenly covered by p if p−1 (U ) is a disjoint union of open sets, each of which is mapped by p homeomorphically onto U . When every point x ∈ B has an open neighborhood that is evenly covered by p, E is called a covering space of B and p is called a covering projection (map).6 The most typical example is the exponential map e : R → S1 defined as follows: e(x) = (cos πx, sin πx). When every (x, y) ∈ R2 is identified with x + iy ∈ C (i.e., we regard R2 = C), we can write S1 = {z ∈ C | |z| = 1} and e(t) = exp(πit) = cos πt + i sin πt for every t ∈ R. One can picture e as a function wrapping the real line R around the circle S1 , where each interval [2n − 1, 2n + 1] makes a circuit of S1 .7 The n-torus (n-dimensional torus)8 Tn is defined as the n-hold product of the unit sphere S1 , that is, Tn = (S1 )n = (S1 × ·)* · · × S+1 . n many

Then, Rn is a covering space of Tn with the covering projection en : Rn → Tn defined by en (x) = (e(x(1)), . . . , e(x(n)) for each x ∈ Rn . This can be shown by using the following fact inductively: Fact For covering projections p : E → B and p : E  → B  , p × p : E × E  → B × B  is also a covering projection.

6 For

a covering projection, the book [GAGT] gives its definition (p. 498) but not any other information. One can refer to Sect. 53 of Munkres’ book [(7)]. 7 It is easy to check details (cf. Theorem 53.1 of Munkres’ book [(7)]). 8 The plural is tori.

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4 Triangulation of Hilbert Cube Manifolds and Related Topics

Let p : E → B be a map. For a map f : X → B, a lifting of f is a map f˜ : X → E such that pf˜ = f , that is, the following diagram commutes: E f˜

X

f

p

B

Then, it is said that f is lifted to f˜. Every covering projection has the unique path lifting property, that is, the following lemma holds: Lemma 4.2.1 (UNIQUE PATH LIFTING) Let p : E → B be a covering projection with a0 ∈ p−1 (b0 ). Every path f : I → B with f (0) = b0 is uniquely lifted to a path f˜ : I → E with f˜(x0 ) = a0 . Proof Let U be an open cover of B such that each open set in U is evenly covered by p. Because of the compactness of I, there is a sequence s0 = 0 < s1 < · · · < sn = 1 such that each [si−1 , si ] is contained in f −1 (U ) for some U ∈ U. When f˜(si−1 ) has been defined, we can take U ∈ U and an open set V in E such that f ([si−1 , si ]) ⊂ U , f˜(si−1 ) ∈ V , and p|V : V → U is a homeomorphism. If f  is a lifting of f |[si−1 , si ] with f  (si−1 ) = f˜(si−1 ), then f  ([si−1 , si ]) ⊂ V because of connectedness. Hence, we have f  = (p|V )−1 f |[si−1 , si ], which means that f˜|[si−1 , si ] is uniquely defined by f˜|[si−1 , si ] = (p|V )−1 f |[si−1 , si ]. So, first define f˜(0) = a0 and iterate this process n times. Then, we can define f˜ on all of I and it is unique. " ! Lemma 4.2.2 Let p : E → B be a covering projection with a0 ∈ p−1 (b0 ). Every map f : I2 → B with f (0, 0) = b0 is uniquely lifted to a map f˜ : I2 → E with f˜(0, 0) = a0 . Proof As in the proof of the previous lemma, take U ∈ cov(B) so that each open set in U is evenly covered by p. Because of the compactness of I2 , there are sequences s0 = 0 < s1 < · · · < sm = 1 and t0 = 0 < t1 < · · · < tn = 1 such that each Di,j = [si−1 , si ] × [tj −1 , tj ] is contained in f −1 (U ) for some U ∈ U. When f˜(si−1 , tj −1 ) has been defined, we can take U ∈ U and an open set V in E such that f (Di,j ) ⊂ U , f˜(si−1 , tj −1 ) ∈ V , and p|V : V → U is a homeomorphism. If f  is a lifting of f |Di,j with f  (si−1 , tj −1 ) = f˜(si−1 , tj −1 ), then f  (Di,j ) ⊂ V because of connectedness. Hence, we have f  = (p|V )−1 f |Di,j . This means that f˜|Di,j is uniquely defined by f˜|Di,j = (p|V )−1 f |Di,j , where f˜|{si−1 } × [tj −1 , ti ] and f˜|[si−1 , si ] × {tj −1 } are also uniquely determined. So, first define f˜(0, 0) = a0 and iterate this process mn times in the following order: (i  , j  ) < (i, j ) ⇐⇒ def.

 j  < j, j  = j, i  < i.

4.2 Covering Spaces and Algebraic Preliminaries

Then, we can define f˜ on all of I and it is unique.

297

" !

Proposition 4.2.3 Let p : E → B be a covering projection and α, β : I → E be paths from a0 to a1 in E. Then, pα  pβ rel. ∂I implies α  β rel. ∂I. Proof Let h : I × I → B be a homotopy with ht (0) = p(a0 ) and ht (1) = p(a1 ) for each t ∈ I, h0 = pα, and h1 = pβ. By Lemma 4.2.2, h is lifted to a homotopy ˜ 0) = a0 . From the uniqueness of liftings of h|{0} × I h˜ : I × I → E such that h(0, ˜ ˜ and h|{1} × I, h0 and h1 , it follows that h({0} × I) = {a0} and h({1} × I) = {a1 }. h˜ 0 = α, and h˜ 1 = β. Hence, α  β rel. ∂I. " ! Recall that a loop in X (at x0 ) is a path α : I → X such that α(0) = α(1) (= x0 ). For the fundamental groups, we use the same notation as Sect. 4.14 of [GAGT]. Proposition 4.2.4 For a covering projection p : E → B with a0 ∈ p−1 (b0 ), the following hold: (1) The induced homomorphism p : π1 (E, a0 ) → π1 (B, b0 ) is injective. (2) A loop α in B at b0 is lifted to a loop in E at a0 if and only if [α] ∈ p (π1 (E, a0 )). Proof In Proposition 4.2.3, replacing paths by loops, we have (1). The “only if” part of (2) is trivial. To see the “if” part of (2), assume that [α] ∈ p (π1 (E, a0 )), that is, there is a loop β : I → E at a0 such that pβ  α rel. ∂I. Let h : I × I → B be a homotopy with h(∂I × I) = {b0 }, h0 = α, and h1 = pβ. By Lemma 4.2.2, h is lifted to a ˜ 0) = a0 . From the uniqueness of liftings of homotopy h˜ : I × I → E with h(0, ˜ h|{0} × I and h|{1} × I, and h1 , it follows that h(∂I × I) = {a0 } and h˜ 1 = β. In particular, h˜ 0 is a loop in E at a0 , which is a lift of h0 = α. " ! The following theorem is a generalization of Lemma 4.2.1: Theorem 4.2.5 (LIFTING CRITERION) Let p : E → B be a covering projection with a0 ∈ p−1 (b0 ) and let X be a connected, locally path-connected space. A map f : X → B with x0 ∈ f −1 (b0 ) can be lifted to a map f˜(x0 ) = a0 if and only if f (π1 (X, x0 )) ⊂ p (π1 (E, a0 )). Furthermore, a lifting f˜ is unique if it exists. Proof If the lifting f˜ exists, then f (π1 (X, x0 )) = p (f˜ (π1 (X, x0 ))) ⊂ p (π1 (E, a0 )). This gives the “only if” part. The uniqueness of f˜ can also be proved as follows: For each x ∈ X, we can take a path α from x0 to x in X.9 By Lemma 4.2.1 above, the path f α can be lifted to a unique path γ in E with γ (0) = a0 . If a lifting f˜ of f exists, then f˜(x) = γ (1).

9A

locally path-connected space is path-connected if it is connected.

298

4 Triangulation of Hilbert Cube Manifolds and Related Topics

Indeed, since f˜α is a lifting of f α, it follows from Lemma 4.2.1 that f˜α = γ , which implies f˜(x) = f˜α(1) = γ (1). Thus, we can see the uniqueness of f˜. Now, we will show the “if” part. By the argument above, if the lifting f˜ exists, then for each x ∈ X, f˜(x) must be defined as follows: Take a path α from x0 to x in X, and lift the path f α to a path γ with γ (0) = a0 , and then define f˜(x) = γ (1). We have to show that this is well-defined, that is, the point γ (1) is uniquely determined independent from the choice of α, where γ is uniquely determined as a lifting of f α. Given x ∈ X, let α and β be paths from x0 to x. Then, f α is lifted to γ with γ (0) = a0 and fβ ← is lifted to ζ with ζ(0) = γ (1), where β ← is the inverse path of β that is defined by β ← (t) = β(1 − t). The join γ ∗ ζ is a lifting of the loop f ◦(α ∗ β ← ) at b0 and γ ∗ ζ = γ (0) = a0 . Due to Proposition 4.2.4, the condition f (π1 (X, x0 )) ⊂ p (π1 (E, a0 )) implies that the loop f ◦(α ∗ β ← ) is lifted to a loop in E at a0 . By uniqueness of liftings, γ ∗ ζ must be a loop at a0 . Hence, ζ ← is a lifting of fβ with ζ ← (0) = ζ(1) = a0 and ζ ← (1) = ζ(0) = γ (1). This means that the point γ (1) is uniquely determined independent from the choice of α. We have to show the continuity of f˜ at x ∈ X. As above, let α be a path from x0 to x in X, and γ be a lifting of the path f α with γ (0) = a0 . Then, f˜(x) = γ (1). Since p is a covering projection and pf˜(x) = f (x), f (x) and f˜(x) have open neighborhoods U0 and V0 respectively such that p|V0 : V0 → U0 is a homeomorphism. For each open neighborhood V of f˜(x) in E, p(V ∩V0 ) is an open neighborhood of f (x) in B. Since f is continuous and X is locally path-connected, x has a path-connected open neighborhood W in X such that f (W ) ⊂ p(V ∩ V0 ). For each y ∈ W , we have a path β from x to y in W . Then, fβ is a path from f (x) to f (y) in p(V ∩ V0 ). Since p|V0 is a homeomorphism, ζ = (p|V0 )−1 fβ is a path in V ∩ V0 , which is a lifting of fβ with ζ(0) = f˜(x) = γ (1). Then, γ ∗ ζ is a lifting of f α ∗ fβ = f ◦(α ∗ β) with γ ∗ ζ(0) = γ (0) = a0 , where α ∗ β is a path from x0 to y in X. Hence f˜(y) = γ ∗ ζ(1) = ζ(1) ∈ V ∩ V0 ⊂ V . Thus, we have f˜(W ) ⊂ V . Therefore, f˜ is continuous. " ! In the rest of this chapter, we will also use elementary results on the homotopy groups and the homology groups. The reader can refer to any standard textbook on Algebraic Topology, e.g., [(22)], [(23)], etc.10 Theorem 4.2.6 (SEIFERT–VAN KAMPEN) Let X = int X1 ∪ int X2 and x0 ∈ X0 = X1 ∩ X2 , where X0 , X1 and X2 are path-connected (hence X is also pathconnected). For i = 1, 2, let ji : π1 (Xi , x0 ) → π1 (X, x0 ) and j0,i : π1 (X0 , x0 ) → π1 (Xi , x0 ) denote the homomorphisms induced by the inclusions. Then, for each group H with homomorphisms fi : π1 (Xi , x0 ) → H , i = 1, 2, such that

10 The Seifert–van Kampen Theorem can be found in not only such textbooks but also Chap. 70 of Munkres’ textbook [(7)].

4.2 Covering Spaces and Algebraic Preliminaries

299

f1 j0,1 = f2 j0,2 , there exists a unique homomorphism f : π1 (X, x0 ) → H such that fj1 = f1 and fj2 = f2 . π1 (X1 , x0 )

f1

j0,1 j1

π1 (X0 , x0 )

π1 (X, x0 )

∃1 f

H

j2 j0,2

f2

π1 (X2 , x0 )

Concerning the homology groups, it is enough to know the simplicial homology groups.11 Indeed, we treat only ANRs and every ANR has the homotopy type of some simplicial complex (Theorem 1.13.22). A sequence of groups and homomorphisms f1

f2

fr

G1 −→ G2 −→ · · · −→ Gr+1 is said to be exact if im fi = ker fi+1 for each i = 1, . . . , r. A homomorphism f : G → H is injective (resp. surjective) if and only if f

0 −→ G −→ H

f

(resp. G −→ H −→ 0)

is exact. An injective homomorphism is called a monomorphism and a surjective homomorphism is called an epimorphism. If H is a subgroup of G then ⊂

p

0 −→ H −→ G −→ G/H −→ 0 is exact, where p is the quotient homomorphism. The following theorem is very useful to calculate the homology of spaces: Theorem 4.2.7 (MAYER–VIETORIS) Let X = int X1 ∪int X2 and X0 = X1 ∩X2 . Moreover, let ϕn : Hn (X0 ) → Hn (X1 ) ⊕ Hn (X2 ), ψn : Hn (X1 ) ⊕ Hn (X2 ) → Hn (X), n ∈ ω be the homomorphisms defined by ϕn (x) = (j0,1 (x), j0,2(x)) and ψn (x1 , x2 ) = j1 (x1 ) − j2 (x2 ),

11 Nevertheless,

ˇ for an arbitrary space, the singular homology or the Cech homology can be used.

300

4 Triangulation of Hilbert Cube Manifolds and Related Topics

where j0,i : Hn (X0 ) → Hn (Xi ), ji : Hn (Xi ) → Hn (X), i = 1, 2, denote the homomorphism induced by the inclusions. Then, there are homomorphisms δn : Hn (X) → Hn−1 (X0 ), n ∈ ω, such that the following sequence is exact: δn+1

ψn

ϕn

· · · −→ Hn (X0 ) −→ Hn (X1 ) ⊕ Hn (X2 ) −→ Hn (X) ϕn−1

δn

ψ1

δ1

−→ Hn−1 (X0 ) −→ · · · −→ H1 (X) −→ H0 (X0 ) ϕ0

ψ0

δ0

−→ H0 (X1 ) ⊕ H0 (X2 ) −→ H0 (X) −→ 0. The exact sequence in Theorem 4.2.7 above is called the Mayer–Vietoris exact sequence for (X; X1 , X2 ). For any space X and n ∈ N, there exists a group homomorphism h∗ : πn (X) → Hn (X) called the Hurewicz homomorphism from the nth homotopy group to the nth homology group (with integer coefficients), which is equivalent to the canonical Abelianization map h∗ : π1 (X) → π1 (X)/[π1 (X), π1 (X)] for n = 1 and a pathconnected space X. The Hurewicz theorem states that if X is (n−1)-connected, then the Hurewicz map is an isomorphism for all k  n when n  2 and Abelianization for n = 1. In particular, this theorem says that the Abelianization of the first homotopy group (the fundamental group) is isomorphic to the first homology group, that is, H1 (X) ∼ = π1 (X)/[π1 (X), π1 (X)]. Theorem 4.2.8 (HUREWICZ) Let X be a path-connected space such that πi (X) = 0 for i = 1, . . . , n − 1, where n  2. Then, Hi (X) = 0 for i = 1, . . . , n − 1 and πn (X) ∼ = Hn (X).

4.3 The Procedure for Killing Homotopy Groups A path-connected space X is said to be n-connected if πi (X) = {0} for every i = 1, . . . , n. This is equivalent to the condition that any map α : Si → X is null-homotopic for every i = 1, . . . , n. When X is 1-connected, it is also said that X is simply connected. A space X is homotopically trivial if and only if X is nconnected for every n ∈ N. Recall that an ANR X is contractible (X  0) if and only if X is homotopically trivial (Theorem 1.13.28). In this section, we will prove the following two theorems. The first theorem is well known but its proof is given for completeness and for the readers’ convenience. And then, using this, we will prove  n and m Bn+1 are the topological the second theorem. In the following, m S i=1 i i=1 i sums of m copies of the unit n-sphere Sn and the unit (n + 1)-ball Bn+1 . Theorem 4.3.1 Let X be a connected ANR that is homotopy dominated by a finite simplicial complex. mn n Then, there exists a tower X = X0 ⊂ X1 ⊂ · · · ⊂ Xk0 and maps ϕn : i=1 Si → Xn−1 , n = 1, . . . , k0 , such that Xk0  0, each Xn is n-

4.3 The Procedure for Killing Homotopy Groups

connected, and Xn = Xn−1 ∪ϕn mn = 0 and Xn = Xn−1 .

mn

n+1 , i=1 Bi

301

where if Xn−1 is n-connected then

Theorem 4.3.2 For each compact connected Q-manifold M, there exists a tower M m=n Mn0 ⊂ M1 ⊂ · · · ⊂ Mk0 of compact Q-manifolds and Z-embeddings ϕn : j =1 (Si × Q) → Xn−1 , n = 1, . . . , k0 , such that Mk0 ≈ Q, each Mn is nconnected, and Mn = Mn−1 ∪ϕn

mn

(Bn+1 × Q), i i=1

where if Mn−1 is n-connected then mn = 0 and Mn = Mn−1 . As mentioned in Remark 1.12, the following is obtained from Hanner’s Characterization of ANRs (Theorem 1.13.20): Proposition 4.3.3 Every compact ANR X is homotopy dominated by a finite simplicial complex. " ! The following is an alternative familiar proof of the above proposition: Proof (Proposition 4.3.3) By the universality of Q, we may assume that X is a closed set in Q. Then, we have an open neighborhood U of X in Q with a retraction r : U → X. Because of the compactness of X, we can easily find n ∈ N and an open set V in [−1, 1]n such that X ⊂ pn−1 (V ) ⊂ U , where pn : Q → [−1, 1]n is the projection defined by pn (x) = (x(1), . . . , x(n)) for each x ∈ Q (cf. Wallace’s Theorem 2.1.2 L of [−1, 1]n so that in [GAGT]). Note that pn (X) ⊂ V . Take a small triangulation  each σ ∈ L[pn (X)] is contained in V . Let K = σ ∈L[pn (X)] F (σ ). Then, K is a subcomplex of L and pn (X) ⊂ |K| ⊂ V . Let f = pn |X : X → |K| and define a map g : |K| → X by g(x) = r(x(1), . . . , x(n), 0, 0, . . . ) for each x ∈ |K| (⊂ [−1, 1]n ). Since gf (x) = r(x(1), . . . , x(n), 0, 0, . . . ) for each x ∈ X, it follows that gf  idX by the homotopy h : X × I → X defined as follows: h(x, t) = r(x(1), . . . , x(n), tx(n + 1), tx(x + 2), . . . ) for each (x, t) ∈ X × I. Thus, X is homotopy dominated by K.

" !

The following is easily proved: Proposition 4.3.4 If a connected (resp. path-connected) X is homotopy dominated by Y , then X is homotopy dominated by a component (resp. path-component) of Y . Proof We have maps f : Y → X and g : X → Y such that f g  idX . Since g(X) is connected (resp. path-connected), it is contained in some component (resp. pathcomponent) Y0 of Y . Then, f |Y0 : Y0 → X is a homotopy domination because (f |Y0 )g = " ! f g  idX .

The following is a combination of Propositions 4.3.3 and 4.3.4: Corollary 4.3.5 Every compact connected ANR X is homotopy dominated by a connected finite simplicial complex. ! "

302

4 Triangulation of Hilbert Cube Manifolds and Related Topics

A group G is said to be finitely generated if G has finite elements g1 , . . . , gn such that every element x ∈ G can be written as a product of powers of gi , i = 1, . . . , n. The following is elementary and well-known: Fact For every connected finite simplicial complex K, the fundamental group π1 (|K|) is finitely generated.12 For the readers’ convenience, we give a proof of the above fact: Proof The 1-skeleton K (1) is a finite connected graph, which has a maximal tree T (i.e., T is a tree with T (0) = K (0) ). Let K (1) \ T = {ei | i = 1, . . . , n} and take v0 ∈ K (0) as a base point. For each i = 1, . . . , n, we have a PL loop αi : I → |K (1) | such that αi (0) = αi (1) = v0 , e1 = αi (1/3), αi (2/3), and αi ([0, i/3]) is the unique arc from v0 to α(i/3) in |T | for i = 1, 2. By the Simplicial Approximation Theorem 1.8.8, for each loop α : I → |K| at v0 , there exist I   I = {0, 1, I} and a simplicial loop α  : I  → |K (1) | at v0 such that α  α  rel. ∂I. Then, α  is homotopic to a join of some members of {αi , αi← | i = 1, . . . , n}, where " ! αi← is the inverse loop of αi . This means that π1 (|K|) is finitely generated.

The following is a direct consequence of the definition of the simplicial homologies: Fact For every finite simplicial complex K, Hk (K) is finitely generated for 0  k  dim K and Hk (K) = 0 if k > dim K. Additionally, H0 (X) and Hdim K (K) is free, where the rank of H0 (X) is equal to the number of components of K. Using the above two facts, we have the following: Proposition 4.3.6 Let X be a connected ANR which is homotopy dominated by a finite simplicial complex K. Then, π1 (X) and Hk (X), 0 < k  dim K, are finitely generated and Hk (X) = 0 for every k > dim K. Additionally, Hdim K (X) is free. Proof There are maps f : |K| → X and g : X → |K| such that fg  id. Then, f induces the epimorphisms f : π1 (|K|) → π1 (X) and f∗ : Hk (K) → Hk (X), k ∈ N, because f g = id and f∗ g∗ = id. For 0 < k  dim k, since Hk (K) is finitely generated, it follows that H∗ (X) is also finitely generated. For every k > dim K, since Hk (K) = 0, we have Hk (X) = 0. Furthermore, since g∗ : Hdim K (X) → Hdim K (K) is a monomorphism and Hdim K (K) is free, it follows that Hdim K (X) is free. " ! Let e1 = (1, 0, . . . , 0) ∈ Sn ⊂ Rn+1 . Observe (In /∂In , ∂In /∂In ) ≈ (Sn , e1 ). Then, we have a map θ : (In , ∂In ) → (Sn , e1 ) such that θ (∂In ) = e1 and θ | rint In → Sn \ {e1 } is a homeomorphism. Each map α : (In , ∂In ) → (X, ∗) induces the map α˜ : (Sn , e1 ) → (X, ∗) such that α = αθ ˜ . In the proof of the following lemma, we use the Seifert–van Kampen Theorem 4.2.6, the Mayer– Vietoris Theorem 4.2.7, and the Hurewicz Isomorphism Theorem 4.2.8.

12 Since

|K| is path-connected, we can omit a base point.

4.3 The Procedure for Killing Homotopy Groups

303

Lemma 4.3.7 Let X be an ANR that is (n − 1)-connected but not n-connected, where n > 0. Suppose that πn (X) is generated by m many and Hn+1 (X)  generators n → X such that the is also finitely generated. Then, there exists a map ϕ : m S i=1 i n adjunction space Xϕ = X ∪ϕ m i=1 Bi is n-connected, πn+1 (Xϕ ) = Hn+1 (Xϕ ) is finitely generated, and Hk (Xϕ ) ∼ = Hk (X) for k > n + 1, where if Hn+1 (X) is free then πn+1 (Xϕ ) = Hn+1 (Xϕ ) is also free, and if πn (X) is free and Hn+1 (X) = 0 then πn+1 (Xϕ ) = Hn+1 (Xϕ ) = 0. Proof Take maps α1 , . . . , αm : (In , ∂In ) → (X, ∗) so that πn (X) is generated by [α1 ], . . . , [αm ]. Let (Bn+1 , Sni , ∗i ), i = 1, . . . , m, be disjoint copies of i (Bn+1 , Sn , e1 ). For each i = 1, . . . , m, there is a map θi : (In , ∂In ) → (Sni , ∗i ) such that θi (∂In ) = ∗i and θi | rint In → Sni \ {∗i } is a homeomorphism. Then, each n αi induces m an map α˜ i : (Si , ∗i ) → (X,n ∗) such that αi = α˜ i θi . The desired map ϕ : ˜ i , i = 1, . . . , m. We will show that i=1 Sj→ X is defined by ϕ|Si = α n satisfies the required conditions. Xϕ = X ∪ϕ m B i=1  i n+1 → Xϕ be the quotient map. For each i = 1, . . . , m, we Let q : X ⊕ m i=1 Bi define    Bi = (1 − t)x + t∗i  x ∈ Sni , 0  t  1/2 ,    Si = ∂Bi = 12 x + 12 ∗i  x ∈ Sni , and    \ rint Bi = (1 − t)x + t∗i  x ∈ Sni , 1/2  t  1 . Ci = Bn+1 i Then, we have retraction ri : Ci → Sni defined by ri ((1 − t)x + t∗i ) = x for x ∈ Sni , 1/2  t  1, where ri  id rel. Sni . Since ri |Si : Si → Sni is a homeomorphism, thereis a map θi : (In , ∂In ) → (Si , ∗i ) such that θi = ri θi , so αi = α˜ i ri θi . Let Y = q( m i=1 Si ), Z = q( m B ), and i=1 i M = Xϕ \ (Z \ Y ) = X ∪ q(

m

i=1 Ci ).

These are subspaces of Xϕ such that Xϕ = M ∪ Z and Y = M ∩ Z. Note that m n m m n+1 13 Y = m . Moreover, we have i=1 Si ≈ i=1 Si and Z = i=1 Bi ≈ i=1 Bi a retraction r : M → X defined by r|X = id and rq|Ci = ϕri |Ci = α˜ i ri |Ci for each i = 1, . . . , m. Then, as is easily observed, r  id rel. X (hence X is a strong deformation retract of M). It follows that r induces an isomorphism r : πn (M) → πn (X), whose inverse is induced by the inclusion X ⊂ M. The map r|Y : Y → X induces an epimorphism (r|Y ) : πn (Y ) → πn (X) because rqθi = α˜ i ri θi = αi for i = 1, . . . , m, that is, the generators [α1 ], . . . , [αm ] ∈ πn (X) are the images

13 For

the wedge (sum)

m

i=1 Xi ,

refer to p. 5.

304

4 Triangulation of Hilbert Cube Manifolds and Related Topics

of [qθ1 ], . . . , [qθm ] ∈ πn (Y ). Since r|Y is the composition of r and the inclusion Y ⊂ M, it follows that the inclusion Y ⊂ M induces an epimorphism from πn (Y ) to πn (M). In the following commutative diagram, the four arrows of the right rectangle are the homomorphisms induced by the inclusions: π1 (Y )

π1 (Z)

π1 (M)

π1 (Xϕ )

=

0

(r|Y )

π1 (X)

∼ = r

By the Seifert–van Kampen Theorem 4.2.6, π1 (Xϕ ) is generated by the union of the images of π1 (M) and π1 (Z). Since π1 (Z) = 0, the bottom arrow of the rectangle is an epimorphism. When n = 1, as observed above, the left vertical arrow is also an epimorphism. Hence, the inclusion Y ⊂ Xϕ induces an epimorphism from π1 (Y ) to π1 (Xϕ ). Since Y ⊂ Z ⊂ Xϕ and π1 (Z) = 0, it follows that π1 (Xϕ ) = 0, that is, Xϕ is 1-connected. When n > 1, since π1 (M) ∼ = π1 (X) = 0, we have π1 (Xϕ ) = 0. Note that H1 (Xϕ ) = 0 by the Hurewicz Theorem 4.2.8. By the Mayer–Vietoris Theorem 4.2.7, we have the following exact sequence of homology groups: i  −j 

(i,j )

· · · −→ Hk (Y ) −→ Hk (M) ⊕ Hk (Z) −→ Hk (Xϕ ) i  −j 

δ

−→ Hk−1 (Y ) −→ · · · −→ H1 (M) ⊕ H1 (Z) −→ H1 (Xϕ ) = 0, where i, i  and j , j  are the homomorphisms induced by the inclusions Y ⊂ M ⊂ Xϕ and Y ⊂ Z ⊂ Xϕ . Since X is (n − 1)-connected and M  X, M is also (n − 1)connected. By the Hurewicz Theorem 4.2.8, we have Hk (M) ∼ = πk (M) = 0 for every 0 < k < n and Hn (M) ∼ = πn (M). Since Hk (Z) = 0 for every k > 0 and Hk (Y ) ∼ = Hk (

m

n i=1 Si )

∼ =

⎧ ⎪ · · ⊕ Z+ if k = n, ⎨Z ( ⊕ ·)* ⎪ ⎩

m many

0

if k = 0, n,

it follows that Hk (Xϕ ) ∼ = Hk (M) = 0 for 0 < k < n and Hk (Xϕ ) ∼ = Hk (M) ∼ = Hk (X) for k > n + 1. Inductively, using the Hurewicz Theorem 4.2.8, we can see that πk (Xϕ ) ∼ = Hk (Xϕ ) = 0 for 0 < k < n, that is, Xϕ is also (n − 1)-connected. In the following diagram, the bottom sequence is exact and two vertical arrows are Hurewicz

4.3 The Procedure for Killing Homotopy Groups

305

isomorphisms: πn (Y )

epi

∼ =

···

πn (M) ∼ =

Hn (Y )

Hn (M)

i

Hn (Xϕ )

0

As already observed, the top arrow of the rectangle is an epimorphism. Hence, the bottom arrow is also an epimorphism, which implies that Hn (Xϕ ) = 0. Again, using the Hurewicz Theorem 4.2.8, we have πn (Xϕ ) ∼ = Hn (Xϕ ) = 0. Consequently, Xϕ is n-connected. Moreover, we have the following short exact sequence: 0

Hn+1 (M)

i

Hn+1 (Xϕ )

im δ

0,

where im δ is free as a subgroup of the free Abelian group Hn (Y ). So, the above short exact sequence is split. Therefore, we have Hn+1 (Xϕ ) ∼ = Hn+1 (M) ⊕ im δ, which is finitely generated. In the above, when Hn+1 (X) is free, since Hn+1 (M) ∼ = Hn+1 (X), it follows that Hn+1 (Xϕ ) is also free. When πn (X) is free, ϕ : πn (Y ) → πn (X) is an isomorphism because the generators [α1 ], . . . , [αm ] ∈ πn (X) are the images of [θ1 ], . . . , [θm ] ∈ πn (Y ). Therefore, the inclusion Y ⊂ M induces an isomorphism from πn (Y ) to πn (M), hence it induces an isomorphism from Hn (Y ) to Hn (M). When Hn+1 (X) = 0, we have im δ = 0. Since Hn+1 (M) ∼ = Hn+1 (X) = 0, it follows that Hn+1 (Xϕ ) = 0. This completes the proof. " ! Now, we can prove Theorem 4.3.1. Proof (Theorem 4.3.1) By virtue of Proposition 4.3.6, we can iteratively apply Lemma to obtain a tower X = X0 ⊂ X1 ⊂ · · · ⊂ Xk0 and maps  4.3.7 n n ϕn : m S → Xn−1 , n = 1, . . . , k0 , such that Xk0  0, each Xn is n-connected, i=1 i  n n+1 , where if Xn−1 is n-connected then mn = 0 and and Xn = Xn−1 ∪ϕn m i=1 Bi Xn = Xn−1 . " ! m n+1 × Q) by a In Theorem 4.3.2, we use an adjunction space X ∪ϕ i=1 (Bi m  m n+1 n Z-embedding ϕ : by a map i=1 (Si × Q) → X instead of X ∪ϕ i=1 Bi  n ϕ: m i=1 Si → X.

306

4 Triangulation of Hilbert Cube Manifolds and Related Topics

Lemma 4.3.8 Let X be an ANR and let pr1 : projection.  n (1) For each map f : m i=1 Si → X, X ∪f pr1

m

m

n i=1 (Si

(Bn+1 × Q)  X ∪f i

i=1

(2) For each map f : X ∪f

m

n i=1 (Si

× Q) →

m

m

n i=1 Si

be the

Bn+1 . i

i=1

× Q) → X,

m m

(Bn+1 × Q) ≈ M ∪ (Bn+1 × Q) × {1}. f i i i=1

(3) For two maps f, g :

i=1

m

n i=1 (Si

f  g ⇒ X ∪f

× Q) → X,

m

(Bn+1 × Q)  X ∪g i

i=1

m

(Bn+1 × Q). i

i=1

Proof (1): We have a cell-like map ψ : X ∪f pr1

m m

(Bn+1 × Q) → X ∪ Bn+1 f i i i=1

i=1

 n+1 defined by ψ|X = id and ψ| m × Q) = pr1 . Then, ψ is a fine i=1 (rint Bi homotopy equivalence by Theorem 1.15.9. (2): It suffices to see that   m m m

n+1 n n (Si × Q × I) ∪ (Bi × Q × {1}), (Si × Q × {0}) i=1

i=1

i=1

≈

 m

Bn+1 × Q, i

i=1

m

i=1

This can be obtained by the following:     n (S × I) ∪ (Bn+1 × {1}), Sn × {0} ≈ Bn+1 , Sn . (3): By Proposition 1.10.1(1), m   Mf  Mg rel. X ∪ (Sn+1 × Q) × {1} . i i=1

 Sni × Q .

4.3 The Procedure for Killing Homotopy Groups

307

Hence, it follows from the above (2) that X ∪f

m

m

(Bn+1 × Q)  Mf ∪ i

i=1

(Bn+1 × Q) × {1} i

i=1

 Mg ∪

m

(Bn+1 × Q) × {1} i i=1 m

 X ∪g

(Bn+1 × Q). i

" !

i=1

In Lemma 4.3.7, assume that X is a Q-manifold. Then, applying the ZEmbedding Approximation Theorem 2.10.11, we have ψ : m ma Z-embedding n × Q) → X, which is homotopic to ϕpr : n × Q) → X. It (S (S 1 i=1 i=1 follows from Lemma 4.3.8 (1) and (3) that X ∪ϕ

m

Bn+1  X ∪ψ

m

i=1

(Bn+1 × Q),

i=1

where  the nright-hand side of the above equation is a Q-manifold because ψ( m i=1 (S × Q)) is collared in X. Using this modification, we can prove Theorem 4.3.2 in the same way as Theorem 4.3.1. Proof (Theorem 4.3.2) By virtue of Proposition 4.3.6, we can iteratively apply the above modification of Lemma 4.3.7 to obtain a tower Mn = nM0 ⊂ M1 ⊂ · · · ⊂ Mk0 of compact Q-manifolds and Z-embeddings ϕn : m i=1 (Si × Q) → Mn−1 , n = 1, . . . , k0 , such that Mk0  0, each Mn is n-connected, and Mn = Mn−1 ∪ϕn

mn

(Bn+1 × Q), i i=1

where if Mn−1 is n-connected then mn = 0 and Mn = Mn−1 . Since a compact contractible Q-manifold is homeomorphic to Q (Theorem 3.7.9), we have Mk0 ≈ Q. " ! In the above proof, since ϕn is a Z-embedding, we have mn  i=1

ϕn (Sni × Q) =

mn

ϕn (Sni × Q),

i=1

which is collared in Mn−1 by the Collaring Theorem 2.11.1. Then, it follows that the above set is the boundary bdMn Mn−1 . Thus, in Theorem 4.3.2, attaching one  n n+1 × Q of m × Q) at a time, we have the following: piece Bn+1 i=1 (Bi i

308

4 Triangulation of Hilbert Cube Manifolds and Related Topics

Corollary 4.3.9 For a compact connected Q-manifold M, there exists a tower M = M0 ⊂ M1 ⊂ · · · ⊂ Mk of compact Q-manifolds and 0 < n1  n2  · · ·  nk such that Mk ≈ Q, each bdMi Mi−1 is a Z-submanifold of Mi−1 and     Mi \ intMi Mi−1 , bdMi Mi−1 ≈ Bni +1 × Q, Sni × Q . Thus, bdMi Mi−1 is collared in both Mi−1 and Mi \ intMi Mi−1 , so it is bi-collared in Mi . " ! Concerning the tower of Corollary 4.3.9 above, assume that Mi = Mi−1 ∪ϕi Bni +1 × Q ≈ Pi × Q for some compact polyhedron Pi . If we could find a compact polyhedron Pi−1 such that Mi−1 ≈ Pi−1 × Q, then the downward induction would provide us with a compact polyhedron P such that M ≈ P × Q. This is our strategy to prove the Triangulation Theorem for compact Q-manifolds. In this process, we need the following: Theorem 4.3.10 Let X be an ANR that is homotopy dominated by a finite simplicial complex. Then, there exists a finite simplicial complex K and a homotopy domination f : |K| → X, which induces a bijection between the components of |K| and X, and which induces isomorphisms between the fundamental groups of their corresponding components. In the rest of this section, we will prove this theorem. Proposition 4.3.11 Let X be homotopy dominated by Y . Then, every pathcomponent of X is homotopy dominated by a path-component of Y . Moreover, let f : Y → X and g : Y → X be maps such that fg  idX (i.e., f is a homotopy domination) and let Y  be the union of path-components meeting g(X). Then, the restriction f  = f |Y  : Y  → X is a homotopy domination that induces the bijection f : π0 (Y  ) → π0 (X) between the sets of path-components. Proof Let X0 ∈ π0 (X) (i.e., X0 is a path-component of X). Since g(X0 ) is pathconnected, it is contained in some path-component Y0 of Y , where Y0 ∈ π0 (Y  ). Note that f (Y0 ) is path-connected and fg(X0 ) ⊂ f (Y0 ), where fg(X0 ) ⊂ X0 because fg  idX . Then, it follows that f (Y0 ) ⊂ X0 and (f |Y0 )(g|X0 ) = fg|X0  idX0 in X0 . Therefore, X0 is homotopy dominated by Y0 . For each Y0 ∈ π0 (Y  ), f (Y0 ) is contained in a unique X0 ∈ π0 (X), which is  f (Y0 ). In the above, we have shown that for each X0 ∈ π0 (X), Y0 ∈ π0 (Y  ) with " ! f (Y0 ) ⊂ X0 is uniquely determined,14 that is, f is bijective. Applying Propositions 4.3.3 and 4.3.11, we can obtain the following: Corollary 4.3.12 For each compact ANR X, there exists a finite simplicial complex K with a homotopy domination f : |K| → X such that f induces f : π0 (|K|) → π0 (X). " !

14 In

general, Y0 ∈ π0 (Y ) with f (Y0 ) ⊂ X0 is not unique for X0 ∈ π0 (X).

4.3 The Procedure for Killing Homotopy Groups

309

By Corollary 4.3.12 and Proposition 4.3.3, Theorem 4.3.10 is reduced to the connected case, that is: Theorem 4.3.13 If a connected ANR X is homotopy dominated by a finite simplicial complex, then there exists a connected finite simplicial complex K and a homotopy domination f : |K| → X, which induces an isomorphism between their fundamental groups. Proof Let f : |L| → X be a homotopy domination, where L is a finite simplicial complex. By Proposition 4.3.12, L can be taken so as to be connected. Then, there is a map g : X → |L| such that fg  idX . It follows that f induces the epimorphism15 f : π1 (|L|) → π1 (X) because f g = id. Since π1 (|L|) is finitely generated, the kernel of f is also finitely generated. Take v0 ∈ L(0) as a base point. Then, there exist loops α1 , . . . , αn : I → |L| at v0 such that for a loop α : I → |L|, if f α  0 rel. ∂I, then α is homotopic to a join of some members of {αi , αi← | i = 1, . . . , n}. Let q : I → I/∂I be the quotient map. For each i = 1, . . . , n, applying the Simplicial Approximation Theorem 1.8.8, we can obtain a simplicial complex Li with a homeomorphism ϕi : I/∂I → |Li | and a simplicial map α˜ i : Li → L such that α˜ i ϕi q  αi rel. ∂I. Since f α˜ i ϕi q  f αi  0 rel. ∂I, it follows that f α˜ i  0, hence f α˜ i extends to the map fi : |vi ∗Li | → X, where vi ∗Li is the simplicial cone over Li defined by vi ∗ Li = σ ∈Li F (vi σ ). Then, we have the following diagram, where the right-hand square is commutative but the left-hand square is commutative up to  rel. ∂I: I

αi

q

I/∂I

f

|L|

X fi

α˜ i ≈ ϕi

|Li |

|vi ∗ Li |

⊂

We define a finite simplicial complex K =

n

i=1 (Zα˜ i

∪ |vi ∗ Li |), where

    Zα˜ i ∪ |vi ∗ Li | ∩ Zα˜ j ∪ |vj ∗ Lj | = |L| if i = j. Since f c¯α˜ i ||Li | = f α˜ i = fi ||Li |, we can extend f to a map f˜ : |K| → X as follows: f˜||Zα˜ i | = f c¯α˜ i and f˜||vi ∗ Li | = fi for each i = 1, . . . , n. Then, f˜ is a homotopy domination because f˜g = fg  idX . So f˜ induces the epimorphism f˜ : π1 (|K|) → π1 (X).

15 A

surjective homomorphism is called an epimorphism.

310

4 Triangulation of Hilbert Cube Manifolds and Related Topics

To see that f˜ is a monomorphism,16 let α : I → |K| be a loop at v0 such that f˜α  0 rel. ∂I. By the Simplicial Approximation Theorem 1.8.8, we may assume α is simplicial with respect to some I   I = F (I) and K. Moreover, it can beassumed that α(I) does not contain any of v1 , . . . , vn , which means that α(I) ⊂ ni=1 |Zα˜ i |. Since |L| is a strong deformation retract of ni=1 |Zα˜ i |, there is α  : I → |L| such that α   α rel. ∂I. Then, f α  = f˜α   f˜α  0 rel. ∂I, which implies that α  is homotopic to a join of some members of {αi , αi← | i = 1, . . . , n}. For each i = 1, . . . , n, αi  0 rel. ∂I in |K| because αi  α˜ i ϕi q rel. ∂I and α˜ i  0 in |K|. Consequently, α  α   0 rel. ∂I in |K|. Therefore, f˜ is a monomorphism. ˜ → π1 (X) is Thus, f˜ : |K| → X is a homotopy domination such that f˜ : π1 (|K|) an isomorphism. " !

4.4 The Splitting Theorem for Q-Manifolds In this section, we proceed to a discussion in the following setting: • Let X and Y be locally compact polyhedra, where X is compact and connected, and let M be an open set in Y × Q that is homeomorphic to X × Q × R by a homeomorphism h : X × Q × R → M. One should keep in mind that X × Q is embedded into Y × Q as a bi-collared set, and M is the image of its bi-collar. For each n ∈ N, we write Q = [−1, 1]n × Qn . Let pn : Y × Q = Y × [−1, 1]n × Qn → Y × [−1, 1]n and in : Y × [−1, 1]n → Y × [−1, 1]n × {0} ⊂ Y × Q be the projection and the natural injection, respectively. In the above setting, we define a splitting of M as a decomposition, M = M1 ∪ M2 (cf. Fig. 4.3), which satisfies the following: (i) M1 and M2 are non-compact Q-manifolds which are closed in M, (ii) M0 = M1 ∩ M2 = pn−1 (pn (M0 )) = pn (M0 ) × Qn for some n ∈ N, (iii) A = pn (M0 ) is a PL bi-collared compact subpolyhedron of Y × [−1, 1]n, where A is PL bi-collared in Y × [−1, 1]n if there is a PL embedding k : A × [−1, 1] → Y × [−1, 1]n that is a closed bi-collar in Y × [−1, 1]n , which is called a

16 An

injective homomorphism is called an monomorphism.

4.4 The Splitting Theorem for Q-Manifolds M0 = M1

311 h(X × Q × {0})

M2

M1

M2

A

Y ×Q

Y × [− 1, 1] n M = h(X × Q × R)

Fig. 4.3 A splitting of M

PL bi-collar of A. Then, we can assume that pn−1 (k(A × [0, 1])) = k(A × [0, 1]) × Qn ⊂ M1 and pn−1 (k(A × [−1, 0])) = k(A × [−1, 0]) × Qn ⊂ M2 . Indeed, since A is compact and pn−1 (k(A × {0})) = k(A × {0}) × Qn = A × Qn = M0 ⊂ M, we can find 0 < δ  1 such that pn−1 (k(A × [−δ, δ])) ⊂ M. Changing the scale of [−1, 1], we can assume that pn−1 (k(A × [−1, 1])) ⊂ M. Let C be a component of A. Then, each of pn−1 (k(C × (0, 1])) and pn−1 (k(C × [−1, 0))) is contained in one of M1 \ M2 or M2 \ M1 and both of them are not contained in the same side. With respect to each component of A, changing the positive side and the negative side if necessary, we make k so as to satisfy the above. In this section, applying Theorem 4.1.6, we will prove the following Splitting Theorem: Theorem 4.4.1 (SPLITTING) If π1 (X) is free or free Abelian, then there exists a splitting M = M1 ∪ M2 such that M0 = M1 ∩ M2 is a strong deformation retract of M, that is, the inclusion M0 ⊂ M is a homotopy equivalence. We start by showing the existence of a splitting of M. Proposition 4.4.2 M has a splitting. Proof Since h(X × Q × {0}) is compact, we can find n ∈ N and an open set U in Y × [−1, 1]n such that h(X × Q × {0}) ⊂ pn−1 (U ) = U × Qn ⊂ M. Take a triangulation K of Y × [−1, 1]n so that st(pn h(X × Q × {0}), K) ⊂ U (⊂ pn (M)),

312

4 Triangulation of Hilbert Cube Manifolds and Related Topics

and let L be the smallest subcomplex of K containing pn h(X × Q × {0}), that is,    L = σ ∈ K  ∃τ ∈ K[pn h(X × Q × {0})] such that σ  τ . Then, |L| ⊂ U and L is finite because pn h(X × Q × {0}) is compact. By Proposition 1.6.8, we can assume that L is a full subcomplex of K1 . Choose 0 < ε < 1 so that |Nε (L, K)| ⊂ U , where Nε (L, K) is an ε-neighborhood of L in K (cf. Sect. 1.9, p. 47). According to Lemma 1.9.3, N = |Nε (L, K)| is a regular neighborhood of |L| in |K|, which is contained in U . Due to Theorem 1.9.6, bd N is bi-collared in Y × [−1, 1]n. Note that bd N ⊂ pn k(X × Q × (−∞, 0)) ∪ pn k(X × Q × (0, ∞)). We have the following compact PL bi-collared subpolyhedron of Y × [−1, 1]n (see Fig. 4.4): A = bd N ∩ pn k(X × Q × (0, ∞)) = h(h−1 (pn−1 (bd N)) ∩ X × Q × (0, ∞)). Then, we have a splitting M = M1 ∪ M2 , where M1 = h(X × Q × (0, ∞)) \ pn−1 (int N) and M2 = h(X × Q × (−∞, 0]) ∪ pn−1 (N). Indeed, M1 and M2 are closed in M and non-compact Q-manifolds. Observe that pn (M1 ∩ M2 ) = A and pn−1 (A) = M1 ∩ M2 . " ! Y ×Q h(X × Q × {0}) R

h(X × Q × R + ) M2

M1

h −1 (p n−1 (A)) Q

h

X × Q × {0}

h −1 (p n−1 (|L |)) [−1, 1] n L Fig. 4.4 Construction of a splitting

A

Y

4.4 The Splitting Theorem for Q-Manifolds

313

Lemma 4.4.3 For any splitting M = M1 ∪ M2 , each of M1 and M2 is homotopy dominated by a finite simplicial complex. Proof Since M0 = M1 ∩ M2 is compact, we can find a > 0 such that M0 ⊂ h(X × Q × (−a, a)). Because of connectedness, each of h(X × Q × [a, ∞)) and h(X × Q × (−∞, −a]) is contained in one of M1 \ M0 or M2 \ M0 . Moreover, both of them are not contained in the same side. Otherwise, either one of M1 and M2 would be contained in h(X × Q × [−a, a]), which contradicts the non-compactness of M1 and M2 . We have a strong deformation retraction r : M = h(X × Q × R) → h(X × Q × [−a, a]) such that, for every (x, y) ∈ X × Q, r(h({(x, y)} × [a, ∞)) = {h(x, y, a)} and r(h({(x, y)} × (−∞, −a]) = {h(x, y, −a)}. Then, the restrictions r|Mi : Mi → Mi ∩ h(X × Q × [−a, a]), i = 1, 2, are strong deformation retractions. Thus, Mi is homotopy equivalent to Mi ∩ h(X × Q × [−a, a]), which is a compact ANR. Since each compact ANR is homotopy dominated by a finite simplicial complex (Proposition 4.3.3), we have the result. " ! Let K and L be locally finite simplicial complexes, where K (0) is given an order so that K is an ordered simplicial complex. For a proper (equiv. perfect) simplicial map f : K → L (i.e., f −1 (v) is compact for every v ∈ L(0) ), the simplicial mapping cylinder Zf is also locally finite (cf. Sect. 1.10). We use the simplicial mapping cylinder in the proof of the following proposition: Lemma 4.4.4 Let M = M1 ∪ M2 be a splitting. Suppose that M1 is homotopy dominated by a finite simplicial complex K with a homotopy domination f : |K| → M1 . Then, there exists a splitting M = N1 ∪ N2 with the following properties: (i) N1 ⊂ int M1 (equiv. int N2 ⊃ M2 ), (ii) N0 = N1 ∩ N2 is strong deformation retract of M1 ∩ N2 = N2 \ int M2 , (iii) there exists a homotopy equivalence α : |K| → N0 such that α  f in M1 . Proof Since M = M1 ∪ M2 is a splitting, pn−1 (pn (M1 ∩ M2 )) = M1 ∩ M2 for some n ∈ N and A = pn (M1 ∩ M2 ) is a PL bi-collared compact subpolyhedron of Y × [−1, 1]n . Since f : |K| → M1 is a homotopy domination, there is a map g : M1 → |K| such that fg  idM1 . Note that in (A) ⊂ M1 . Applying the Simplicial Approximation Theorem 1.8.8, we have a triangulation L of A and a simplicial map ψ : L → K that is homotopic to gin . By giving an order on L(0) such that L is considered an ordered simplicial complex, we can define the simplicial mapping cylinder Zψ of ψ with the simplicial collapsing c¯ψ : Zψ → K (cf. Sect. 1.10).

314

4 Triangulation of Hilbert Cube Manifolds and Related Topics M p n−11 (U ) M2

Y ×Q M1

[−1, 1] n 1 − n Y × [− 1, 1] n U 2

A

|K |

U1

f (|K |)

Fig. 4.5 N and bd N

Then, A = |L| ⊂ |Zψ | and f c¯ψ |A = f ψ  fgin |A  in |A. We can apply the Homotopy Extension Theorem 1.5.7 or 1.13.11 to obtain a map f  : |Zψ | → M1 such that f  |A = in and f   f c¯ψ . It follows that f  g  f c¯ψ g = f (c¯ψ ||K|)g = fg  idM1 , which means that f  is a homotopy domination. Replacing K and f by Zψ and f  , we may assume that A is a subpolyhedron of |K| and f |A = in |A. Since f (|K|) ∪ pn−1 (A) is a compact subset of the open set M in Y × Q, we can find n1  n and an open set U in Y × [−1, 1]n1 such that (U ) ⊂ M. f (|K|) ∪ pn−1 (A) ⊂ pn−1 1 Note that U ⊂ pn1 (M). Let Ui = U ∩ pn1 (Mi ), i = 1, 2. Refer to Fig. 4.5. Then, U = U1 ∪ U2 , pn−1 (Ui ) ⊂ Mi , i = 1, 2, and 1 U1 ∩ U2 = A × [−1, 1]n1 −n ⊂ Y × [−1, 1]n1 . Moreover, A×[−1, 1]n1−n is PL bi-collared in Y ×[−1, 1]n1 , so in U , which implies that A × [−1, 1]n1 −n is PL collared in both U1 and U2 . Let m = 2 dim K + 1 and n2 = n1 + m. Note that pn2 f |A = pn2 in |A is a PL embedding of A into U1 × [−1, 1]m (⊂ Y × [−1, 1]n2 ). Applying the PL Embedding Approximation Theorem 1.8.11, we can obtain a PL embedding ϕ : |K| → U1 × [−1, 1]m (⊂ Y × [−1, 1]n2 ) such that ϕ|A = pn2 f |A and ϕ  pn2 f rel. A in U1 × [−1, 1]m . Since A × [−1, 1]n2−n is PL collared in U1 × [−1, 1]m , we can push ϕ(|K| \ A) off A × [−1, 1]n2 −n so as to satisfy ϕ(|K|) ∩ (A × [−1, 1]n2 −n ) = ϕ(A) = A × {0}, where 0 ∈ [−1, 1]n2 −n . Consider the following subpolyhedron of Y × [−1, 1]n2 : P = (A × [−1, 1]n2−n+1 ) ∪ (ϕ(|K|) × {1}).

4.4 The Splitting Theorem for Q-Manifolds (|K |) × {1}

315 N

Y × [− 1, 1] n 2 +1

1 bd N p n 2 (M 1 )

p n 2 (M 2 ) −1 A × [− 1, 1] n 2 − n +1

A

[−1, 1] n 2 − n

Y × [− 1, 1] n

Fig. 4.6 P and bd N

Then, ϕ(|K|)×{1} is a strong deformation retract of P , hence the inclusion ϕ(|K|)× {1} ⊂ P is a homotopy equivalence. Let N be a regular neighborhood of P in U1 × [−1, 1]m+1. Then, P is a strong deformation retract of N, hence the inclusion P ⊂ N is a homotopy equivalence. Moreover, pn−1 (N) ∩ M2 = M1 ∩ M2 = 2 +1 pn−1 (A). Refer to Fig. 4.6. Now, the desired splitting M = N1 ∪ N2 can be defined as follows: (N) and N1 = M \ int N2 . N2 = M2 ∪ pn−1 2 +1 Then, N1 ⊂ M \ M2 = int M1 , that is, (i) holds. Observe that N1 ∩ N2 = pn−1 (bd N) and N2 \ int M2 = pn−1 (N). Hence, to see (ii), we have to show 2 +1 2 +1 that bd N is a strong deformation retract of N. Since N is a regular neighborhood of P , bd N is a strong deformation retract of N \ P (Theorem 1.9.6). Since P is a Z-set in N, N \ P is homotopy dense in N. Then, it is easy to see that bd N is a strong deformation retract of N. Thus, we have (ii). For (iii), recall that N1 ∩N2 = pn−1 (bd N). Let α : |K| → pn−1 (bd N) be the following composition 2 +1 2 +1 of homotopy equivalences: ϕ

in2 +1

|K| −→ ϕ(|K|) = ϕ(|K|) × {1} ⊂ P ⊂ N −→ bd N −→ pn−1 (bd N), 2 +1 r

where r : N → bd N is a strong deformation retraction.

" !

Proposition 4.4.5 There exists a splitting M = M1 ∪ M2 such that M0 = M1 ∩ M2 , M1 , and M2 are connected and the inclusions M0 ⊂ Mi , i = 1, 2, induce isomorphisms between their fundamental groups. Proof Let M = M1 ∪ M2 be any splitting (the existence is guaranteed by Proposition 4.4.2). Then, M1 is homotopy dominated by a finite simplicial complex by Proposition 4.4.3. By Theorem 4.3.10, we have a finite simplicial complex K and a homotopy domination f : |K| → M1 that induces a bijection between their components and isomorphisms between the fundamental groups of their

316

4 Triangulation of Hilbert Cube Manifolds and Related Topics

corresponding components. Applying Lemma 4.4.4, we can obtain a splitting M = N1 ∪ N2 with the following properties: (1) N1 ⊂ int M1 , (2) N0 = N1 ∩ N2 is a strong deformation retract of M1 ∩ N2 = N2 \ int M2 , (3) there exists a homotopy equivalence α : |K| → N0 such that α  f in M1 . It follows from (2) that N1 is a strong deformation retract of M1 , that is, there is a strong deformation retraction r : M1 → N1 . By (3), α has a homotopy inverse β : N0 → |K|. See the following diagram: N0 β

⊂

α

|K|

N1 ∩ r

f

M1

Observe that rfβ  rαβ  r|N0 = id in N1 , that is, rfβ is homotopic to the inclusion N0 ⊂ N1 . Since r and β are homotopy equivalences and f induces a bijection between the components and isomorphisms between the fundamental groups of these corresponding components, the inclusion N0 ⊂ N1 induces a bijection between the components and isomorphisms between the fundamental groups of these corresponding components. Using the same procedure as above, we can obtain a finite simplicial complex L with a homotopy domination g : |L| → N2 that induces a bijection between their components and isomorphisms between the fundamental groups of their corresponding components, and a splitting M = M1 ∪ M2 with the following properties: (4) M2 ⊂ int N2 , i.e., int M1 ⊃ N1 , (5) M0 = M1 ∩ M2 is a strong deformation retract of N2 ∩ M1 = M1 \ int N1 , (6) there exists a homotopy equivalence β : |L| → M0 such that β  g in N2 . Then, by the same argument as above, the inclusion M0 ⊂ M2 induces a bijection between the components and isomorphisms between the fundamental groups of these corresponding components. We show that the inclusion M0 ⊂ M1 induces a bijection between the components, that is, if C be a component of M1 , then C ∩ M0 is a component of M0 . Since the inclusion M0 ⊂ M1 \ int N1 is a homotopy equivalence by (5) and C ∩ M0 = (C \ int N1 ) ∩ M0 , it suffices to show that C \ int N1 is a component of M1 \ int N1 . If C ⊂ M1 \ N1 , then C = C \ int N1 is a component of M1 \ int N1 . When C ∩ N1 = ∅, each component of C ∩ N1 is a component of N1 , which meets N0 because the inclusion N0 ⊂ N1 induces a surjection between the components. Therefore, C \ int N1 = C ∩ (M1 \ int N1 ) ⊃ C ∩ N0 = ∅.

4.4 The Splitting Theorem for Q-Manifolds

317

Assume that C \ int N1 is disconnected. Since C is a path-component of M1 because M1 is locally path-connected, we have a path α : I → C such that α(0) and α(1) belong to different path-components of C \ int N1 . Then, we can assume that α(I) ⊂ C ∩N1 . Indeed, let C0 be a path-component of C \int N1 with α(0) ∈ C0 and replace I with [t0 , t1 ], where       t0 = sup t ∈ I  α(t) ∈ C0 and t1 = inf t > t0  α(t) ∈ C \ int N1 . Since the inclusion N0 ⊂ N1 induces an injection between the components (= the path-components), we have a path β : I → N0 such that β(0) = α(0) and β(1) = α(1), where β(I) ⊂ C because C is a path-component of M1 . So, α(0) and α(1) are connected by a path in C ∩ N0 ⊂ C \ int N1 , hence α(1) ∈ C0 which is a contradiction. Consequently, C \ int N1 is connected. Then, it follows that C \ int N1 is a component of M1 \ int N1 . Now, we show that M0 , M1 , and M2 are connected. Since the inclusions M0 ⊂ Mi , i = 1, 2, induce bijections between the components, it suffices to show that M0 is path-connected. Let x, y ∈ M0 . Since M = h(X × Q × R), there is a path γ : I → M such that γ (0) = x and γ (1) = y. Because M0 is bi-collared in M, we have a neighborhood U of M0 in M and a strong deformation retraction r : U → M0 (i.e., r is a retraction such that r  id rel. M0 in U ). Then, we have 0 = t0 < t1 < · · · < tn = 1 such that if γ ([ti−1 , ti ]) ⊂ U , then γ ([ti−1 , ti ]) ⊂ M1 or γ ([ti−1 , ti ]) ⊂ M2 . When γ ([ti−1 , ti ]) ⊂ U , since the inclusions M0 ⊂ M1 and M0 ⊂ M2 induce injections between the components, then there exists a path γi : [ti−1 , ti ] → M0 with γi (ti−1 ) = γ (ti−1 ) and γi (ti ) = γ (ti ). We can define a path γ  : I → M0 as follows: 

γ |[ti−1 , ti ] =

 rγ |[ti−1 , ti ] γi

if γ ([ti−1 , ti ]) ⊂ U, otherwise.

Then, γ  (0) = γ (0) = x and γ  (1) = γ (1) = y. Thus, M0 is path-connected, so connected. It remains to prove that the inclusion M0 ⊂ M1 induces an isomorphism between the fundamental groups. Note that the inclusion M0 ⊂ M1 is a composition of two inclusions M0 ⊂ M1 \ int N1 ⊂ M1 , where the first inclusion is homotopy equivalence by (5). Thus, it suffices to show that the inclusion M1 \ int N1 ⊂ M1 induces an isomorphism between the fundamental groups. Observe that N1 ∩ (M1 \ int N1 ) = N1 ∩ N2 = N0 and N1 ∪ (M1 \ int N1 ) = M1 . Recall that the inclusion N0 ⊂ N1 induces a bijection between the components and isomorphisms between the fundamental groups of these corresponding components. By connecting all components of N0 by a tree in M1 \ int N1 , we can reduce N0 and N1 to connected, where the above equalities are kept. Then, their fundamental groups are the free products of the fundamental groups of their components and

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4 Triangulation of Hilbert Cube Manifolds and Related Topics

the new inclusion induces an isomorphism between their fundamental groups. Thus, it can be assumed that N0 and N1 are connected. Applying the Seifert–van Kampen Theorem 4.2.6, we can obtain a unique homomorphism φ : π1 (M1 ) → π1 (M1 \ int N1 ) such that φj2 = id and φj1 = i2 i1−1 , which implies that j2 is a monomorphism. See the following commutative diagram: i1 ∼ =

π1 (N0 )

π1 (N1 )

i2 i1−1

j1

i2

π1 (M1 \ int N1 )

j2

π1 (M1 )

φ

π1 (M1 \ int N1 )

= id

where i1 , i2 , j1 , and j2 are homomorphisms induced by the inclusions. Moreover, π1 (M1 ) is generated by j1 (π1 (N1 )) ∪ j2 (π1 (M1 \ int N1 )). Since i1 is an isomorphism, it follows that j1 (π1 (N1 )) = j1 i1 (π1 (N0 )) = j2 i2 (π1 (N0 )) ⊂ j2 (π1 (M1 \ int N1 )). Hence, π1 (M1 ) = j2 (π1 (M1 \ int N1 )), which means that j2 is an epimorphism. Thus, j2 is an isomorphism. " ! Now, using Theorem 4.1.6, we can prove the Splitting Theorem 4.4.1. Proof (Splitting Theorem 4.4.1) By Proposition 4.4.5, we have a splitting M = M1 ∪ M2 such that M0 = M1 ∩ M2 , M1 , and M2 are connected and the inclusions M0 ⊂ Mi , i = 1, 2, induce isomorphisms between their fundamental groups. Then, we apply the Seifert–van Kampen Theorem 4.2.6 to show that the inclusions Mi ⊂ M, i = 1, 2, induce isomorphisms between their fundamental groups. Indeed, using the same argument as in the final paragraph of the proof of Proposition 4.4.5, since i1 (resp. i2 ) is an isomorphism, it follows that j1 (resp. j2 ) is also an isomorphism. See the following commutative diagram: π1 (M0 )

i1 ∼ =

i2 ∼ =

π1 (M2 )

π1 (M1 ) j1

j2

π1 (M)

∼ π1 (M) = ∼ π1 (X) (M ≈ X × Q × R), it follows that π1 (M  ) Since π1 (M1 ) = 1 is also free or free Abelian. Due to Lemma 4.4.3, M1 is homotopy dominated by a finite simplicial complex. We can apply Theorem 4.1.6 to obtain a finite complex K and a homotopy equivalence f : |K| → M1 . Applying Lemma 4.4.4, we can obtain a splitting M = N1 ∪ N2 with the following properties:

4.5 An Immersion of a Punctured n-Torus into Rn

319

(1) N1 ⊂ int M1 , (2) N0 = N1 ∩ N2 is a strong deformation retract of M1 ∩ N2 = N2 \ int M2 , (3) there exists a homotopy equivalence α : |K| → N0 such that α  f in M1 . By the same the same argument as in the proof of Proposition 4.4.5, we can see that the inclusion N0 ⊂ N1 is homotopic to rfβ, where β : N0 → |K| is a homotopy inverse of α and r : M1 → N1 is a strong deformation retraction. Since f is a homotopy equivalence, the inclusion N0 ⊂ N1 is a homotopy equivalence, which means that N0 is a strong deformation retract of N1 (Proposition 1.13.15). This implies that N2 is a strong deformation retract of M, that is, the inclusion N2 ⊂ M is a homotopy equivalence. Therefore, N2  X, so N2 has the homotopy type of a finite simplicial complex. Again using the above procedure, we can obtain a new splitting, M = M1 ∪ M2 , such that (4) M2 ⊂ int N2 , i.e., int M1 ⊃ N1 , (5) M0 = M1 ∩ M2 ⊂ N2 ∩ M1 = M1 \ int N1 is a homotopy equivalence, (6) M0 ⊂ M2 is a homotopy equivalence, equivalently M0 is a strong deformation retract of M2 . As observed above, N0 is a strong deformation retract of N1 , from which it follows that M1 \ int N1 is a strong deformation retract of M1 . Thus, M1 \ int N1 ⊂ M1 is a homotopy equivalence. Combining this with (5), M0 ⊂ M1 is a homotopy equivalence, which means that M0 is a strong deformation retract of M1 (Proposition 1.13.15). It follows from this with (6) that M0 is a strong deformation retract of M, that is, the inclusion M0 ⊂ M is a homotopy equivalence. " !

4.5 An Immersion of a Punctured n-Torus into Rn A map α : X → Y is called an immersion if α is locally an open embedding, that is, each x ∈ X has an open neighborhood U in X such that α|U is an open embedding. The n-torus (n-dimensional torus)17 Tn is defined as the n-hold product of the unit sphere S1 , that is, Tn = (S1 )n = (S1 × ·)* · · × S+1 . n many

Let e : R → S1 be the covering projection defined by e(t) = (cos πt, sin πt) for every t ∈ R. When every (x, y) ∈ R2 is identified with x + iy ∈ C (i.e., we regard R2 = C), we can describe as follows: e(t) = exp(πit) = cos πt + i sin πt for every t ∈ R.

17 The

plural is tori.

320

4 Triangulation of Hilbert Cube Manifolds and Related Topics

Note that e([−1, 1]) = S1 and e|(−1, 1) is injective. The covering projection en : Rn → Tn is defined by en (x) = (e(x(1)), . . . , e(x(n))), where en ([−1, 1]n ) = Tn and en |(−1, 1)n is injective. We will use the punctured torus Tn0 = Tn \ {p0 }, where p0 = en (1, . . . , 1) ∈ en ([−1/4, 1/4]n). Immersions of Tn0 play a key role in our proof of the Handle Straightening Theorem in the next section. In this section, we will prove the following: Theorem 4.5.1 There exists an immersion α : Tn0 → Rn such that αen |[−1/4, 1/4]n = id. For the two-dimensional punctured torus T20 , such an immersion can be constructed as indicated in Fig. 4.7. But, it is hard to extend such a construction to the n-dimensional punctured tori Tn0 , n > 2. To prove Theorem 4.5.1, we start at the following lemma, which can be easily proved: S

[−1, 1]2 Sm

Sm

[−1/ 2, 1/ 2]2

S e2 Sm

T20

≈

p0

S The punctured torus

[−1/ 2, 1/ 2]2

Fig. 4.7 An immersion α : T20 → R2

4.5 An Immersion of a Punctured n-Torus into Rn

321

Lemma 4.5.2 Let f : Tn × [−1, 1] → Rn+1 be an embedding such that 0 < prn+1 f (x, t) < c for every (x, t) ∈ Tn × [−1, 1]. Then, the map f ∗ : Tn+1 × [−1, 1] → Rn+2 defined in the following formula is an embedding such that prn+2 f ∗ (x, t) > 0 for every (x, t) ∈ Tn+1 × [−1, 1]: f ∗ (x, e(s), t) = (pr1 f (x, t), . . . , prn f (x, t), prn+1 f (x, t) cos πs, prn+1 f (x, t) sin πs + c), where (x, e(s), t) ∈ Tn × S1 × [−1, 1] = Tn+1 × [−1, 1] and s ∈ [−1, 1]. Proof Since Tn+1 × [−1, 1] is compact, it suffices to show that f ∗ is a continuous injection. The continuity of f ∗ is obvious from the definition. To see that f ∗ is injective, assume that f ∗ (x, e(s), t) = f ∗ (x  , e(s  ), t  ), where (x, e(s), t), (x  , e(s  ), t  ) ∈ Tn × S1 × [−1, 1] and s, s  ∈ [−1, 1]. Then, pri f (x, t) = pri f (x  , t  ) for each i = 1, . . . , n and prn+1 f (x, t) = =

, ,

(prn+1 f (x, t) cos πs)2 + (prn+1 f (x, t) sin πs)2 (prn+1 f (x  , t  ) cos πs  )2 + (prn+1 f (x  , t  ) sin πs  )2

= prn+1 f (x  , t  ), hence f (x, t) = f (x  , t  ). Because f is injective, we have (x, t) = (x  , t  ). Since prn+1 f (x, t) = prn+1 f (x  , t  ), it follows that e(s) = (cos πs, sin πs) = (cos πs  , sin πs  ) = e(s  ). Therefore, (x, e(s), t) = (x  , e(s  ), t  ). Thus, f ∗ is an injection. Moreover, prn+2 f ∗ (x, e(s), t) = prn+1 f (x, t) sin πs + c  c − prn+1 f (x, t) > 0. Hence, prn+2 f ∗ (x, t) > 0 for every (x, t) ∈ Tn+1 × [−1, 1].

" !

We will define immersions hn : Rn × [−1, 1] → Rn+1 , n ∈ N, which induce embeddings h˜ n : Tn × [−1, 1] → Rn+1 such that hn = h˜ n (en × id). First, we inductively define maps fn , gn : Rn × [−1, 1] → R, n ∈ N, as follows: f1 (x1 , t) = (2 + t) cos πx1 , g1 (x1 , t) = (2 + t) sin πx1 + 22 , fn (x1 , . . . , xn , t) = gn−1 (x1 , . . . , xn−1 , t) cos πxn , and gn (x1 , . . . , xn , t) = gn−1 (x1 , . . . , xn−1 , t) sin πxn + 2n+1 .

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4 Triangulation of Hilbert Cube Manifolds and Related Topics

Let hn : Rn × [−1, 1] → Rn+1 , n ∈ N, be maps defined as follows:  hn (x1 , . . . , xn , t) = f1 (x1 , t), f2 (x1 , x2 , t),

 . . . , fn (x1 , . . . , xn , t), gn (x1 , . . . , xn , t) .

In other words, hn can be described as follows: h1 (x1 , t) = ((2 + t) cos πx1 , (2 + t) sin πx2 + 4); h2 (x1 , x2 , t) = ((2 + t) cos πx1 , ((2 + t) sin πx1 + 4) cos πx2 , ((2 + t) sin πx1 + 4) sin πx2 + 8); h3 (x1 , x2 , x3 , t) = ((2 + t) cos πx1 , ((2 + t) sin πx1 + 4) cos πx2 , (((2 + t) sin πx1 + 4) sin πx2 + 8) cos πx3 , (((2 + t) sin πx1 + 4) sin πx2 + 8) sin πx3 + 16); ··· .

Then, as is easily observed, hn induces the map h˜ n : Tn × [−1, 1] → Rn+1 such that hn = h˜ n (en × id), that is, hn (x1 , . . . , xn , t) = h˜ n (en (x1 , . . . , xn ), t) = h˜ n (e(x1 ), . . . , e(xn ), t). By using Lemma 4.5.2, we can easily show that h˜ n is an embedding, which is called the standard embedding of Tn × [−1, 1] into Rn+1 (Fig. 4.8). Hence, hn is an immersion. We have also the following: g1 (x1 , t) = (2 + t) sin πx1 + 22 ; g2 (x1 , x2 , t) = (2 + t) sin πx1 sin πx2 + 22 sin πx2 + 23 ; g3 (x1 , x2 , x3 , t) = (2 + t) sin πx1 sin πx2 sin πx3 + 22 sin πx2 sin πx3 + 23 sin πx3 + 24 ; ··· . We are ready to start proving Theorem 4.5.1. Proof (Theorem 4.5.1) Recall p0 = en (1, . . . , 1) (= en (±1, . . . , ±1)) and let  S = x ∈ Rn  Sε = x ∈ Rn

   x(i) = 0 for some i = 1, . . . , n and    |x(i)| < ε for some i = 1, . . . , n , 0 < ε < 1.

4.5 An Immersion of a Punctured n-Torus into Rn

323

T1 × [− 1, 1] R2

1 7 h˜ 1 5

S1 = T1

4

−1

3 2 1 0

15 T2 × [− 1, 1]

R3

h˜ 2

8 7

1 0 Fig. 4.8 The standard embedding h˜ n : Tn × [−1, 1] → Rn+1

Then, Tn0 = Tn \ {p0 } ≈ en (Sε ). Indeed,    Tn \ en (Sε ) = en (x) ∈ Tn  |x(i)|  ε for each i = 1, . . . , n   = en [1 − ε, 1 + ε]n . Let ϕ : Rn → R be the map defined by ϕ(x) = 2−n sin πx1 · · · sin πxn + 2−n+1 sin πx2 · · · sin πxn + · · · + 2−2 sin πxn−1 sin πxn + 2−1 sin πxn = 2−(n+1) gn (x, 0) − 1,

324

4 Triangulation of Hilbert Cube Manifolds and Related Topics

where x = (x1 , . . . , xn ) ∈ Rn . Using maps f1 , . . . , fn defined above and ϕ, we define fϕ : Rn → Rn as follows:   fϕ (x) = f1 (x1 , ϕ(x)), f2 (x1 , x2 , ϕ(x)), . . . , fn (x1 , . . . , xn , ϕ(x)) for each x = (x1 , . . . , xn ) ∈ Rn . In other words, fϕ (x) = phn (x), where p : Rn+1 = Rn ×R → Rn is the projection and hn : Rn × [−1, 1] → Rn+1 is the immersion defined above. Now, using elementary properties of determinations, we can compute the Jacobian determinant of h as follows:   ∂fi ∂fi ∂ϕ det Jfϕ = det + ∂xj ∂t ∂xj ⎛ ⎞ ⎞ ⎛ ∂fi ∂fi ∂fi ∂ϕ ∂fi + 0 − ⎜ ⎟ ⎜ ∂xj ∂t ∂xj ∂t ⎟ ⎟ = det ⎜ ∂xj ⎟ = det ⎜ ⎝ ⎠ ⎠ ⎝ ∂ϕ ∂ϕ 1 1 ∂xj ∂xj ⎛ ⎞ ⎛ ⎞ ∂fi ∂fi ∂fi ∂fi ⎜ ∂xj − ∂t ⎟ − ⎟ + det ⎝ ∂xj ∂t ⎠ . = det ⎜ ⎝ ∂ϕ ⎠ 0 0 1 ∂xj Observe that ∂fi /∂xj = 0 if i < j , and ∂fi /∂xi has a factor of sin πxi . In the last part of the above equality, on S, the upper left-hand corner of the second matrix is triangular with at least one zero on the diagonal, hence the determinant of this matrix is zero. Since ϕ(x) = 2−(n+1) gn (x, 0), it follows that ∂ϕ/∂xj = 2−(n+1) ∂gn /∂xj at x ∈ S and t = 0. Moreover, ∂gn /∂t = sin πx1 · · · sin πxn = 0 on S. Hence, at x ∈ S and t = 0, ⎞ ⎞ ⎛ ⎛ ∂fi ∂fi ∂fi ∂fi − ⎟ ⎜ ∂xj ⎜ ∂t ⎟ ⎟ = −2−(n+1) det ⎜ ∂xj ∂t ⎟ = − det Jhn . det Jfϕ = det ⎜ ⎠ ⎝ ∂ϕ ⎝ ∂gn ∂gn ⎠ 2n+1 0 ∂xj ∂xj ∂t Since hn is an immersion, it follows that det Jfϕ = −2−(n+1) det Jhn = 0 at x ∈ S and t = 0. Then, each point x ∈ S has a neighborhood Ux in Rn such that fϕ |Ux is an embedding. Since S is compact, we can choose small ε > 0 so that fϕ |Sε is an immersion. Recall that T0 ≈ Sε . Hence, we have an immersion α  : T0 → Rn . Finally, we modify this immersion α  to an immersion α : Tn0 → Rn so that n αe |[−1/4, 1/4]n = id. Choose a sufficiently small δ > 0 so that α  en |[−δ, δ]n : [−δ, δ]n → Rn is an embedding. We have a homeomorphism β : [−1, 1]n → [−1, 1]n such that β|∂[−1, 1]n = id and β([−1/4, 1/4]n) = [−δ/2, δ/2]n. This β ˜ n |[−1, 1]n = en β, where induces a homeomorphism β˜ : Tn → Tn such that βe

4.6 The Handle Straightening Theorem

325

˜ n ) = Tn because β(T 0 0 ˜ n (1, . . . , 1) = en β(1, . . . , 1) = en (1, . . . , 1) = p0 . ˜ 0 ) = βe β(p ˜ n : Tn → Rn is an immersion and the following is an embedding: Then, α  β|T 0 0 ˜ n |[−1/4, 1/4]n = α  en β|[−1/4, 1/4]n : [−1/4, 1/4]n → Rn . α  βe Since α  en (∂[−δ/2, δ/2]n) is bi-collared in Rn , the generalized Schoenflies Theorem 2.7.7 provides a homeomorphism ψ : Rn → Rn such that ψ([−1/4, 1/4]n) = ˜ n (∂[−1/4, 1/4]n) = ∂[−1/4, 1/4]n, the homeα  en ([−δ/2, δ/2]n). Since ψ −1 α  βe omorphism ˜ n |[−1/4, 1/4]n : [−1/4, 1/4]n → [−1/4, 1/4]n ψ −1 α  βe can be extended to a homeomorphism ϕ : Rn → Rn by   ˜ n (4x∞ )−1 x for x ∈ Rn \ [−1/4, 1/4]n, ϕ(x) = 4x∞ ψ −1 α  βe ˜ n : Tn → Rn is the where x∞ = max{|x(1)|, · · · , |x(n)|}. Then, ϕ −1 ψ −1 α  β|T 0 0 desired immersion. ! "

4.6 The Handle Straightening Theorem A compact subpolyhedron Y of a polyhedron X is said to be straight in X provided that bd Y is PL collared in both Y and X \ int Y , that is, there exists a PL bi-collar k : bd Y × [−1, 1] → X such that k(bd Y × [−1, 0]) ⊂ Y and k(bd Y × [0, 1]) ⊂ X \ int Y . We use the same notational convention as Sect. 4.4, that is, for each n ∈ N, we write Q = [−1, 1]n × Qn and pn : X × Q → X × [−1, 1]n denotes the projection defined as follows: pn (x, y) = (x, y(1), . . . , y(n)) for each (x, y) ∈ X × Q. Lemma 4.6.1 Let X be a polyhedron. Given a compactum A in X × Q and an open set U in X × Q with A ⊂ U , there exist n ∈ N and a straight compact subpolyhedron Y of X × [−1, 1]n such that A ⊂ int pn−1 (Y ) ⊂ pn−1 (Y ) ⊂ U . Proof For each x ∈ A, we can take n(x) ∈ N and an open neighborhood Vx of −1 pn(x) (x) in X × [−1, 1]n(x) so that pn(x) (Vx ) ⊂ U . Since A is compact, there are k −1 x1 , . . . , xk ∈ A such that A ⊂ i=1 pn(xi ) (Vxi ). Let n = max{n(x1 ), . . . , n(xk )}.

326

4 Triangulation of Hilbert Cube Manifolds and Related Topics

Then we have the following open set in X × [−1, 1]n : V =

k 

Vxi × [−1, 1]n−n(xi ) ,

i=1

where A ⊂ pn−1 (V ) ⊂ U . Take a triangulation K of X × [−1, 1]n so that   K ≺ V , (X × [−1, 1]n ) \ pn (A) . Let L be the subcomplex of K with |L| = |K[pn (A)]|. Then, we have |L| ⊂ V . Choose 0 < ε < 1 so that |Nε (L, K)| ⊂ V , where Nε (L, K) is the ε-neighborhood of L in K.18 Then, Y = |Nε (L, K)| is the desired one. In fact, since Y is a regular neighborhood of |L| in |K| = X × [−1, 1]n by Lemma 1.9.4, it follows from Theorem 1.9.6 that bd Y is PL collared in both Y and (X × [−1, 1]n ) \ int Y . " ! The following Handle Straightening Theorem is the main theorem of this section, which we will apply to prove the topological invariance of Whitehead torsion mentioned in Sect. 4.1. Theorem 4.6.2 (HANDLE STRAIGHTENING) Let X be a polyhedron and n ∈ N. Given an open embedding h : Rn × Q → X × Q, there exists a homeomorphism g : X × Q → X × Q and a straight subpolyhedron Y of X × [−1, 1]m for some m ∈ N such that g|(X × Q) \ h([−1/4, 1/4]n × Q) = id and −1 gh([−1/8, 1/8]n × Q) = pm (Y ) = Y × Qm .

We first treat only the case n = 1 that is simpler than the case n  2. Proof (The Case n = 1 of Theorem 4.6.2) Let us consider the restriction of the open embedding h : R × Q → X × Q to the following open subspace:       (−1/4, 1/4) \ [−1/8, 1/8] × Q = (−1/4, −1/8) × Q ∪ (1/8, 1/4) × Q     ≈ {−1/4} × Q × R ∪ {1/4} × Q × R . Applying the Splitting Theorem 4.4.1 to each component, we can obtain splittings h((−1/4, −1/8) × Q) = M−1 ∪ M−2 and h((1/8, 1/4) × Q) = M1 ∪ M2 such that M±1 ∩ M±2 are strong deformation retracts of M±1 ∪ M±2 (so they are contractible) and M±1 ∩ M±2 = A± × Qm for some compact contractible PL bi-

18 For

the ε-neighborhood Nε (L, K), see p. 47.

4.6 The Handle Straightening Theorem

327

A− × Q m

h({−1/ 4} × Q)

M −2

A+ × Q m

Y × Qm

M −1

h([−1/ 8, 1/ 8] × Q)

M1

M2

h({1/ 4} × Q)

h([−1/ 4, 1/ 4] × Q) Fig. 4.9 The polyhedron Y ⊂ X × [−1, 1]m when n = 1

collared subpolyhedra A± ⊂ X × [−1, 1]m , where we can assume that cl M±1 ⊃ h({±1/8} × Q)



equiv. cl M±2 ⊃ h({±1/4} × Q) .

See Fig. 4.9. Then, it follows that (A− ∪ A+ ) × Qm = bd(h([−1/8, 1/8] × Q) ∪ M−1 ∪ M1 ). Thus, we have a straight subpolyhedron Y = pm (h([−1/8, 1/8] × Q) ∪ M−1 ∪ M1 ) ⊂ X × [−1, 1]m , where bd Y = A− ∪ A+ and h([−1/8, 1/8] × Q) ∪ M−1 ∪ M1 = Y × Qm ⊂ h([−1/4, 1/4] × Q). To construct a homeomorphism g : X × Q → X × Q, observe that h([−1/4, 1/4] × Q) \ (int Y × Qm ) = M−2 ∪ M2 and bd M±2 = (A± × Qm ) ∪ h({±1/4} × Q). Refer to Fig. 4.9. Since M±2 are compact contractible Q-manifolds, we have M±2 ≈ Q by Theorem 3.7.9. Using the Z-Set Unknotting Theorem 2.11.6, we can easily construct homeomorphisms g− : h([−1/4, −1/8] × Q) → M−2 and g+ : h([1/8, 1/4] × Q) → M2 so that g± |h({±1/4} × Q) = id and g± (h({±1/8} × Q) = A± × Qm . Since Y ×Qm ≈ Q, we use the Z-Set Unknotting Theorem 2.11.6 again to construct g0 : h([−1/8, 1/8] × Q) → Y × Qm so that g0 |h({±1/8} × Q) = g± |h({±1/8} × Q).

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4 Triangulation of Hilbert Cube Manifolds and Related Topics

Then, pasting g0 , g± , and id, we can obtain the desired homeomorphism g : X × Q → X × Q such that g|(X × Q) \ h([−1/4, 1/4] × Q) = id and −1 gh([−1/8, 1/8] × Q) = pm (Y ) = Y × Qm .

The proof of the case n = 1 is completed.

" !

→ constructed For the case n  2, we will use the immersion α : in the previous section and Theorem 4.1.7. To apply Theorem 4.1.7, we need the following: Tn0

Rn

Lemma 4.6.3 Let X be a polyhedron and Y a subpolyhedron of X × [−1, 1]k , and let α : M → X×Q be an immersion of a Q-manifold M such that α|α −1 (Y ×Qk ) : α −1 (Y × Qk ) → Y × Qk is a homeomorphism. Then, for each compactum C in M, there exist a compact polyhedron P , a PL immersion f : P → X × [−1, 1]m , m  k, and an embedding g : P ×Qm → M such that g(P ×Qm ) is a neighborhood of C in M, Y × [−1, 1]k is a subpolyhedron of P , f |Y × [−1, 1]k = id, and αg = f × id, that is, the following diagram commutes: C

g

∩

g(P ×

α

M

⊂

m)

≈ g

P×

X× f ×id

m

⊃

∪

Y×

k

Proof Since α is an immersion, M has an open cover U such that α|U is an open embedding for each U ∈ U. Since C is compact,  we can easily find finite open sets V1 , . . . , Vn in M and m  k such that C ⊂ ni=1 Vi , each cl Vi is compact and contained in some Ui ∈ U, and α(Vi ) = int Ai × Qm for some compact straight subpolyhedron Ai of X × [−1, 1]m , where each α| cl Vi : cl Vi → Ai × Qm is a homeomorphism. If cl Vi ∩ cl Vj = ∅, then Ai ∩ Aj = pm (α(cl Vi ) ∩ α(cl Vj )) ⊃ pm α(cl Vi ∩ cl Vj ) = ∅, where it should be remarked that Ai ∩ Aj = ∅ does not imply cl Vi ∩ cl Vj = ∅. However, the following holds: (Y × [−1, 1]m−k ) ∩ Ai = ∅ ⇔ α −1 (Y × Qk ) ∩ cl Vi = ∅. Indeed, the left-hand side is equivalent to the following: (Y × Qk ) ∩ (Ai × Qm ) = ∅,

4.6 The Handle Straightening Theorem

329

which is equivalent to the right-hand side because α −1 (Ai × Qm ) = cl Vi . For each i = 1, . . . , n, let J (i) = {j < i | cl Vi ∩ cl Vj = ∅} and    m−k Bi = Ai ∩ (Y × [−1, 1] )∪ Aj . j ∈J (i)

We can inductively define polyhedra P0 ⊂ P1 ⊂ · · · ⊂ Pn as follows: P0 = Y × [−1, 1]m−k and Pi = Pi−1 ∪Bi Ai , 19 where Pi−1 ∪Bi Ai = Pi−1 ⊕Ai if Bi = ∅. Then, P = Pn is the desired polyhedron. The required PL immersion f : P → X × [−1, 1]m can be defined by f |P0 = id (i.e., f |Y × [−1, 1]m−k = id) and f |Ai = id for each i = 1, . . . , n. We define an embedding g : P × Qm → M as follows: g|P0 × Qm = (α|α −1 (Y × Qk ))−1 , g|Ai × Q = (α| cl Vi )−1 : Ai × Qm → cl Vi for each i = 1, . . . , n. Then, n αg = f × id. Moreover, g(P × Qm ) is a neighborhood of C in M because " ! i=1 Vi ⊂ g(P × Qm ). Let us start to prove the case n  2. Proof (The Case n  2 of Theorem 4.6.2) We write B1 = [−1/8, 1/8]n, B2 = [−1/4, 1/4]n ⊂ Rn , where (Bi , bd Bi ) ≈ (Bn , Sn−1 ) and B1 ⊂ int B2 . Consider the restriction of the open embedding h : Rn × Q → X × Q to the following subspace: (int B2 \ B1 ) × Q ≈ bd B1 × Q × R, which is an open embedding into X × Q. Since π1 (bd B1 ) = Z or 0 depending on n = 2 or n > 2, we can apply the Splitting Theorem 4.4.1 to obtain a splitting h((int B2 \ B1 ) × Q) = M1 ∪ M2 such that the inclusion M1 ∩ M2 ⊂ M1 ∪ M2 is a homotopy equivalence and M1 ∩ M2 = A × Qm for some compact PL bi-collared subpolyhedron A ⊂ X × [−1, 1]m ,

X and Y containing a common subspace A = ∅, X∪A Y denotes the adjunction space X ∪idA Y , where idA : A → X.

19 Given two spaces

330

4 Triangulation of Hilbert Cube Manifolds and Related Topics

where we may assume that cl Mi ⊃ h((bd Bi ) × Q), that is, cl Mi = Mi ∪ h((bd Bi ) × Q), i = 1, 2. Then, Mi is homotopy dense in cl Mi because h((bd Bi ) × Q) is collared in h((B2 \ int B1 ). Thus, the following inclusion is also a homotopy equivalence: A × Qm = cl M1 ∩ cl M2 ⊂ cl M1 ∪ cl M2 , which implies that A × Qm is a strong deformation retract of cl M1 ∪ cl M2 . Hence, A × Qm is a strong deformation retract of each of cl M1 and cl M2 . Let r : cl M1 → A × Qm be a strong deformation retraction. Note that the following inclusion is a homotopy equivalence h(bd B1 × Q) ⊂ h((B2 \ int B1 ) × Q) = cl M1 ∪ cl M2 , where cl M1 is a strong deformation retract of cl M1 ∪ cl M2 because A × Qm = cl M1 ∩ cl M2 is a strong deformation retract of cl M2 . Hence, the inclusion h(bd B1 × Q) ⊂ cl M1 is a homotopy equivalence. Thus, the following restriction is a homotopy equivalence: r|h(bd B1 × Q) : h(bd B1 × Q) → A × Qm . Observe that A is the boundary of the following straight subpolyhedron: Y = pm (h(B1 × Q) ∪ M1 ) ⊂ X × [−1, 1]m . Then, it follows that h(B1 × Q) ∪ M1 = Y × Qm , h(B2 × Q) = (Y × Qm ) ∪ cl M2 and cl M1 = (Y × Qm ) \ h(int B1 × Q). Since bd Y × Qm = A × Qm is a strong deformation retract of cl M2 , it follows that Y × Qm is a strong deformation retract of h(B2 × Q), which is contractible. As a compact contractible Q-manifold is homeomorphic to the Hilbert cube (Theorem 3.7.9), we have Y × Qm ≈ Q. We have to construct a homeomorphism g of X × Q onto itself so that −1 g|(X × Q) \ h(B2 × Q) = id and gh(B1 × Q) = pm (Y ) = Y × Qm .

As before, let Tn0 = Tn \ {p0 }, where p0 = en (1, . . . , 1) ∈ Tn \ en (B2 ),

4.6 The Handle Straightening Theorem

331

Rn × Q

Rn × Q

[−1, 1] n × Q

[−1, 1] n × Q B2 × Q h −1 (Y ) × Q m B1 × Q

× id

e n × id

× id

e n × id

Tn × Q

Tn × Q p0

p0

Y1

Y2 e n (B 2 ) × Q

Fig. 4.10 Y1 and Y2

and let α : Tn0 → Rn be the immersion such that αen |B2 = id, which had been obtained in Theorem 4.5.1. Let Y1 = en (B1 ) × Q and Y2 = (en × id)h−1 (Y × Qm ). These are both subsets of Tn0 × Q, which are homeomorphic to the Hilbert cube Q. Then, Y1 ⊂ int Y2 ⊂ Y2 ⊂ en (B2 ) × Q (cf. Fig. 4.10). Since en α|en (B2 ) = id, we can define a strong deformation retraction r0 so that the following diagram commutes: r0

Y2 \ int Y1

bd Y2 (en ×id)h−1

h(α×id)

cl M1

r

bd Y × Q m

Then, the restriction r0 | bd Y1 is a homotopy equivalence because h(α × id)(bd Y1 ) = h(bd B1 × Q) and r|h(bd B1 × Q) is a homotopy equivalence.

332

4 Triangulation of Hilbert Cube Manifolds and Related Topics

∼ π1 (Sn−1 ) ∼ Since π1 (bd Y1 ) = = Z or 0, we can apply Theorem 4.1.7 to obtain a homeomorphism f0 : bd Y1 → bd Y2 such that f0  r0 | bd Y1 . Recall that Y1 ≈ Y2 ≈ Q. Since bd Y1 and bd Y2 are Z-sets in Y1 and Y2 , respectively, f0 extends to a homeomorphism f0 : Y1 → Y2 (Corollary 2.10.7). Then, f0 (bd Y1 ) = bd Y2 and f0 | bd Y1  r0 | bd Y1  id in Y2 \ int Y1 . In the following diagram, we have just defined the bottom homeomorphism f0 . We will construct upper homeomorphisms step-by-step so that each ladder of the diagram commutes with the bottom. Finally, f4 will be constructed so that f4 | bd B2n × Q = id. Then, hf4 h−1 can extend to the desired homeomorphism g. X×Q

g

X×Q

h

h

B2 × Q

f4

∪

int B2 × Q

B2 × Q ∪

f3

int B2 × Q

ξ ×id

Rn × Q

ξ ×id f2

Rn × Q

en ×id

en ×id

Tn × Q

f1

∪

Y1

Tn × Q ∪

f0

Y2

Step 1 We will extend f0 to a homeomorphism f1 : Tn × Q → Tn × Q so that f1  id. Here, we want to use Theorem 4.1.7, but it can be applied to Q-manifolds being the products of compact polyhedra with Q whose fundamental groups are free or free Abelian. Consider the following Q-manifolds (cf. Fig. 4.10):    Tn × Q \ int Y1 = Tn \ en (int B1 ) × Q and     n T × Q \ int Y2 = Tn × Q \ int(en × id)h−1 (Y × Qm ). 

Here, Tn \ en (int B1 ) can be regarded as a compact polyhedron and the first one is the product of this polyhedron and Q. However, for now, it is not known whether the second one can be regarded as the product of a polyhedron and Q or not. We take the following neighborhoods of p0 = en (1, . . . , 1) ∈ Tn : D1 = en ([7/8, 9/8]n), D2 = en ([3/4, 5/4]n) ⊂ Tn \ en (B2 ),

4.6 The Handle Straightening Theorem

333

where (Di , bd Di ) ≈ (Bn , Sn−1 ) and D1 ⊂ int D2 . Then, we have a polyhedron P = Tn \ (en (int B1 ) ∪ int D1 ) such that     P × Q = Tn × Q \ int Y1 ∪ (int D1 × Q) , where it should be remarked that P  Tn \ {p0 , 0}.  Z∗Z∗Z π1 (P ) ∼ = Z ⊕ ··· ⊕Z

if n = 2, (n copies)

if n > 2.

Consider the following Q-manifold:     W = Tn × Q \ int Y2 ∪ ({p0 } × Q) . Then, bd Y2 ⊂ W , W \ (int D1 × Q) is compact and the following composition is an immersion: W ⊂ Tn0 × Q

α×id

Rn × Q

h

X×Q,

where h(α × id)(bd Y2 ) = A × Qm and A = bd Y is a compact PL bi-collared subpolyhedron of X × [−1, 1]m . Applying Lemma 4.6.3, we can obtain a compact  polyhedron Z, a PL immersion α  : Z → X×[−1, 1]m , m  m, and an embedding ϕ : Z × Qm → W such that ϕ(Z × Qm ) is a neighborhood of W \ (int D1 × Q),  A × [−1, 1]m −m is a subpolyhedron of Z, α  |A × Qm = id, and h(α × id)ϕ = α  × id : Z × Qm → X × [−1, 1]m × Qm . Then, it follows that ϕ(A × Qm ) = (en × id)h−1 (bd Y × Qm ) = bd Y2 , (int D2 \ D1 ) × Q ⊂ ϕ(Z × Qm ), and ψ = ϕ −1 |(int D2 \ D1 ) × Q is an open embedding into Z × Qm , where (int D2 \ D1 ) × Q ≈ bd D1 × Q × R ≈ Sn−1 × Q × R. Since π1 (Sn−1 ) ∼ = Z or 0, we can apply the Splitting Theorem 4.4.1 to obtain a splitting ψ((int D2 \ D1 ) × Q) = A1 ∪ A2 such that the inclusion A1 ∩A2 ⊂ A1 ∪A2 is a homotopy equivalence and A1 ∩A2 = B ×Qk for some PL bi-collared subpolyhedron B of Z ×[−1, 1]k for some k  m ,

334

4 Triangulation of Hilbert Cube Manifolds and Related Topics Tn × Q

D2 × Q Y1

{p 0 } × Q

P×Q bd Y1

f0

D1 × Q f1

(cl A 1 )

Y2 int Y1 W ( int D 2 × Q)

Y2

{p 0 } × Q

(C × Q k ) bd Y2

Tn × Q

(B × Q k ) (Z × Q m ) Z × Qm

A × Qm

(W ( int D 2 × Q)) C × Qk

A2

A1

B × Qk Fig. 4.11 Y1 and Y2

where we may assume cl A1 ⊃ ψ(bd D1 × Q), that is, cl Ai = Ai ∪ ψ(bd Di × Q), i = 1, 2. Then, B × Qk is a strong deformation retract of each of cl A1 and cl A2 . See Fig. 4.11. Let r  : cl A1 → B × Qk be a strong deformation retraction. Then, by the same argument as r, we can see that the following restriction is a homotopy equivalence: r  |ψ(bd D1 × Q) : ψ(bd D1 × Q) → B × Qk . Since ϕψ|ϕ(int D2 \ D1 ) × Q = id, we can define a strong deformation retraction r0 so that the following diagram commutes: r1

ϕ(cl A1 )

ϕ

ψ

cl A1

ϕ(B × Q k )

r

B ×Qk

4.6 The Handle Straightening Theorem

335

Then, the restriction r1 | bd D1 × Q is a homotopy equivalence. In the same way as f0 , we can apply Theorem 4.1.7 to obtain a homeomorphism f1 : bd D1 × Q → ϕ(B × Qk ) such that f1  r1 | bd D1 × Q. Since B × Qk is a strong deformation retract of cl A2 , it follows that cl A1 ∪ (D1 × Q) is a strong deformation retract of D2 × Q, so it is contractible. Hence, we have cl A1 ∪ (D1 × Q) ≈ Q by Theorem 3.7.9. Observe that B × Qk is a Z-set in cl A1 ∪ (D1 × Q). Since D1 × Q ≈ Q and bd D1 × Q is a Z-set in D1 × Q, f1 extends to a homeomorphism f1 : D1 ×Q → cl A1 ∪(D1 ×Q) by Corollary 2.10.7. Again see Fig. 4.11. Observe that B is the boundary of the following subpolyhedron:   C = pk ψ(W \ (int D2 × Q)) ∪ A2 ⊂ Z × [−1, 1]k , where we have   ϕ(C × Qk ) = W \ (int D2 × Q) ∪ ϕ(cl A2 )   = W \ (int D1 × Q) ∪ ϕ(int A1 )   = (Tn × Q) \ int Y2 ∪ (int D1 × Q) ∪ ϕ(int A1 ) . Then, it follows that (Y2 \ int Y1 ) ∪ ϕ(C × Qk ) ∪ ϕ(cl A1 ) = (Tn × Q) \ (int Y1 ∪ (int D1 × Q)) = P × Q. Moreover, we have (Y2 \ int Y1 ) ∩ ϕ(C × Qk ) = bd Y2 and ϕ(C × Qk ) ∩ ϕ(cl A1 ) = ϕ(B × Qk ), where bd Y2 and ϕ(B × Qk ) are Z-sets in ϕ(C × Qk ). Already we obtained two strong deformation retractions r0 : Y2 \ int Y1 → bd Y2 and r1 : ϕ(cl A1 ) → ϕ(B × Qk ), which extend to a strong deformation retraction r2 : P × Q → ϕ(C × Qk ). Then, we can apply Theorem 4.1.7 to see that r2 is homotopic to a homeomorphism f1 . Already, we obtained two homeomorphisms f0 : bd Y1 → bd Y2 and f1 : bd D1 × Q → ϕ(B × Qk )

336

4 Triangulation of Hilbert Cube Manifolds and Related Topics

such that f0  r0 | bd Y1 and f1  r1 | bd D1 × Q, where bd Y1 and bd D1 × Q are Z-sets in P × Q. Since f0  f1 | bd Y1 and f1  f1 | bd D1 × Q, we can apply the Z-set Unknotting Theorem 2.11.6 to obtain a homeomorphism f1∗ : P × Q → ϕ(C × Qk ) such that f1∗ | bd Y1 = f0 , f1∗ | bd D1 × Q = f1 , and f1∗  f1  r2 . Finally, we can extend f1∗ to a homeomorphism f1 : Tn → Tn by f1 |Y1 = f0 and f1 |D1 × Q = f1 .20 Step 2 (Construction of f2 ) Note that π1 (Rn × Q, (0, 0)) = 0. By the Lifting Criterion Theorem 4.2.5, f1 (en ×id) can be lifted to a map f2 with f2 (0, 0) = (0, 0). Namely, f2 is a unique map such that the following diagram commutes: Rn × Q

f2

Rn × Q en ×id

en ×id

Tn × Q

f1

Tn × Q

Then, f2 is a homeomorphism. Indeed, f1−1 (en × id) can be lifted to a map f2 with f2 (0, 0) = (0, 0). Since f2 f2 and f2 f2 are liftings of id, it follows from uniqueness of liftings that f2 f2 = id and f2 f2 = id. By uniqueness of liftings, we have f2 |B1 × Q = (α × id)f1 (en × id)|B1 × Q because both are liftings of f1 (en × id)|B1 × Q and f2 (0, 0) = (0, 0) = (α × id)f1 (en × id)(0, 0). Then, observe f2 (B1 × Q) = (α × id)f1 (en (B1 ) × Q) = (α × id)f1 (Y1 ) = (α × id)(Y2 ) = (α × id)(en × id)h−1 (Y × Qm ) = h−1 (Y × Qm ).

Chapman’s Lecture Notes [(2)], it is asserted that f1  id and its proof is left to the reader as an exercise. However, the author doesn’t know its proof.

20 In

4.6 The Handle Straightening Theorem

337

Note that 2Zn = {v ∈ Rn | en (v) = en (0)} (= (en )−1 (en (0))) and Rn =

n  #

[v(i) − 1, v(i) + 1].

v∈2Zn i=1

 Let x ∈ ni=1 [v(i) − 1, v(i) + 1], v ∈ 2Zn , and y ∈ Q. Then, x − v ∈ [−1, 1]n . Since v(i) ∈ 2Z, it follows that e(x(i) − v(i)) = (cos π(x(i) − v(i)), sin π(x(i) − v(i))) = (cos πx(i), sin πx(i)) = e(x(i)), hence en (x − v) = en (x). Replacing x by prRn f2 (x, y), we also have en (prRn f2 (x, y) − v) = en prRn f2 (x, y). Since f2 is a lifting of f1 (en × id), it follows that (en × id)f2 (x − v, y) = f1 (en × id)(x − v, y) = f1 (en × id)(x, y) and (en × id)(prRn f2 (x, y) − v, prQ f2 (x, y)) = (en × id)f2 (x, y) = f1 (en × id)(x, y). By uniqueness of liftings, we have f2 (x − v, y) = f2 (x, y) = (prRn f2 (x, y) − v, prQ f2 (x, y)). Then, it follows that prRn f2 (x, y) − x∞ = (prRn f2 (x, y) − v) − (x − v)∞ = prRn f2 (x − v, y) − (x − v)∞ . Consequently, we have sup

(x,y)∈Rn×Q

prRn f2 (x, y) − x∞ =

sup

(x,y)∈[−1,1]n×Q

prRn f2 (x, y) − x∞ .

Because of the compactness of [−1, 1]n × Q, we have c0 =

sup

(x,y)∈[−1,1]n×Q

prRn f2 (x, y) − x∞ < ∞.

338

4 Triangulation of Hilbert Cube Manifolds and Related Topics

Step 3 (Construction of f3 ) Choose 1/8 < t0 < 1/4 so that f2 (B1 × Q) = h−1 (Y × Qm ) ⊂ [−t0 , t0 ]n × Q. We define a homeomorphism γ : [0, 1/4) → R+ as follows: ⎧ ⎨t γ (t) = t (1 − 4t0 ) ⎩ 1 − 4t

if t  t0 , if t  t0 .

Let ξ : int B2 → Rn be a homeomorphism defined by ξ(x) = γ (x∞ )x−1 ∞ x, where x∞ = max{|x(1), . . . , |x(n)|}. Note that ξ(x)∞ = γ (x∞ ) for each x ∈ Rn . The homeomorphism f3 is defined by the following commutative diagram: int B2 × Q

f3

int B2 × Q ξ ×id

ξ ×id

Rn × Q

f2

Rn × Q

Since ξ |[−t0 , t0 ]n = id, we have f3 |B1 × Q = f2 |B1 × Q = (α × id)f1 (en × id)|B1 × Q. Observe that ξ −1 is defined by ξ −1 (y) = γ −1 (y∞ )y−1 ∞ y, where γ −1 (s) =

⎧ ⎨s s ⎩ 1 + 4(s − t0 )

if s  t0 , if s  t0 .

For each s  t0 and s   1, since 1/8 < t0 < 1/4, it follows that    −1    s s γ (s) − γ −1 (s  ) =    1 + 4(s − t ) − 1 + 4(s  − t )  0 0 =

|s − s  |(1 − 4t0 ) (1 + 4(s − t0 ))(1 + 4(s  − t0 ))


0. For each and (x  , y  ) ∈ int B2 × Q with x − x  ∞ < δ, x  ∞  x∞ − x − x  ∞ > 1/4 − δ  γ −1 (max{1 + c0 , 2c0 /ε}),

the reduced product, refer to p. 108. In this case, (B2 × Q)bd B2 ≈ bd B2 ∪prbd B B2 × Q, 2 where prbd B2 : bd B2 × Q → bd B2 is the projection.

21 For

340

4 Triangulation of Hilbert Cube Manifolds and Related Topics

that is, γ (x  ∞ )  max{1 + c0 , 2c0 /ε}, so c0 /γ (x  ∞ )  ε/2. Therefore, print B2 f3 (x  , y  ) − x∞  print B2 f3 (x  , y  ) − x  ∞ + x  − x∞
0 and m ∈ N, choose n ∈ N so that n > m and 1/n < ε. We define an embedding h : Q → [−n/(n + 1), n/(n + 1)]N as follows: ⎧ n−1 ⎪ ⎪ x(i) if |x(i)|  , ⎪ ⎪ ⎪ n ⎪ ⎪ ⎨n − 1 1 n − 1

n−1 h(x)(i) = + x(n) − if x(i)  , ⎪ n n+1 n n ⎪ ⎪ ⎪ ⎪ ⎪ 1 n − 1

n−1 n−1 ⎪ ⎩− + x(n) + if x(i)  − . n n+1 n n Then, h|[−m/(m + 1), m/(m + 1)]N = id. For each x ∈ Q and i ∈ N, |h(x)(i) − x(i)| < 1/n < ε, so d(h, id) < ε. Hence, rint Q is a cap set for Q. (2): For each n ∈ N, let Mn =

n 

n pr−1 i ({±1}) = ∂[−1, 1] × Q.

i=1

 Then, M1 ⊂ M2 ⊂ · · · is a tower of compact Z-sets in Q with B(Q) = n∈N Mn . We show that this has the cap for Q. Let C be a compact set in Q. For each m ∈ N and ε > 0, choose n ∈ N so that n > m and n−1 < ε. As in Fig. 5.1, we define a retraction r : [−1, 1]n = [−1, 1]n−1 × [−1, 1] → ∂[−1, 1]n−1 × [−1, 1] ∪ [−1, 1]n−1 × {1} = ∂[−1, 1]n \ rint[−1, 1]n−1 × {−1}

358

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces [−1, 1] n 1 [−1, 1]

r

y= −1 [−1 + n −1, 1 − n −1 ] n −1

[−1, 1] n −1

x

1 2n

+1

( x = 2n(y − 1) + 1)

Fig. 5.1 The retraction r

such that pr1 r(x, y) − x < n−1 (< ε) for each (x, y) ∈ [−1, 1]n−1 × [−1, 1], where pr1 : [−1, 1]n−1 × [−1, 1] → [−1, 1]n−1 is the projection. Indeed, such r can be defined as follows:  ⎧ 2nx y−1 ⎪ ⎪ , 1 if x  + 1, ⎨ y − 1 + 2n 2n r(x, y) =   ⎪ y−1 x ⎪ ⎩ , y + 2n(1 − x) if x  + 1, x 2n Let θ : Q2 ×I → Q be the coordinate-switching pseudo-isotopy defined in Note on p. 118 (where X is now considered as a singleton or Q). Then, θ0 = pr1 and θt : Q2 → Q is a homeomorphism for each t > 0. Let pn : Q = [−1, 1]n × Q → [−1, 1]n and qn : Q = [−1, 1]n × Q → Q be the projections, that is, pn (x) = (x(1), . . . , x(n)) and qn (x) = (x(n + i))i∈N , and define a map α : C → I by α(x) = 1 − pn−1 (x). Then, α −1 (0) = C ∩ (∂[−1, 1]n−1 × Q) = C ∩ Mn−1 . By the universality of Q, we have an embedding g : C → Q. Then, we define a map h : C → Mn = ∂[−1, 1]n × Q as follows: h(x) = (rpn (x), θα(x)(qn (x), g(x))). For each x ∈ C ∩ Mn−1 , since α(x) = 0, h(x) = (rpn (x), θ0 (qn (x), g(x))) = (pn (x), qn (x)) = x.

5.1 Cap Sets and F.D.Cap Sets in 2 and Q

359

Therefore, h|C ∩ Mn−1 = id. Because Mm ⊂ Mn−1 , we have h|C ∩ Mm = id. Moreover, for every x ∈ C, pn−1 h(x) − pn−1 (x) = rpn (x) − pn−1 (x)  n−1 , which means that d(h, id)  n−1 < ε. To verify that h is an embedding, let x = y ∈ C. If α(x) = α(y) = 0, then h(x) = x = y = h(y) as seen above. When α(x) = α(y) = t > 0, since θt is a homeomorphism and g is an embedding, it follows that qn h(x) = θt (qn (x), g(x)) = θt (qn (y), g(y)) = qn h(y), which implies h(x) = h(y). If α(x) = α(y), then pn−1 h(x) = pn−1 (x) = 1 − α(x) = 1 − α(y) = pn−1 (y) = pn−1 h(y), hence h(x) = h(y). (3): Each [−1, 1]n is identified with {x ∈ Q | x(i) = 0 if i > n }, which is a compact Z-set in Q by Theorem 2.10.6. We may show that the tower ([−1, 1]n )n∈N has the f.d.cap for Q. Let C be a finite-dimensional compact set in Q and k = 2 dim C + 1. For each ε > 0 and m ∈ N, choose n ∈ N so that n > m and n−1 < ε. By the Embedding Theorem 1.12.14, there exists an embedding g : C → Ik . We define an embedding h : C → [−1, 1]n−1 × I × Ik ⊂ [−1, 1]n+k−1 as follows: h(x) = (x(1), . . . , x(n − 1), α(x), α(x)g(x)), where α : C → I is the map defined by α(x) = d(x, C ∩ [−1, 1]n−1 ). For each x ∈ C ∩ [−1, 1]n−1 , since α(x) = 0, we have h(x) = (x(1), . . . , x(n − 1), 0, 0, . . . ) = x, hence h|C ∩ [−1, 1]n−1 = id, so h|C ∩ [−1, 1]m = id. Since pn−1 h(x) = pn−1 (x) for every x ∈ C, it follows that d(h, id)  n−1 < ε. To see that h is an embedding, let x = y ∈ C. If α(x) = α(y) = 0, then h(x) = x = y = h(y). If α(x) = α(y) = t > 0, then tg(x) = tg(y), which implies that h(x) = h(y). If α(x) = α(y), then prn h(x) = α(x) = α(y) = prn h(y), hence h(x) = h(y). (4): As in the proof of (3), we regard [−1, 1]n ⊂ Q. We show that the tower ([−1, 1]n × Q)n∈N has the cap for Q2 . We use the metric ρ ∈ Metr(Q2 ) defined as follows:   ρ((x, y), (x  , y  )) = max d(x, x  ), d(y, y  ) . Let C be a compact set in Q2 . For each ε > 0 and m ∈ N, choose n ∈ N so that n > m and n−1 < ε. As in the proof of (2), take the coordinate-switching

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pseudo-isotopy θ : Q2 × I → Q defined in Note on p. 118. By the universality of Q, we have an embedding g : C → Q. We define an embedding h : C → [−1, 1]n−1 × I × Q ⊂ Q2 as follows: h(x, y) = (x(1), . . . , x(n − 1), α(x, y), θα(x,y)(y, g(x, y))), where α : C → I is the map defined by   α(x, y) = min 2−n , ρ((x, y), C ∩ ([−1, 1]n−1 × Q)) . Since α(x, y)  2−n , it follows from () on p. 119 that pn θα(x,y)(y, g(x, y)) = pn (y). For each (x, y) ∈ C ∩ ([−1, 1]n−1 × Q), since α(x, y) = 0, we have h(x, y) = (x(1), . . . , x(n − 1), 0, θ0 (y, g(x, y))) = (x(1), . . . , x(n − 1), 0, 0, . . . , y) = (x, y), hence h|C ∩ ([−1, 1]n−1 × Q) = id, so h|C ∩ ([−1, 1]m × Q) = id. For every (x, y) ∈ C, since pn−1 pr1 h(x, y) = pn−1 (x) and pn θα(x,y)(y, g(x, y)) = pn (y),   ρ(h(x, y), (x, y))  max n−1 , (n + 1)−1 )  n−1 < ε. To see that h is an embedding, let (x, y) = (x  , y  ) ∈ C. If α(x, y) = α(x  , y  ) = 0, then h(x, y) = (x, y) = (x  , y  ) = h(x  , y  ). When α(x, y) = α(x  , y  ) = t > 0, since θt is a homeomorphism and g is an embedding, it follows that pr2 h(x, y) = θt (y, g(x, y)) = θt (y  , g(x  , y  )) = pr2 h(x  , y  ), which implies that h(x, y) = h(x  , y  ). If α(x, y) = α(x  , y  ), then prn h(x, y) = α(x, y) = α(x  , y  ) = prn h(x  , y  ), so we have h(x, y) = h(x  , y  ). (5): Recall that every compact set in s is a Z-set. It suffices to show that the tower ([−n, n]N )n∈N has the cap for s, where [−1, 1]n is identified with {x ∈ s | x(i) = 0 if i > n }. Let C be a compact set in s, ε > 0, and m ∈ N. Choose n0 ∈ N so that n0 > m and n−1 0 < ε. Furthermore, choose n1 ∈ N so that n1  n0 and pri (C) ⊂ [−n1 , n1 ] for each i < n0 , where pri : s = RN → R is the projection onto the i-th factor. We define an embedding h : C → [−n1 −1, n1 +1]N as follows: ⎧ ⎪ x(i) ⎪ ⎪ ⎪ ⎪ x(i) − n1 ⎨ n1 + h(x) = x(i) ⎪ ⎪ ⎪ x(i) + n1 ⎪ ⎪ ⎩−n1 − x(i)

if i < n0 or |x(i)|  n1 , if i  n0 and x(i)  n1 , if i  n0 and x(i)  −n1 .

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Then, h|C ∩ [−n0 , n0 ]N = id, so h|C ∩ [−m, m]N = id. Since h(x)(i) = x(i) for every x ∈ C and i < n0 , it follows that d(h, id)  n−1 0 < ε. Hence, s Q is a cap set for s. (6): As with (5), it suffices to show that the tower ([−n, n]n )n∈N has the f.d.cap for s. Let C be a finite-dimensional compact set in s and k = 2 dim C + 1. For each ε > 0 and m ∈ N, take the same n0 , n1 ∈ N as chosen in (5) and an embedding g : C → Ik as in (3) (Theorem 1.12.14). Let n = max{n1 , n0 + k}. We define an embedding h : C → [−n1 , n1 ]n0 × I × Ik ⊂ [−n, n]n as follows: h(x) = (x(1), . . . , x(n0 ), α(x), α(x)g(x)), where α : C → I is a map defined by α(x) = d(x, C ∩ [−n0 , n0 ]n0 ). Then, h|C ∩ [−n0 , n0 ]n0 = id, so h|C ∩ [−m, m]m = id. Since h(x)(i) = x(i) for every x ∈ C and i < n0 , it follows that d(h, id)  n−1 0 < ε. Hence, s f is an f.d.cap set for s. (7): We may show that the tower ([−n, n]n × Q)n∈N has the cap for s × Q. The proof is the same as (4), where Q × Q and [−1, 1]n−1 should be replaced with s × Q and [−(n − 1), n − 1]n−1 , respectively. Details are left to the reader.  (8): It is enough to prove that the tower (n i∈N [−2−i , 2−i ])n∈N has the cap for 1 . Let C be a compact set in 1 , 0 m and i>n0 |x(i)| < ε for every x ∈ C. Moreover, choose n1 ∈ N so that n1  n0 and pri (C) ⊂ n1 [−2−i , 2−i ] for each i  n0 , where pri : 1 → R is the projection onto the i-th factor. Note that if x ∈ C and i > n0 , then|x(i)| < ε < 1 and 2−i m < 2−n0 n0 < 1. We define an embedding h : C → n1 i∈N [−2−i , 2−i ] as follows: ⎧ ⎪ x(i) if i  n0 or |x(i)|  2−i m, ⎪ ⎪ ⎪ −i −i ⎪ ⎪ ⎪2−i m + 2 (n1 − m)(x(i) − 2 m) ⎪ ⎪ ⎨ 1 − 2−i m h(x)(i) = if i > n0 and 2−i m  x(i) < 1, ⎪ ⎪ −i −i ⎪ ⎪ −i m + 2 (n1 − m)(x(i) + 2 m) ⎪ −2 ⎪ ⎪ ⎪ 1 − 2−i m ⎪ ⎩ if i > n0 and − 1 < x(i)  −2−i m. Then, h|C ∩ m



i∈N [−2

h(x) − x =

−i , 2−i ]



= id. For each x ∈ C,

|h(x)(i) − x(i)| =

i>n0





  |x(i)| − |h(x)(i)| i>n0

|x(i)| < ε,

i>n0

hence d(h, id) < ε. Thus, 1Q is a cap set for 1 .

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(9): We show that the tower ([−n, n]n )n∈N has the f.d.cap for 1 . Let C be a finite-dimensional compact set in 1 and k = 2 dim C + 1. For each 0 < ε < 1 and m ∈ N, choose n0 ∈ N so that n0 > m and i>n0 |x(i)| < ε/2 for every x ∈ C (the same as in (8)). Moreover, choose n1 ∈ N so that n1  n0 + k + 1 and pri (C) ⊂ [−n1 , n1 ] for each i = 1, . . . , n0 . Let α : C → I be a map defined by   α(x) = min 1, d(x, C ∩ [−m, m]m ) , and let δ = ε/2(k +1). By the Embedding Theorem 1.12.14, we have an embedding g : C → [0, δ]k . We define an embedding h : C → [−n1 , n1 ]n0 × [0, δ]k+1 ⊂ [−n1 , n1 ]n1 as follows: h(x) = (x(1), . . . , x(n0 ), α(x)δ, g(x)). Then, h|C ∩ [−m, m]m = id. For each x ∈ C, h(x) − x =



|h(x)(i) − x(i)| 

i>n0



|h(x)(i)| +

i>n0



|x(i)|

i>n0

 (k + 1)δ + ε/2 < ε, hence d(h, id) < ε. Thus, 1f is an f.d.cap set for 1 . " !  −1 Remark 5.1 (1): Recall that B(Q) = n∈N prn ({±1}), where prn : Q = N [−1, 1] → [−1, 1] is the projection onto the n-th factor. The proof of (2) can  be adjusted to see that the subset n∈N pr−1 n (1) ⊂ Q is also a cap set for Q. Details are left as an exercise. (2): We can show that I (Q)f = (−1, 1)N f is an f.d.cap set for Q, that is, the following tower has the f.d.cap for Q: 7

−

n 8n n , n+1 n+1

 . n∈N

Indeed, combining methods of (1) and (3), for each m ∈ N and ε > 0, we can choose m > n and construct an embedding h : C → [−n/(n + 1), n/(n + 1)]n such that h|C ∩ [−m/(m + 1), m/(m + 1)]m = id and d(h, id) < ε. Filling in details is a good exercise for the reader. N n (3): It can be shown that QN f is a cap set for Q , that is, the tower (Q )n∈N has the cap for QN , where Qn is identified with {x ∈ QN | x(i) = 0 if i > n } ⊂ QN . The proof is left to the reader. p

p

(4): Similar to (8) and (9), for every p ∈ N, Q is a cap set and f is an f.d.cap  set for p , where the copy n∈N [−2−n , 2−n ] ⊂ p of the Hilbert cube Q can be replaced with another suitable copy. The proof is a good exercise for the reader.

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363

As we saw above, there are various (f.d.)cap sets. However, (f.d.)cap sets for an 2 -manifold and a Q-manifold are topologically unique, that is: Theorem 5.1.4 Let X be an 2 -manifold or a Q-manifold, and let M and N be (f.d.)cap sets for X. For each open cover U of X, there exists a homeomorphism h : X → X such that h(M) = N and h is U-close to id.   Proof Let M = i∈N Mi and N = j ∈N Nj , where M1 ⊂ M2 ⊂ · · · and N1 ⊂ N2 ⊂ · · · are towers of (finite-dimensional) compact Z-sets in X and they have the (f.d.)cap. For each open cover U ∈ cov(X), we have a complete metric d ∈ Metr(X) such that {Bd (x, 1) | x ∈ X} ≺ U (Proposition 1.3.22 (1)). We will inductively construct increasing sequences 1 = i(1) < i(2) < · · · , j (1) < j (2) < · · · ∈ N, and homeomorphisms hn : X → X, n ∈ N, so that the following conditions are satisfied: −1 −n (1) d(hn , hn−1 ) < 2−n , d(h−1 n , hn−1 ) < 2 , (2) hn |Mi(n−1) = hn−1 |Mi(n−1) , and −1 (3) Nj (n−1) ⊂ hn (Mi(n) ) ⊂ Nj (n) , so h−1 n (Nj (n−1) ) ⊂ Mi(n) ⊂ hn (Nj (n) ),

where j (0) = 0, N0 = ∅, and h0 = id. Then, (1) implies that (hn )n∈N and (h−1 n )n∈N uniformly converge to maps h, h : X → X such that d(h, id) < 1 and d(h , id) < −1 1, hence h and h are U-close to id. Since h−1 n+1 |hn+1 (Mi(n) ) = hn |hn (Mi(n) ) by −1 −1 (2), it follows from (3) that hn+1 |Nj (n−1) = hn |Nj (n−1) . Hence, for each n ∈ N, h|Mi(n) = hn |Mi(n) and h |Nj (n−1) = h−1 n |Nj (n−1) , which implies that h h|Mi(n) = (h |Nj (n) )(hn |Mi(n) ) = (h−1 n+1 |Nj (n) )(hn+1 |Mi(n) ) = id and −1 hh |Nj (n−1) = (h|Mi(n) )(h−1 n |Nj (n−1) ) = (hn |Mi(n) )(hn |Nj (n−1) ) = id.

Moreover, it follows from (3) that h(M) =



hn (Mi(n) ) =

n∈N 

h (N) =



n∈N



Nj (n) = N and

n∈N 

h (Nj (n) ) =



Mi(n) = M.

n∈N

Thus, h h|M = id and hh |N = id, which means that h h = id and hh = id because M and N are dense in X. Therefore, h : X → X is a homeomorphism with h−1 = h that is the desired one. Now, assume that h0 , . . . , hn−1 : X → X, 1 = i(1) < · · · < i(n − 1), and 0 < j (1) < · · · < j (n − 1) have been obtained so as to satisfy the conditions (1), (2), and (3). As is easily observed, (hn−1 (Mi ))i∈N is a tower of (finite-dimensional) compact Z-sets in X with the (f.d.)cap. Take V ∈ cov(X) with mesh st V < 2n−1 −n−1 . Since N and mesh h−1 j (n−1) is a (finite-dimensional) compact Zn−1 (st V) < 2 set in X and Nj (n−1) ∩ hn−1 (Mi(n−1) ) = hn−1 (Mi(n−1) ), there exist i(n) > i(n − 1) and an embedding g : Nj (n−1) → hn−1 (Mi(n) ) such that g|hn−1 (Mi(n−1) ) = id and

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g is V-homotopic to id. Then, g(Nj (n−1) ) is a Z-set in X. By the Z-Set Unknotting Theorem 2.5.12 or 2.11.6, g can be extended to a homeomorphism g˜ : X → X that is st V-isotopic to id. Then, Nj (n−1) ⊂ g˜ −1 (hn−1 (Mi(n) )), g˜ −1 |hn−1 (Mi(n−1) ) = id, −n−1 ˜ h−1 . d(g˜ −1 , id) < 2−n−1 , and d(h−1 n−1 g, n−1 ) < 2

Note that g˜ −1 (hn−1 (Mi(n) )) is a (finite-dimensional) compact Z-set in X. Take W ∈ cov(X) with mesh st W < 2n−1 and mesh h−1 ˜ W) < 2−n−1 . Now, n−1 g(st by the (f.d.)cap of (Nn )n∈N , we have j (n) > j (i − 1) and an embedding f : g˜ −1 (hn−1 (Mi(n) )) → Nj (n) such that f |Nj (n−1) = id and f is W-homotopic to id. Again using the Z-Set Unknotting Theorem 2.5.12 or 2.11.6, we can extend f to a homeomorphism f˜ : X → X that is st W-isotopic to id. Then, Nj (n−1) = f˜(Nj (n−1) ) ⊂ f˜g˜ −1 (hn−1 (Mi(n) )) ⊂ Nj (n) , f˜g˜ −1 |hn−1 (Mi(n−1) ) = f˜|hn−1 (Mi(n−1) ) = id, −n−1 ˜−1 , h−1 g) . d(f˜, id) < 2−n−1 , and d(h−1 n−1 g˜ f n−1 ˜ < 2

Moreover, it follows that d(f˜g˜ −1 , id)  d(f˜, id) + d(g˜ −1 , id) < 2−n and −1 −n ˜−1 , h−1 )  d(h−1 g˜ f˜−1 , h−1 g) ˜ h−1 d(h−1 n−1 g˜ f n−1 n−1 n−1 ˜ + d(hn−1 g, n−1 ) < 2 .

Thus, the homeomorphism hn = f˜g˜ −1 hn−1 : X → X satisfies the conditions (1), (2), and (3). By induction, the proof is completed. " ! The following is a direct consequence of the above theorem: Corollary 5.1.5 Let X and Y be together 2 - or Q-manifolds such that X ≈ Y , and let M and N be (f.d.)cap sets for X and Y , respectively. Then, (X, M) ≈ (Y, N), that is, there exists a homeomorphism h : X → Y with h(M) = N, where h can be taken arbitrarily close to a given homeomorphism f : X → Y . " ! Since s ≈ p , p ∈ N, by the Mazur Theorem 1.2.3 and Kadec–Anderson Theorem 3.5.6, the following is obtained from Proposition 5.1.3 with Remark 5.1(4) and Corollary 5.1.5: p

Corollary 5.1.6 The pairs (p , Q ), p ∈ N, are homeomorphic to (s, s Q ) and the p " ! pairs (p , f ), p ∈ N, are homeomorphic to (s, s f ). By Proposition 5.1.3 and Remark 5.1, we have the following homeomorphisms among pairs:

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365

Corollary 5.1.7 (1) (Q, rint Q) ≈ (Q, B(Q)) ≈ (Q2 , Qf × Q) ≈ (QN , QN f ); p p (2) (Q, Qf ) ≈ (Q, I (Q)f ); (3) (s ×Q, s f ×Q) ≈ (s, s Q ) ≈ ( ×Q, f ×Q) ≈ p " ! (p , Q ), p ∈ N. The above homeomorphisms give the following homeomorphisms: Corollary 5.1.8 (1) rint Q ≈ B(Q) ≈ Qf × Q ≈ QN f ≈ sf × Q ≈ sQ ≈ p p p f × Q ≈ Q , p ∈ N; (2) Qf ≈ s f ≈ f , p ∈ N. " ! In Corollary 5.1.7 (1), the first homeomorphism gives (Q, ∂Q) ≈ (Q, I (Q)).5 Hence, we have the following: Corollary 5.1.9 ∂Q ≈ I (Q) ≈ s ≈ p , p ∈ N. It is easy to see that ≈ 2f . Thus, 2Q and

RN f

N N N N N (QN f )f ≈ Qf and (Rf )f ≈ Rf , where 2f are normed linear spaces such that

" ! QN f

≈

2Q

and

2 2 N 2 (2Q )N f ≈ Q and (f )f ≈ f ,

which means the following: Theorem 5.1.10 All results on E-manifolds in Chap. 2 are valid for 2Q - and 2f manifolds. ! " Since every 2f -manifold has the homotopy type of a countable locally finite simplicial complex by Theorem 1.13.22, the following Triangulation Theorem for 2f -manifolds is obtained as a combination of the Classification Theorem 2.6.1 and Proposition 2.2.17: Theorem 5.1.11 (TRIANGULATION) Every 2f -manifold is homeomorphic to " ! |K| × 2f for some countable locally finite simplicial complex K. Concerning the Triangulation Theorem for 2Q -manifolds, we discuss the proof in the next section. In Theorem 3.9.2, it was proved that every infinite-dimensional compact convex set C in a metrizable topological linear space is homeomorphic to the Hilbert cube Q if C is an AR. Concerning the radial interior rint C, we have the following: Theorem 5.1.12 Let C be an infinite-dimensional compact convex set C in a metrizable topological linear space E. If C is an AR and rint C = ∅, then (C, rint C) ≈ (Q, rint Q). Proof As mentioned above, C ≈ Q by Theorem 3.9.2. By virtue of Corollary 5.1.5, it is enough to show that rint C is a cap set for C. By a translation, we may assume that 0 ∈ rint C without loss of generality. Due to Theorem 1.4.15, E has an admissible F -norm  · . 5 ∂Q

= Q \ rint Q is the radial boundary (cf. p. 22).

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First, we will show that aC is a Z-set in C for each 0 < a < 1. For each map f : In → C and ε > 0, we just have to construct a map f  : In → ∂C ε-close to f (cf. Theorem 2.8.6). Since C is convex, we can define maps ϑk : C k+1 × k → C, k = 1, . . . , n, as follows: ϑk (x1 . . . , xk+1 ; z) =

k+1 

z(i)xi for each (x1 . . . , xk+1 ; z) ∈ C k+1 × k ,

i=1

where k is the standard k-simplex. Note that ϑk (x, . . . , x; z) = x for each x ∈ C and z ∈ k . Each x ∈ C has an open neighborhood Ux in C such that diam ϑ(Uxk+1 × k ) < ε/4 for every k = 1, . . . , n. Let δ > 0 be a Lebesgue number for the open cover {Ux | x ∈ C} of C, where it should be noted that δ < ε/4 because diam Ux < ε/4 for each x ∈ C. Take a triangulation K of In (i.e., In = |K|) so that diam f (σ ) < δ for each σ ∈ K. Then, f |K (0) is uniquely extends to a map g : In = |K| → C that is affine on each simplex of K, that is, g

 k+1 i=1

  z(i)vi = k+1 i=1 z(i)f (vi ) = ϑ(f (v1 ), . . . , f (vk+1 ); z) for each v1 , . . . , vk+1  ∈ K and z ∈ k .

For each σ = v1 , . . . , vk+1  ∈ K, since diam g(σ (0) ) = diam f (σ (0) ) < δ, it follows that g(σ (0) ) is contained in some Ux , whence   diam g(σ ) = diam ϑ {(f (v1 ), . . . , f (vk+1 ))} × k  diam ϑ(Uxk+1 × k ) < ε/4. Hence, we can easily see that g is ε/2-close to f . Each x ∈ C has an open neighborhood Ux in E with an open circled neighborhood Vx of 0 in E such that diam(Ux +V is compact, x )∩C < ε/2. Since C  m there are finite x1 , . . . , xm ∈ C such that C ⊂ m U . Then, V = x 0 i i=1 i=1 Vxi is an open circled neighborhood of 0 in E and diam(x + V0 ) ∩ C < ε/2 for each x ∈ C. Since C − C is compact and fl C is an infinite-dimensional linear subspace of E, V0 ∩ fl C ⊂ C − C, that is, there exists v ∈ V0 ∩ fl C \ (C − C). Then, x + v ∈ C for each x ∈ C. Indeed, if x + v = y ∈ C, then v = y − x ∈ C − C. Let F = fl(g(K (0) ) ∪ {0, v}). Note that F is a finite-dimensional linear subspace of E and F = fl(F ∩ C) because   F = fl g(K (0) ) ∪ {0, v} ⊂ fl(F ∩ C) ⊂ F.

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367

Then, applying Propositions 1.4.3 and 1.4.4, we have 0 ∈ F ∩ rint C = rint(F ∩ C) = coreF (F ∩ C). Since the Minkowski functional p = pF ∩C : F → R+ for F ∩ C is continuous (Proposition 1.4.6), we can define a map λ : In × I → R as follows: λ(x, t) = p(gϕ(x) + tv) for each (x, t) ∈ In × I. Then, λ(x, 0) = p(gϕ(x)) < 1 and λ(x, 1) = p(gϕ(x) + v) > 1 because g(In ) ⊂ F ∩ rint C = rint(F ∩ C) and g(x) + v ∈ C = rcl C for each x ∈ In . By the intermediate value theorem, we have 0 < α(x) < 1 such that λ(g(x), α(x)) = 1, that is, gϕ(x) + α(x)v ∈ ∂(F ∩ C). Due to Proposition 1.4.2, gϕ(x) + sv ∈ ∂(F ∩ C), 0 < s < 1 ⇒ ∀t ∈ [0, s), gϕ(x) + tv ∈ rint(F ∩ C) = p−1 ([0, 1)). Hence, α(x) is uniquely determined for each x ∈ In . Then, we can define f  : In → ∂(F ∩ C) ⊂ ∂C by f  (x) = g(x) + α(x)v. Since V0 is circled and v ∈ V0 , g(x), f  (x) = g(x) + α(x)v ∈ (g(x) + V0 ) ∩ C, which means that f  is ε/2-close to g because diam(g(x) + V0 ) ∩ C < ε/2. Thus, f is ε-close to f  . For the continuity of f  , we just have to show that α : In → R is continuous. Suppose α(x) < s. Then, g(x) + sv ∈ F ∩ C. Since C is closed, if x  ∈ In is sufficiently close to x, then g(x  )+sv ∈ F ∩C, which implies that α(x  ) < s. Hence, α is upper semi-continuous. Now, suppose α(x) > s. Then, g(x)+sv ∈ rint(F ∩C). Note that g(In ) + Rv ⊂ F . As observed above, g(x) + sv ∈ rint(F ∩ C) = intF (F ∩ C). So, if x  ∈ In is sufficiently close to x then g(x  ) + sv ∈ intF (F ∩ C), which implies that α(x  ) > s. Hence, α is lower semi-continuous. Consequently, α is continuous. Now, we have − 2−n )C, n ∈ N, in C, which are homeomorphic to  Z-sets (1−n C ≈ Q. Then, n∈N (1 − 2 )C = rint C. Indeed, for each x ∈ rint C, we can find t > 1 so that tx = y ∈ C. Choose n ∈ N so that 2−n < 1 − 1/t, i.e., 1/t < 1 − 2−n . Then, we have x = (1/t)y ∈ 0, (1 − 2−n )y ⊂ (1 − 2−n )C.

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Note that (1 − 2−m )C is a Z-set in (1 − 2−n )C for any m > n because (1 − 2−m )C = (1 − 2−n )

1 − 2−m C. 1 − 2−n

To see that the tower ((1 − 2−n )C)n∈N has the property (cap), let ε > 0 and m ∈ N. We may construct an embedding h : C → (1 − 2−n )C for some n > m so that h|(1 − 2−m )C = f |(1 − 2−m )C = id and h is ε-close to id. Since C is compact, we can choose n ∈ N so that n > m and x − (1 − t)x = tx < ε/2 for every x ∈ C and 0  t  2−n . Let ϕ : C → (1 − 2−n )C be a map defined by ϕ(x) = (1 − 2−n )x for each x ∈ C. Then, ϕ|(1 − 2−m )C ε/2 id in (1 − 2−n )C by the straight line homotopy (x, t) → (1 − t)ϕ(x) + tx = (1 − 2−n )(1 − t)x + tx = (1 − 2−n + 2−n t)x. Applying the Homotopy Extension Theorem 1.13.11, we can obtain a map f : C → (1 − 2−n )C such that f |(1 − 2−m )C = id and f ε/2 ϕ. By the strong M0 universality of Q (Theorem 2.10.10), f is ε/3-close to a Z-embedding h : C → (1 − 2−n )C such that h|(1 − 2−m )C = f |(1 − 2−m )C = id. Then, h is ε-close to id. This completes the proof. " !

5.2 Manifold Pairs Modeled on the Pair of 2 (or Q) and its (F.D.)Cap Set At the end of the previous section, we saw that all results in Chap. 2 are valid for 2Q - and 2f -manifolds. In this section, using (f.d.)cap sets, we characterize manifold pairs modeled on the pairs (2 , 2f ), (2 , 2Q ), (Q, Qf ), and (Q, rint Q), and then prove the Open Embedding Theorem, the Classification Theorem, and the Triangulation Theorem for these manifold pairs. The Triangulation Theorem for 2Q -manifolds will be obtained as a corollary of the one for (2 , 2Q )-manifold pairs. First, we show that every 2 -manifold and Q-manifold has (f.d.)cap sets, and that every 2Q -manifold (resp. 2f -manifold) M can be embedded in an 2 -manifold and a Q-manifold as a cap set (resp. an f.d.cap set). To this end, we need the following: Lemma 5.2.1 Let M be an (f.d.)cap set for a metrizable space X. For every open set U in X, M ∩ U is an (f.d.)cap set for U .  Proof Let M = n∈N Mn , where M1 ⊂ M2 ⊂ · · · is a tower of (finitedimensional) compact Z-sets in X that has the (f.d.)cap for X. We can write  U = n∈N Fn , where each Fn is closed in X and Fn ⊂ int Fn+1 . Then, M ∩ U =

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369



n∈N (Mn ∩ Fn ) and each Mn ∩ Fn is a (finite-dimensional) compact Z-set in U . We show that the tower (Mn ∩ Fn )n∈N has the (f.d.)cap for U . Let C be a (finite-dimensional) compact set in U . Because of compactness, C ⊂ int Fk for some k ∈ N. For each U ∈ cov(U ), let

V = U ∪ {X \ Fk } ∈ cov(X). For each m ∈ N, there exists n ∈ N and an embedding h : C → Mn such that n > m, h|C ∩ Mm = id, and h is V-close to id. Then, h(C) ⊂ U . Otherwise, there is some x ∈ C such that h(x) ∈ U , hence x, h(x) ∈ X \ Fk , which contradicts C ⊂ int Fk . Because of compactness, h(C) ⊂ int Fn0 for some n0  max{n, k}. Thus, we have embedding h : C → Mn0 ∩ Fn0 that is U-close to id. " ! Proposition 5.2.2 Every 2 -manifold and Q-manifold has (f.d.)cap sets. Proof By the Open Embedding Theorem 2.5.10, every 2 -manifold M is homeomorphic to an open set U in 2 , that is, there is a homeomorphism f : M → U . Due to Lemma 5.2.1, 2Q ∩ U (or 2f ∩ U ) is an (f.d.)cap set for U . Then, f −1 (2Q ∩ U ) (or f −1 (2f ∩ U )) is an (f.d.)cap set for M. For a Q-manifold M, M × [0, 1) is homeomorphic to an open set U in Q by the Open  Embedding Theorem 2.11.4. As above, M × [0, 1) has an (f.d.)cap set N = n∈N Nn , where N1 ⊂ N2 ⊂ · · · is a tower of (finite-dimensional) compact Z-sets in M × [0, 1) that has the (f.d.)cap for M × [0, 1). Then, each Nn is a Z-set in M × I and the tower (Nn )n∈N has the (f.d.)cap for M × I. Indeed, for each (finitedimensional) compact Z-set C in M × I, m ∈ N, and U ∈ cov(M × I), choose V ∈ cov(M × I) and k ∈ N so that st V ≺ U, Nm ⊂ M × [0, 1 − 2−k ], and 

  {x} × [1 − 2−k , 1]  ({x} × [1 − 2−k , 1]) ∩ C = ∅ ≺ V.

Let f : I → [0, 1 −2−k−1] be a PL homeomorphism defined by f |[0, 1 −2−k ] = id and f (t) =

t + 1 − 2−k for 1 − 2−k  t  1. 2

Then, (idM × f )|C is V-close to id, (idM × f )|C ∩ Nm = id, and (idM × f )(C) is a (finite-dimensional) compact Z-set in M × [0, 1). Hence, there exist n ∈ N and an embedding g : (idM × f )(C) → Nn such that g|(idM × f )(C) ∩ Nm = id and g is V-close to id. Note that C ∩ Nm ⊂ (idM × f )(C) ∩ Nm . Thus, we have an embedding g(idM × f )|C : C → Nn that is U-close to id and g(idM × f )|C ∩ Nm = (g|(idM × f )(C) ∩ Nm )(idM × f )|C ∩ Nm = id. Therefore, M × I has an (f.d.)cap set N. Since M ≈ M × I, M has an (f.d.)cap set. " !

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Theorem 5.2.3 Every 2Q -manifold (resp. 2f -manifold) M can be embedded in an 2 -manifold and a Q-manifold as a cap set (resp. an f.d.cap set). Proof First, recall that 2Q ≈ rint Q and Qf ≈ 2f (Corollary 5.1.8), and also that 2 ≈ s (the Kadec–Anderson Theorem 3.5.6). Now, as mentioned at the end of previous section, the Open Embedding Theorem 2.5.10 is valid for 2Q -manifolds (resp. 2f -manifolds). Hence, M is homeomorphic to an open set U in rint Q (resp. in s f ). Take an open set V in Q such that U = V ∩ rint Q (resp. U = V ∩ s f ). Then, V is a Q-manifold and V ∩ s is an 2 -manifold. Due to Lemma 5.2.1, U is a cap set (resp. an f.d.cap set) in V and V ∩ s. " ! We show that a subset of an 2 - or Q-manifold is an (f.d.)cap set if it is an (f.d.)cap set locally, that is: Proposition 5.2.4 Let X be an 2 - or Q-manifold and M ⊂ X. If every x ∈ X has an open neighborhood U in X such that M ∩ U is an (f.d.)cap set for U , then M is an (f.d.)cap set for X. Proof We apply Michael’s Theorem 1.3.20 on local properties. Let P be the property of open sets in X defined as follows: U has property P ⇔ M ∩ U is an (f.d.)cap set for U. def

It suffices to show that property P is G-hereditary, where (G-1) is Lemma 5.2.1. Because of separability of X, any discrete collection of open sets is countable. Then, (G-3) can be easily proved. To see (G-2), let U and V be open sets in X such that M ∩ U and M ∩ V are (f.d.)cap sets for U and V , respectively. On the other hand, U ∪V has an (f.d.)cap set N by Proposition 5.2.2. We will construct a homeomorphism h : U ∪ V → U ∪ V such that h(N) = M ∩ (U ∪ V ), which implies that M ∩ (U ∪ V ) is an (f.d.)cap set for U ∪ V . Take an open cover U ∈ cov(U ) that is fitting in U ∪ V (Lemma 2.1.16). Since N ∩ U is an (f.d.)cap set for V by Lemma 5.2.1, we can apply Theorem 5.1.4 to obtain a homeomorphism f : U → U such that f (N ∩ U ) = M ∩ U and f is U-close to id. Since U is fitting in U ∪ V , f can be extended to a homeomorphism f˜ : U ∪ V → U ∪ V by f˜|V \ U = id. Then, we have f˜(N) = (M ∩ U ) ∪ (N ∩ V \ U ), which is an (f.d.)cap set for U ∪ V . In the same way as above, we can obtain a homeomorphism g˜ : U ∪ V → U ∪ V such that g|U ˜ \ V = id. g˜ f˜(N) = (M ∩ V ) ∪ (f˜(N) ∩ U \ V ) = (M ∩ V ) ∪ (M ∩ U \ V ) = M ∩ (U ∪ V ). Then, h = g˜ f˜ is the desired one.

" !

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371

Now we can prove the following characterization: Theorem 5.2.5 Let M be an 2 -manifold (resp. a Q-manifold) and N ⊂ M. (1) (M, N) is an (2 , 2f )-manifold pair (resp. a (Q, Qf )-manifold pair) if and only if N is an f.d.cap set for M. (2) (M, N) is an (2 , 2Q )-manifold pair (resp. a (Q, rint Q)-manifold pair) if and only if N is a cap set for M. Proof Because of similarity, we give only the proof of (1). First, assume that (M, N) is an (2 , 2f )-manifold pair (resp. a Q-manifold). Then, every x ∈ M has an open neighborhood U in M such that N ∩ U is an f.d.cap set for U . It follows from Proposition 5.2.4 that N is an f.d.cap set for M. Conversely, assume that N is an f.d.cap set for M. Then, every x ∈ M has an open neighborhood U in M that is homeomorphic to an open set V in 2 (resp. Q). Then, 2f ∩ V (resp. Qf ∩ V ) is an f.d.cap set for V by Lemma 5.2.1. By virtue of Theorem 5.1.4, (U, N ∩ U ) ≈ (V , 2f ∩ V ) (resp. (U, N ∩ U ) ≈ (V , Qf ∩ V )), which means that (M, N) is an (2 , 2f )-manifold pair (resp. a (Q, Qf )-manifold pair). " ! Using the Open Embedding Theorem 2.5.10 and the Topological Uniqueness of (f.d.)cap sets for 2 -manifolds (Theorem 5.1.4), we can easily obtain the following: Theorem 5.2.6 (OPEN EMBEDDING) For each (2 , 2f )-manifold pair (resp. (2 , 2Q )-manifold pair) (M, N), there exists an open embedding g : M → 2 such that g(N) = g(M) ∩ 2f (resp. g(N) = g(M) ∩ 2Q )). " ! Combining the Classification Theorem 2.6.1 and Corollary 5.1.5, we also have the following: Theorem 5.2.7 (CLASSIFICATION) Let (M, N) and (X, Y ) be (2 , 2f )-manifold pairs (or (2 , 2Q )-manifold pairs). Then, (M, N) ≈ (X, Y ) if and only if M  X or N  Y . " ! The following is a combination of the Classification Theorem 3.8.8 and Corollary 5.1.5: Theorem 5.2.8 (CLASSIFICATION) Let (M, N) and (X, Y ) be (Q, Qf )manifold pairs (or (Q, rint Q)-manifold pairs). Then, (M, N) ≈ (X, Y ) if and only if M and X are CE equivalent. " ! For the products, we have the following: Lemma 5.2.9 Let M be a cap set for a metrizable space X. If Y is a separable locally compact ANR, then M × Y is a cap set for X × Y .  ⊂ · · · is a tower of compact Z-sets in Proof Let M = n∈N Mn , where M1 ⊂ M2  X that has the cap for X. We can write Y = n∈N Yn , where each Yn is a compact set in Y and Yn ⊂ int Yn+1 . Then, M × Y = n∈N (Mn ∩ Yn ) and each Mn × Yn is a compact Z-set in X × Y . We show that the tower (Mn ∩ Yn )n∈N has the cap for

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X × Y . Here, we use the metric d for X × Y defined as follows: d((x, y), (x  , y  )) = max{dX (x, x  ), dY (y, y  )}, where dX ∈ Metr(X) and dY ∈ Metr(Y ). Let C be a compact set in X × Y , m ∈ N, and ε > 0. Then, we have n1 ∈ N and an embedding f : pr1 (C) → Mn1 such that n1 > m, f |pr1 (C) ∩ Mm = id and dX (f, id) < ε. Choose n ∈ N so that n  n1 and pr2 (C) ⊂ Yn . Since C ⊂ pr1 (C) × pr2 (C), we have an embedding f × id|C : C → Mn × Yn , where f × id|C ∩ (Mm × Ym ) = id and d(f × id, id) < ε. " ! The above proof cannot be applied to the f.d.cap set case because the finitedimensionality of C does not imply that of pr1 (C) even if Y is finite-dimensional. Lemma 5.2.10 Let M be an f.d.cap set for an 2 - or Q-manifold X. If Y is a separable locally finite-dimensional locally compact ANR, then M × Y is an f.d.cap set for X × Y .  Proof As in the proof of Lemma 5.2.9, let M = n∈N Mn , where M1 ⊂ M2 ⊂ · · · is a tower of finite-dimensional compact Z-sets in X that has the f.d.cap for X. We  can write Y = n∈N Yn , where each Yn is a finite-dimensional compact set in Y and Yn ⊂ int Yn+1 . Then, M × Y = n∈N (Mn ∩ Yn ) and each Mn × Yn is a finitedimensional compact Z-set in X × Y (the Product Theorem 1.12.12). We show that the tower (Mn ∩ Yn )n∈N has the f.d.cap for X × Y . We use the same metric as in the proof of Lemma 5.2.9. Let C be a finite-dimensional compact set in X × Y , m ∈ N, and ε > 0. By Proposition 2.1.17, we have an open cover U of X \ Mm such that mesh U < ε and U is fitting in X. Since X \ Mm is an 2 - or Q-manifold, there exists an closed embedding f : C \ (Mm × Y ) → X \ Mm that is U-close to pr1 (the Strong Universality Theorem 2.3.15 or 2.10.11). Since U is fitting in X, f can be extended to a map f˜ : C → X by f˜|C∩(Mm ×Y ) = pr1 |C∩(Mm ×Y ). Then, dX (f˜, pr1 ) < ε and f˜(C) = f (C \ (Mm × Y )) ∪ (pr1 (C) ∩ Xm ). From the Addition Theorem 1.12.11, it follows that f˜(C) is a finite-dimensional. By the f.d.cap of (Mn )n∈N for X, there exist n1 ∈ N and an embedding g : f˜(C) → Mn1 such that n1 > m and g|f˜(C) \ Mm = id. Choose n ∈ N so that n  n1 and C ⊂ X × Yn . We can define an embedding h : C → Mn × Yn by h(x) = (g f˜(x), pr2 (x)) for each x ∈ C. Obviously, d(h, id) < ε. For each x ∈ C ∩ (Mm × Ym ), it follows that h(x) = (g f˜(x), pr2 (x)) = (f˜(x), pr2 (x)) = (pr1 (x), pr2 (x)) = x,

5.2 Manifold Pairs Modeled on the Pair of 2 (or Q) andits (F.D.)Cap Set

hence h|C ∩ (Mm × Ym ) = id.

373

" !

The following is obtained from Lemmas 5.2.9 and 5.2.10 above: Proposition 5.2.11 Let K be a countable locally finite simplicial complex and let M be an (f.d.)cap set for 2 or Q. Then, |K| × M is an (f.d.)cap set for |K| × 2 or |K| × Q. " ! In the above, |K| × 2 and |K| × Q are respectively an 2 -manifold and a Q-manifold. Combining Proposition 5.2.11, Theorem 5.1.4, and the Triangulation Theorems 2.6.6 and 4.7.1, we have the following Triangulation Theorem: Theorem 5.2.12 (TRIANGULATION) Let (E, F ) be one of (2 , 2f ), (2 , 2Q ), (Q, Qf ), or (Q, rint Q). For each (E, F )-manifold pair (M, N), there exists a countable locally finite simplicial complex K such that (M, N) ≈ (|K| × E, |K| × F ).

" !

The following Triangulation Theorem for 2Q -manifolds can also be obtained by combining Theorems 5.2.12 and 5.2.3. Theorem 5.2.13 (TRIANGULATION) Each 2Q -manifold is homeomorphic to |K| × 2Q for some countable locally finite simplicial complex K. " ! The Triangulation Theorem for 2f -manifolds (Theorem 5.1.11) can also be proved in the same way. We can generalize Proposition 5.1.3(4), (7), and Corollary 5.1.7(1), (3) as follows: Theorem 5.2.14 If M is an f.d.cap set for an 2 - or Q-manifold X, then M × Q is a cap set for X × Q. In other words, if (X, M) is an (2 , 2f )- or (Q, Qf )-manifold pair, then (X × Q, M × Q) is an (2 , 2Q )- or (Q, rint Q)-manifold pair. Proof Because of the similarity, we only give the proof for the 2 -manifold case. Each point x ∈ X has an open neighborhood U in X that is homeomorphic to an open set in 2 . Due to Lemma 5.2.1, M ∩ U is an f.d.cap set for U . Because 2 ≈ s, we have an open embedding f : U → s. Then, s f ∩ f (U ) is an f.d.cap set for f (U ) by Lemma 5.2.1. By virtue of Corollary 5.1.5, we have (U, M ∩ U ) ≈ (f (U ), s f ∩ f (U )), hence     U × Q, (M ∩ U ) × Q ≈ f (U ) × Q, (s f ∩ f (U )) × Q . Since s f × Q is a cap set for s × Q by Proposition 5.1.3(7), the following is a cap set for f (U ) × Q by Lemma 5.2.1: (s f × Q) ∩ (f (U ) × Q) = (s f ∩ f (U )) × Q.

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Therefore, the following is a cap set for U × Q: (M × Q) ∩ (U × Q) = (M ∩ U ) × Q. We can apply Proposition 5.2.4 to obtain the result.

" !

2 -manifold

Note In the above theorem, the case can also be obtained by combining the Open Embedding Theorem 5.2.6, Lemma 5.2.1, Corollary 5.1.5, and Proposition 5.1.3(7).

Combining Theorems 5.2.14 and 5.2.3, we have the following: Corollary 5.2.15 For every 2f -manifold M, M × Q is an 2Q -manifold.

" !

In the rest of this section, we show the following: Proposition 5.2.16 Let M be an (f.d.)cap set for an 2 - or Q-manifold X. (1) For any Z-set A, M \ A is an (f.d.)cap set for X. (2) If L is a countable union of (finite-dimensional) compact Z-sets and M ⊂ L, then L is an (f.d.)cap set for X.  Proof Let M = n∈N Mn , where M1 ⊂ M2 ⊂ · · · is a tower of (finitedimensional) compact Z-sets in X that has the (f.d.)cap for X. Take d ∈ Metr(X). −1  (1): For each n ∈ N, let Ln = {x ∈ Mn | d(x, A)  n }. Then, M \ A = n∈N Ln and each Ln is a (finite-dimensional) compact Z-set in X as a closed subset of Mn . Let C be a (finite-dimensional) compact Z-set in X, m ∈ N, and ε > 0. Since A is a Z-set in X, X \ A is homotopy dense in X (Theorem 2.8.6), that is, there is a homotopy h : X × I → X such that h0 = id and h(X × (0, 1]) ⊂ X \ A. Since C is compact, we can find δ > 0 such that diam h({x} × [0, δ]) < ε/2 for every x ∈ C. We define a map λ : C → [0, δ] as follows: λ(x) = min{δ, d(x, Lm )} for each x ∈ C. Then, we have a map f  : C → X\A defined by f  (x) = h(x, λ(x)) for each x ∈ C, where f  ε/2 id and f  |C ∩ Lm = id. By the Strong Universality Theorem 2.9.6 or 2.10.10, we can easily obtain a Z-embedding f : C → X \ A such that f |C ∩ Lm = id and d(f, f  ) < ε/2. Using the (f.d.)cap of (Mn )n∈N , we can obtain n > m and an embedding g : f (C) → Mn such that g|f (C) ∩ Mm = id and d(g, id) < min{ε/2, dist(f (C), A)}. Then, it follows that d(gf, id)  d(g, id) + d(f, id) < ε/2 + ε/2 = ε.

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Since C ∩ Lm ⊂ f (C) ∩ Mm , we have gf |C ∩ Lm = g|C ∩ Lm = id. For each x ∈ C, d(gf (x), f (x))  d(g, id) < dist(f (C), A)  d(f (x), A)  d(f (x), gf (x)) + d(gf (x), A), hence d(gf (x), A) > 0. Since C is compact, we can find n  n such that d(fg(x), A) > n−1 for every x ∈ C. Then, fg(C) ⊂ Ln . Thus, we have an embedding gf : C → Ln such that gf |C ∩ Lm = id and d(gf, id) < ε. (2): First, we will show that if C is a (finite-dimensional) compact Z-set in X, then M ∪ C is an (f.d.)cap set for X. From the (f.d.)cap of (Mn )n∈N , there exists an embedding h : C → Mn for some n ∈ N such that h  id. Since Mn is a Z-set in X, h is a Z-embedding. By the Z-Set Unknotting Theorem 2.9.7 or 2.11.6, we ˜ can extend h to a homeomorphism h˜ : X → X. It follows from (2) that M \ h(C) ˜ ˜ ˜ and h(M) \ h(C) are (f.d.)cap sets for X, which are (f.d.)cap sets for X \ h(C) by ˜ Lemma 5.2.1. Taking an open cover of X \ h(C) fitting in X (Lemma 2.1.16), we can apply the Topological Uniqueness of (f.d.)cap sets (Theorem 5.1.4) to obtain a ˜ ˜ ˜ ˜ homeomorphism g : X → X such that g(h(M) \ h(C)) = M \ h(C) and g|h(C) = id. Thus, we have a homeomorphism g h˜ : X → X such that ˜ ˜ ˜ ˜ g h(M ∪ C) = g(h(M) \ h(C)) ∪ g h(C)) ˜ ˜ ˜ = (M \ h(C)) ∪ h(C) = M ∪ h(C) = M, ˜ where the last equality comes from h(C) ⊂ M. Consequently, M ∪ C is an (f.d.)cap set for X.  Using the above special case, we show the general case. Let L = n∈N Ln , where N1 ⊂ N2 ⊂ · · · are (finite-dimensional) compact Z-sets in X. We will construct homeomorphisms hn : X → X, n ∈ N, so as to satisfy the following conditions: (1) hn (M) = M ∪ Ln , −1 −n (2) d(hn , hn−1 ) < 2−n and d(h−1 n , hn−1 ) < 2 , −1 (3) hn |Mn−1 ∪ hn−1 (Ln−1 ) = hn−1 |Mn−1 ∪ h−1 n−1 (Ln−1 ), where d ∈ Metr(X) is complete and M0 = L0 = ∅. Then, a homeomorphism h : X → X can be obtained as the uniform limit of (hn )n∈N , whose inverse h−1 is the uniform limit of (h−1 n )n∈N . Observe that   hn (Mn ∪ h−1 h(M) = h n (Ln )) n∈N Mn ) = =

 n∈N

n∈N

hn (Mn ) ∪



n∈N

Ln = L.

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Hence, L is an (f.d.)cap set for X. Now, assuming that hn−1 has been obtained, we define hn as follows: For the sake of simplicity, let A = Mn−1 ∪ h−1 n−1 (Ln−1 ) and B = hn−1 (Mn−1 ) ∪ Ln−1 = hn−1 (A). From Lemma 5.2.1 and Proposition 5.2.16, M \A and (M ∪Ln )\B are (f.d.)cap sets for X \ A and X \ B, respectively. Let U be an open cover of X \ B fitting in X that refines the covers {Bd (x, 2−n−1 ) | x ∈ X\A} and {hn−1 (Bd (x, 2−n−1 )) | x ∈ X\B} −n (cf. Proposition 2.1.17), which means mesh U < 2−n and mesh h−1 n−1 (U) < 2 .  By Corollary 5.1.5, we can obtain a homeomorphism h : X \ A → X \ B such that h (M \ A) = (M ∪ Ln ) \ B and h is U-close to hn−1 . Since U is fitting in X, we can extend h to a homeomorphism hn : X → X so as to satisfy (3), that is, hn |A = hn−1 |A. Then, we show (1) as follows: hn (M) = h (M \ A) ∪ hn−1 (A) = ((M ∪ Ln ) \ B) ∪ B = M ∪ Ln . Moreover, d(hn , hn−1 ) < 2−n and −1 −1 −1 −1 −n d(h−1 n , hn−1 ) = d(id, hn−1 hn ) = d(hn−1 hn−1 , hn−1 hn ) < 2 .

Thus, (2) is also satisfied.

" !

5.3 Absorption Property and Absorption Bases In the previous two sections, introducing (f.d.)cap sets for Hilbert space 2 or 2 manifolds, we characterized the pairs (2 , 2f ) and (2 , 2Q ), and manifold pairs modeled on these pairs. We want to generalize results to the pairs (2 (), 2f ()) and (2 () × 2 , 2f () × 2Q ), and manifold pairs modeled on these pairs, where it should be noted that (2 × 2 , 2f × 2Q ) ≈ (2 , 2Q ). An (f.d.)cap set is defined as a set with a tower of members of M0 (or Mfd 0 ) that has the (finite-dimensional) compact absorption property (cap) (or (f.d.cap)). Although  (cap) and (f.d.cap) are properties of a tower M1 ⊂ M2 ⊂ · · · in X with M = n∈N Mn , we can translate them into properties of a subset M ⊂ X itself. In this section, we introduce the absorption property for a general class C other than M0 (or Mfd 0 ). As mentioned in Fact (3) on p. 354, it should be noted that any non-separable metrizable space has no (f.d.)cap sets. Hereafter, in this chapter: •

let C be a closed hereditary, additive topological class of metrizable spaces,

where C is said to be:

5.3 Absorption Property and Absorption Bases

377

– topological if C contains any space homeomorphic to a member of C; – closed hereditary if C contains any closed subspace of a member of C; – additive if C contains any space that is the union of two closed subspaces belonging to C. It should be remarked that C contains singletons because it is closed hereditary. Hence, C also contains every finite discrete space because it is additive. For a class C (⊂ M), we will use the following notation: • C(τ ) — the subclass of C consisting of spaces of weight  τ ; • C(ℵ0 ) — the subclass of C consisting of separable spaces; • Cσ — the class consisting of all metrizable spaces that can be expressed as countable unions of closed subspaces belonging to C; • C(n) — the subclass of C consisting of spaces of dimension  n; • Cfd — the subclass of C consisting of finite-dimensional spaces, i.e., Cfd = n∈ω C(n) ; • Cscd — the subclass of C consisting of strongly countable-dimensional spaces; • ⊕C — the class consisting of topological sums (spaces expressed as discrete unions) of members of C; • τ C — the class of topological sums (spaces expressed as discrete unions) of at most τ many members of C.6 These classes are closed hereditary and topological. They are also additive, except for the last two classes, ⊕C and τ C. In fact, ℵ0 M0 is not additive. For example, X=

 n∈ω

[2n, 2n + 1], Y =



[2n − 1, 2n] ∈ ℵ0 M0 ,

n∈N

but X ∪ Y = R+ ∈ ℵ0 M0 . Evidently, Cσ , τ C, (τ C)σ ⊂ M(τ ) if C ⊂ M(τ ). The class (M0 )σ is the class consisting of all σ -compact metrizable spaces, M0 (ℵ0 ) = M0 and (M0 )σ (ℵ0 ) = (M0 )σ . It should be noted that (Cσ )fd = (Cfd )σ = (Cσ )scd . The last equality can be shown asfollows: Evidently, (Cfd )σ ⊂ (Cσ )scd . Every X ∈ (Cσ )scd can be written  as X = i∈N Xi = n∈NAn , where each Xi and An is closed in X, Xi ∈ C, and dim An < ∞. Then, X = i,n∈N (Xi ∩ An ), where each Xi ∩ An is closed in X and dim Xi ∩An  dim An < ∞ by the Subset Theorem 1.12.7. Moreover, Xi ∩An ∈ C because C is closed hereditary. Thus, Xi ∩An ∈ Cfd for each i, n ∈ N, which implies that X ∈ (Cfd )σ . As is easily observed, (Cσ )scd = (Cscd )σ , which allows us to write this without parentheses, Cscd σ .

6 In

general, (⊕C)(τ ) ⊂ τ C. Note that (⊕C)(τ ) = τ C if and only if C = C(τ ).

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

For a metrizable space X, we define    CZ (X) = C ∈ C  C is a Z-set in X . It is said that M ⊂ X is C-absorptive in X if it satisfies the following condition called the C-absorption property: (abC )

If C ∈ CZ (X), D ⊂ M ∩ C is closed in X and U ∈ cov(X), then there is a Z-embedding f : C → X such that f is U-close to id, f |D = id, and f (C) ⊂ M.7

The following can be easily shown: Fact (1) If every singleton in X is a Z-set, then every C-absorptive set in X is dense in X.8 (2) If M ⊂ X is C-absorptive and f : X → Y is a homeomorphism, then f (M) is C-absorptive in Y . Note that the M0 - (or Mfd 0 -)absorption property is a property for subsets but the (finite-dimensional) compact absorption property (cap) (or (f.d.cap)) is a property for towers of subsets. However, we have the following: Proposition 5.3.1 Let X be an 2 - or Q-manifold. Then, every (f.d.)cap set M in X is M0 - (or Mfd 0 -)absorptive in X.  Proof Let M = n∈N Mn , where M1 ⊂ M2 ⊂ · · · is a tower of (finitedimensional) compact Z-sets in X that has the (f.d.)cap for X. Let C ∈ M0 (or ∈ Mfd 0 ) be a Z-set in X and D ⊂ M ∩ C a closed subset. For each U ∈ cov(X), choose d ∈ Metr(X) so that {Bd (x, 1) | x ∈ X} ≺ U (Proposition 1.3.22(1)). We  can take a tower C1 ⊂ C2 ⊂ · · · of closed sets in C such that C \ D = n∈N Cn . Then, we can inductively obtain 1 = n(1) < n(2) < · · · ∈ N and Z-embeddings hi : D ∪ Ci → X, i ∈ N, such that hi (Ci ) ⊂ Mn(i) , hi |D = id, hi |Ci−1 = hi−1 |Ci−1 , and d(hi , hi−1 ) < 2−i , where h0 = id. Indeed, assume that hi−1 has been obtained. Since hi−1 (Ci ) misses D = hi−1 (D), we can choose   0 < δ < min 2−(i+1) , dist(hi−1 (Ci ), D) .

the C-absorption property was defined by replacing a Z-embedding f : C → X in (abC ) with a homeomorphism f : X → X. However, when X is an 2 ()-manifold, it will be shown in Lemma 5.3.2 that these are coincident. 8 It will be shown in Proposition 5.3.5 that if Mfd ⊂ C, then every C-absorptive set in X is 0 homotopy dense in X. 7 In [150],

5.3 Absorption Property and Absorption Bases

379

Applying the (f.d.)cap for X, we have n(i) > n(i − 1) and an embedding f : hi−1 (Ci ) → Mn(i) such that f |hi−1 (Ci ) ∩ Mn(i−1) = id and f δ id. Because hi−1 (Ci−1 ) ⊂ Mn(i−1) , we have f |hi−1 (Ci−1 ) = id. Since d(f, id) < δ, it follows that D∩f (hi−1 (Ci )) = ∅. Hence, we can define a Z-embedding g : B ∪Ci → X by g|B = id and g|Ci = f hi−1 |Ci . Since g δ hi−1 |D ∪ Ci , applying the Homotopy Extension Theorem 1.13.11, we can extend g to a map g˜ : hi−1 (C) → X such that g˜ δ hi−1 . By the Strong Universality of 2 - or Q-manifolds (Theorem 2.9.6 or 2.10.10), we can obtain a Z-embedding hi : C → X such that hi |D∪Ci = g|D∪ ˜ Ci = g and d(hi , hi−1 ) < δ  2−i . Then, hi (Ci ) = g(Ci ) = f hi−1 (Ci ) ⊂ Mn(i) , hi |D = g|D = id, and hi |Ci−1 = g|Ci−1 = f hi−1 |Ci−1 = hi−1 |Ci−1 . Because C = D ∪ i∈N Ci and hj |Ci = hi |Ci for j > i, we can define a map h : C → X by h|D = id and h|Ci = hi for i ∈ N. The continuity follows from the fact that h is the uniform limit of hi . In fact, d(h, hi ) < 2−i for each i ∈ N because if x ∈ D ∪ Ci then h(x) = hi (x), and if x ∈ Cj , j > i, then h(x) = hj (x) and d(hj (x), hi (x))  d(hj , hj −1 ) + · · · + d(hi+1 , hi ) < 2−j + · · · + 2−(i+1) < 2−i . In particular, d(h, id) = d(h, h0 ) < 1, which implies that h is U-close to id. For each x = y ∈ A, we can find i ∈ N such that x, y ∈ D ∪ Ci . Then, h(x) = hi (x) = hi (y) = h(y). Hence, h is injective, which means that h is an embedding because C is compact. Moreover, h(C \ D) =

 i∈N

h(Ci ) ⊂



Mn(i) = M.

i∈N

" !

Thus, we have the result.

Remark 5.2 In the above proof, when M is an f.d.cap set for X, it is enough to assume that dim C \ D < ∞ instead of dim C < ∞. It is said that M ⊂ X is strongly C-absorptive in X if it satisfies the following stronger condition than (abC ), which is called the strong C-absorption property: (sabC )

If C ∈ CZ (X), D ⊂ M ∩ C is closed in X, and U is an open neighborhood of C in X with U ∈ cov(U ), then there is a homeomorphism f : X → X such that f is U-isotopic to id (hence f |X \ U = id), f |D = id, and f (C) ⊂ M.

Lemma 5.3.2 Every C-absorptive set M in an 2 ()-manifold X is strongly Cabsorptive in X. Proof Let C, D, U , and U be as in (sabC ). Take V ∈ cov(U ) so that st V ≺ U and st V is fitting in X (cf. Proposition 2.1.17). Moreover, take an open neighborhood V of C in X so that cl V ⊂ U , and W ∈ cov(X) so that W ≺ V[C] ∪ {X \ cl V }. h¯

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Since M is C-absorptive, we have a Z-embedding f : C → X such that f is Wclose to id, f |D = id, and f (C) ⊂ M. Then, f V id. We can apply the Z-Set Unknotting Theorem 2.9.7 to extend f to a homeomorphism f˜ : U → U such that f˜ is st V-isotopic to id. Since st V is fitting in X, f˜ can be extended over X. Thus, we have the desired homeomorphism of X onto itself. " ! Remark 5.3 Lemma 5.3.2 is valid even when X is a Q-manifold. Indeed, in the above proof, V can be taken so that V is locally finite in U and each member of V has the compact closure in X. Then, the homotopy f V id is proper by p Proposition 1.3.9, that is, f V id. Hence, we can apply the Z-Set Unknotting Theorem 2.11.6 instead of 2.9.7. The following lemma is useful and will be used frequently: Lemma 5.3.3 Let X = (X, d) be a complete metric space. Given a sequence (fn )n∈N of homeomorphisms of X onto itself and a sequence (Un )n∈N of open covers of X with the following conditions: (i) fn is Un -close to idX , (ii) mesh Un < 2−n and mesh f1−1 · · · fn−1 (Un+1 ) < 2−n , the sequence (fn · · · f1 )n∈N converges to a homeomorphism f : X → X with d(f, idX ) < 1. Proof Recall that C(X, X) is complete with respect to the sup-metric. Since (fn · · · f1 )n∈N is uniformly Cauchy, it converges to a map f ∈ C(X, X), where d(f, idX ) 

∞  n=1

d(fn , idX )
0, by the continuity of f at g(x), there is δ > 0 so that d(g(x), x  ) < δ implies d(fg(x), f (x  )) < ε/2. Choose m ∈ N so that n  m implies d(f, fn · · · f1 ) < ε/2 and d(g, f1−1 · · · fn−1 ) < δ.

5.3 Absorption Property and Absorption Bases

381

Then, it follows that d(fg(x), x)  d(fg(x), f (f1−1 · · · fn−1 (x))) + d(f (f1−1 · · · fn−1 (x)), fn . . . f1 (f1−1 · · · fn−1 (x))) < ε/2 + ε/2 = ε. This means that fg(x) = x. Thus, we have fg = idX . In the same way as above, we can see that gf = idX . " ! Lemma 5.3.4 Let M be a C-absorptive set in an 2 ()-manifold X. For each Zσ -set C ∈ Cσ in X, each open set W in X and U ∈ cov(W ), there exists a homeomorphism f : X → X such that f is U-close to id (so f |X \ W = id) and f (C ∩ W ) ⊂ M. Proof By Proposition 2.1.17, we may assume that U is fitting in X. Since W is completely metrizable (1.3.16), W has an admissible complete metric such that {Bd (x, 1) | x ∈ W } ≺ V (1.3.22(1)). Then, it suffices to construct a homeomorphism f : W → W such that f (C ∩ W ) ⊂ M ∩ W and d(f, id) < 1. Since C ∈ Cσ is a Zσ -set in X, C is closed hereditary and additive, and W is Fσ in X, we can write C ∩ W = n∈N Cn , where C1 ⊂ C2 ⊂ · · · and Cn ∈ CZ (X) for every n ∈ N. Because the class C is topological, using Lemma 5.3.2, we can inductively define homeomorphisms fn : X → X and Un ∈ cov(W ), n ∈ N, so as to satisfy the following conditions: (1) fn is Un -close to id (which implies that fn |X \ U = id), (2) fn |fn−1 · · · f1 (Cn−1 ) = id, fn (fn−1 · · · f1 (Cn )) ⊂ M, (3) mesh Un < 2−n and mesh f1−1 · · · fn−1 (Un+1 ) < 2−n , where C0 = ∅ and f0 = id, i.e., the second condition for n = 1 means f1 (C1 ) ⊂ M. Then, by Lemma 5.3.3, the sequence (fn · · · f1 |U )n∈N converges to a homeomorphism f : W → W with d(f, id) < 1. For each m > n, since fm · · · fn+1 |fn · · · f1 (Cn ) = id, it follows that f |Cn = fn· · · f1 |Cn . Hence, f (Cn ) = fn · · · f1 (Cn ) ∩ f (W ) ⊂ M ∩ W . Since C ∩ W = n∈N Cn , we have f (C ∩ W ) ⊂ M ∩ W . " ! Remark 5.4 Lemma 5.3.4 is valid even when X is a Q-manifold because so is Lemma 5.3.2 (cf. Remark 5.3). Proposition 5.3.5 Let X be an 2 ()-manifold. If Mfd 0 ⊂ C, then every Cabsorptive set M in X is homotopy dense in X. Proof To show that M is homotopy dense in X, we verify condition (b) in Corollary 1.15.5. Let U be an open set in X and let α : Sn−1 → U ∩ M be a map that extends to a map β : Bn → U . Since α(Sn−1 ) is a Z-set in U , the complement U \ α(Sn−1 ) is homotopy dense in U . Hence, we can assume that β(Bn \ Sn−1 ) ⊂ U \ α(Sn−1 ). Take an open cover V of U \ α(Sn−1 ) so that st V is fitting in U (cf. Proposition 2.1.17). By the strong M1 -universality of 2 ()-

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

manifolds, we have a (strong) Z-embedding γ : Bn \ Sn−1 → U \ α(Sn−1 ) that is V-close to β|Bn \ Sn−1 . Using Lemma 5.3.4, we can obtain a homeomorphism f : U \ α(Sn−1 ) → U \ α(Sn−1 ) such that f γ (Bn \ Sn−1 ) ⊂ U ∩ M and f is V-close to id. Since f γ : Bn \ Sn−1 → U ∩ M is st V-close to β, f γ can be extended to a map α˜ : Bn → U ∩ M defined by α|S ˜ n−1 = β|Sn−1 = α, which is an extension of α. Thus, it follows from Corollary 1.15.5 that M is homotopy dense in X. " ! Remark 5.5 In the above proof, the strong M1 -universality can be replaced with the proper L-universality. Hence, Proposition 5.3.5 is valid even when X is a Qmanifold because so is Lemma 5.3.4 (cf. Remark 5.4). Remark 5.6 Since every n-dimensional compactum can be embedded into I2n+1 (Embedding Theorem 1.12.14), we have the following: n Mfd 0 ⊂ C ⇔ C contains n-cubes I , n ∈ N. 2 Corollary 5.3.6 Suppose that Mfd 0 ⊂ C. Every C-absorptive set M in an  ()manifold X is an ANR, every Z-set in M is a strong Z-set in M and every compact set in M is a strong Z-set in M.

Proof By Proposition 5.3.5 above, M is homotopy dense in X. Since X is an ANR, it follows from Theorem 1.13.26 that M is an ANR. Since every Z-set in X is a strong Z-set in X (Theorem 2.9.3), it follows from Proposition 2.9.4 that every Zset in M is a strong Z-set in M. Recall that every compact set in an 2 ()-manifold is a strong Z-set (Proposition 3.2.10 or Corollary 3.3.14 with Toru´nczyk Characterization Theorem 3.4.1). Hence, we can apply the argument in the proof of Proposition 2.9.4 to prove that every compact set in M is a strong Z-set. The details are left to the reader. " ! Recall that every Z-set in each Q-manifold is a strong Z-set because of the local compactness of Q-manifolds (Proposition 2.8.1). As seen in Remark 5.5, Proposition 5.3.5 is valid for a Q-manifold. Moreover, applying Theorem 1.13.26 as in the above proof, we can prove the following: Proposition 5.3.7 Suppose that Mfd 0 ⊂ C. Every C-absorptive set M in a Qmanifold X is homotopy dense in X and it is an ANR. " ! Now, we can prove the following characterization of C-absorptive sets in 2 ()manifolds: Theorem 5.3.8 Let X be an 2 ()-manifold. For M ⊂ X, the following are equivalent: (a) M is C-absorptive in X. (b) M ∩ W is C-absorptive in every open set W ⊂ X.

5.3 Absorption Property and Absorption Bases

383

(c) For each open set U in X, V ∈ cov(U ), and each Zσ -set C in U with C ∈ Cσ , there exists a homeomorphism f : U → U such that f is V-close to id and f (C) ⊂ M ∩ U . (d) For each C ∈ CZ (X), each open set U in X and V ∈ cov(U ), there exists a homeomorphism f : U → U such that f is V-close to id and f (C ∩ U ) ⊂ M ∩ U. Proof The implication (b) ⇒ (a) is obvious. The implications (a) ⇒ (d) and (b) ⇒ (c) are direct consequences of Lemma 5.3.4 above. Since every Zσ -set in an open set U ⊂ X is a Zσ -set in X, the implication (a) ⇒ (c) also follows from Lemma 5.3.4. Moreover, in condition (d), C ∩ U is a Zσ -set in U and C ∩ U ∈ Cσ because U is Fσ in X and C is closed hereditary. So, (c) ⇒ (d) is trivial. Thus, we only have to show the implications (d) ⇒ (a) and (c) ⇒ (b). (a) (5.3.4)

triv.

(b) (5.3.4)

(5.3.4)

(d)

triv.

(c)

(d) ⇒ (a) (or (c) ⇒ (b)): Let C ∈ CZ (X) (or ∈ CZ (W )) with D ⊂ M ∩ C a closed set in X (where C \ D ∈ Cσ is Fσ in X). For each open neighborhood U of C in X (or in W ) with V ∈ cov(U ), take an open cover W of the open set U \ D such that W ≺ V|(U \ D) and W is fitting in X (Proposition 2.1.17). By (d) (or (c)), we have a homeomorphism f : U \ D → U \ D such that f is W-close to id and f (C ∩ (U \ D)) ⊂ M. Then, f extends to a homeomorphism f˜ : X → X (or f˜ : W → W ) by f˜|X \ (U \ D) = id (or f˜|W \ (U \ D) = id), whence f˜ is V-close to id, f |D = id and f (C) ⊂ M. " ! Remark 5.7 Theorem 5.3.8 is also valid for a Q-manifold X because Lemma 5.3.4 is valid for a Q-manifold X (Remark 5.4). Corollary 5.3.9 Every C-absorptive set M in an 2 ()-manifold X is τ Cabsorptive in X.  Proof Let C = λ∈ Cλ ∈ (τ C)Z (X), where card   τ and Cλ ∈ C for each λ ∈ . For a closed subset D ⊂ C ∩ M and each λ ∈ , let Dλ = D ∩ Cλ . Then, Cλ ∈ CZ (X) and Dλ is a closed subset of Cλ ∩ M. We have open sets Wλ , λ ∈ , in X such that (Wλ )λ∈ is discrete in X. By Theorem 5.3.8, M ∩ Wλ is C-absorptive in Wλ . For every U ∈ cov(X), there are Z-embeddings fλ : Cλ → X, λ ∈ , such that fλ is U-close to id, fλ |Dλ = id, and fλ (Cλ ) ⊂ M ∩ Wλ . We can define a Z-embedding f : C → X by f |Cλ = fλ for each λ ∈ . Then, f is U-close to id, f |D = id, and fλ (Cλ ) ⊂ M, Thus, M is τ C-absorptive in X. " !

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

To treat non-separable spaces, we need the following: Proposition 5.3.10 Let M be a C-absorptive set in an 2 ()-manifold X. Suppose that  {Uλ | λ ∈ } is a pairwise disjoint open collection in X, L is a Zσ -set in λ∈ Uλ , and L ∩ Uλ ∈ Cσ for each λ ∈ . Then, for every open set W in X and V ∈ cov(W ), there is a homeomorphism f : X → X such that f is V-close to id (so f |X \ W = id) and f (L ∩ W ∩ Uλ ) ⊂ M ∩ W ∩ Uλ for each λ ∈ . Proof Due to Proposition 2.1.17, there exist Vλ ∈ cov(W ∩ Uλ ), λ ∈ , such that Vλ ≺ V and Vλ is fitting in X. Then, 

   Vλ ∈ cov W ∩ λ∈ Uλ ,

λ∈

 which is fitting in X. Since L is a Zσ -set in λ∈ Uλ , each L ∩ Uλ is a Zσ -set in Uλ (Corollary 2.8.7(2)). Applying Lemma 5.3.4, we can obtain homeomorphisms fλ : W ∩ Uλ → W ∩ Uλ , λ  ∈ , such that fλ is Vλ -close to id and fλ (L ∩ W ∩ Uλ ) ⊂ M ∩ W ∩ Uλ . Since λ∈ Vλ is fitting in X, the desired homeomorphism f : X → X can be defined by    f |W ∩ Uλ = fλ , λ ∈ , and f |X \ W ∩ λ∈ Uλ = id.

" !

A C-absorption base M for X is defined as a C-absorptive set M in X that is a  σ -discrete union of members of CZ (X), that is, M = n∈N Mn and each Mn is a discrete union of members of CZ (X). Then, each Mn is a Z-set in X as a discrete union of Z-sets (Corollary 2.8.7(1)), which means that Mn ∈ ⊕CZ (X), so Mn ∈ τ CZ (X) if w(X)  τ . Thus, every M is a Zσ -set in X. Evidently, if M ⊂ X is a C-absorption base for X and f : X → Y is a homeomorphism, then f (M) is a C-absorption base for Y . It is easy to show that if each Mn is a discrete union of at most countably many members of CZ (X), then M is a countable union of members of CZ (X). In particular, when X is separable, then M is a countable union of members of CZ (X). Thus, we have the following: Proposition 5.3.11 Let X be a separable metrizable space and M ⊂ X. Then, M is a C-absorption base for X if and only if M is C-absorptive in X and written as a countable union of members of CZ (X). " ! By the following proposition, a C-absorption base can be defined as a Cabsorptive Zσ -set that belongs to (τ C)σ : Proposition 5.3.12 Let M be a C-absorptive set in a (separable) metrizable space X. Then, M is a C-absorption base for X if and only if M is a Zσ -set in X and M ∈ (τ C)σ (or M ∈ Cσ ). Proof The “only if” part is obvious. We show the “if”  part. Since M is a Zσ -set in X and M ∈ (τ C)σ , we can write M = i∈N Ki = n∈N Mn , where each Ki is a Z-set in X and Mn ∈ τ C for each n ∈ N. Then, each Ki ∩Mn is a Z-set in X because

5.3 Absorption Property and Absorption Bases

385

 it is closed in Ki . For each n ∈ N, Mn = α∈A(n) Cα , where Cα ∈ C, α ∈ A(n), and card A(n)  τ . Then, {Cα | α ∈ A(n)} is discrete in X because it is discrete in Mn and Mn is closed in X. Since each Cα is closed in M, Ki ∩ Cα is closed in Ki , so a Z-set in X. Since C is closed hereditary, it follows that Ki ∩ Cα ∈ CZ (X). Thus, M is a σ -discrete union of members of CZ (X). " ! Combining the above with Corollary 5.3.9, we have: Corollary 5.3.13 Every C-absorption base M for an 2 ()-manifold X is a τ Cabsorption base for X. ! " When an 2 ()-manifold X has a C-absorption base, it is topologically unique, that is: Theorem 5.3.14 (UNIQUENESS OF ABSORPTION BASES) Let M and N be Cabsorption bases for an 2 ()-manifold X. For each open cover U of X, there exists a homeomorphism f : X → X such that f (M) = N and f is U-close to id. Proof First, let M=

 n∈N

Mn , Mn =

 α∈A(n)

Cα , N =

 n∈N

Nn , and Nn =



Dβ ,

β∈B(n)

where {Cα | α ∈ A(n)}, ⊂CZ (X), n ∈ N, are discrete in X. For {Dβ | β ∈ B(n)} ∗ = ∗ = each n ∈ N, let M M and N n n in i  in Ni . As discrete unions of Z-sets,  Mn = α∈A(n) Cα and Nn = β∈B(n) Dβ are Z-sets in X (Corollary 2.8.7(1)). Hence, Mn∗ and Nn∗ are also Z-sets in X. We use a complete metric d ∈ Metr(X) such that {Bd (x, 1) | x ∈ U } ≺ U (1.3.22(1)). By induction, we shall construct homeomorphisms fn , gn : X → X and Un ∈ cov(X), n ∈ N, so as to satisfy the following conditions: −1 fn−1 · · · g0−1 f0 (Mn∗ )) ⊂ N, (1) fn (gn−1 −1 fn−1 · · · g0−1 f0 (M)) (2) gn (Nn∗ ) ⊂ fn (gn−1   −1 fn−1 · · · g0−1 f0 (M)) , i.e., Nn∗ ⊂ gn−1 fn (gn−1 −1 ∗ ) ∪ N∗ (3) fn |gn−1 fn−1 · · · g0−1 f0 (Mn−1 n−1 = id, −1 −1 ∗ = id, (4) gn |fn gn−1 fn−1 · · · g0 f0 (Mn∗ ) ∪ Nn−1 −1 (5) gn fn is st Un -close to id, −1 (6) mesh st Un < 2−n and mesh f1−1 g1 · · · fn−1 gn−1 (st Un ) < 2−n ,

where f0 = g0 = id and M0∗ = N0∗ = ∅. When n = 1, the condition (3) is nothing, whereas the conditions (1), (2), (4) and (6) mean f1 (M1 ) ⊂ N, g1 (N1 ) ⊂ f1 (M), g1 |f1 (M1 ) = id, and mesh st U1 < 1/2, respectively. Suppose that f0 , g0 , . . . , fn−1 , gn−1 have been obtained. Then, we can take Un ∈ cov(X) so as to satisfy the condition (6). Since {Cα | α ∈ A(n)} and {Dβ | β ∈ B(n)} are discrete in X, we have pairwise disjoint open collections {Uα | α ∈ A(n)}

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

and {Vβ | β ∈ B(n)} in X such that Cα ⊂ Uα and Dβ ⊂ Vβ . For each α ∈ A(n), let  −1  ∗ ∗ Uα∗ = Uα \ gn−1 fn−1 · · · g0−1 f0 (Mn−1 ) ∪ Nn−1 . Since N is a C-absorptive in X, we can apply Proposition 5.3.10 to obtain a homeomorphism fn : X → X such that −1 fn (gn−1 fn−1 · · · g0−1 f0 (Cα ) ∩ Uα∗ )) ⊂ N ∩ Uα∗ for each α ∈ A(n),

 fn |X \ α∈A(n) Uα∗ = id, and fn is Un -close to id. Then, fn satisfies (1) and (3). −1 fn−1 · · · g0−1 f0 (M) is C-absorptive. For each β ∈ B(n), let Note that fn gn−1   −1 ∗ . fn−1 · · · g0−1 f0 (Mn∗ ) ∪ Nn−1 Vβ∗ = Vβ \ fn gn−1 Again, applying Proposition 5.3.10, we can obtain a homeomorphism gn : X → X such that −1 gn (Dβ ∩ Vβ∗ ) ⊂ fn gn−1 fn−1 · · · g0−1 f0 (M) ∩ Vβ∗ for each β ∈ B(n),

 gn |X \ β∈B(n) Vβ∗ = id, and gn is Un -close to id. Then, gn satisfies (2) and (4). Since fn and gn is Un -close to id, we have the condition (5). −1 Now, due to Lemma 5.3.3, the sequence (gn−1 fn gn−1 fn−1 · · · g0−1 f0 )n∈N converges to a homeomorphism h : X → X with d(h, id) < 1. Since {Bd (x, 1) | x ∈ U } ≺ U, h is U-close to id. It remains to show that h(M) = N. By (1), (4) and (3), we have −1 h(Mn∗ ) = fn gn−1 fn−1 · · · g0−1 f0 (Mn∗ ) ⊂ N for each n ∈ ω.

Hence, h(M) =

 n∈ω

h(Mn∗ ) ⊂ N. Moreover, by (2), (3) and (4), we have

−1 Nn∗ ⊂ gn−1 fn gn−1 fn−1 · · · g0−1 f0 (M) −1 = gm fm · · · g0−1 f0 (M) for each m > n ∈ N.

Then, it follows that N =



∗ n∈N Nn

⊂ h(M). Consequently, we have h(M) = N. " !

Remark 5.8 When X is a Q-manifold, Theorem 5.3.14 can be proved by the same proof, that is, an M0 - (or Mfd 0 -)absorption base for a Q-manifold X is topologically unique. Using the above theorem, we can show that an M0 - (or Mfd 0 -)absorption base for an 2 -manifold X is none other than an (f.d.)cap set for X, that is:

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387

Proposition 5.3.15 Let X be an 2 - or Q-manifold. Then, M ⊂ X is an M0 - (or Mfd 0 -)absorption base for X if and only if M is an (f.d.)cap set for X. Proof First, we show the “if” part. As seen in Proposition 5.3.1, an (f.d.)cap set M for X is M0 - (or Mfd 0 -)absorptive in X. Since M is a countable union of (finitedimensional) compact Z-sets in X, M is an M0 - (or Mfd 0 -)absorption base for X by Proposition 5.3.11. The “only if” part follows from Theorem 5.3.14 with Remark 5.8. Indeed, X has an (f.d.)cap set N (Proposition 5.2.2). Due to the “if” part proved above, N is an fd M0 - (or Mfd 0 -)absorption base for X. If M is an M0 - (or M0 -)absorption base for X, then there is a homeomorphism h : X → X such that f (N) = M. Therefore, M is also an (f.d.)cap set for X. " !

5.4 Absorption Bases and Strong Universality In Sect. 3.1, we introduced two kinds of universality for a class C of spaces. Recall that X is said to be universal for C or C-universal if any map f : Y → X from Y ∈ C can be approximated by closed embeddings. And, it is said that X is strongly universal for C or strongly C-universal when the following condition is satisfied: (SUC )

For each C ∈ C and each closed set D ⊂ C, if f : C → X is a map such that f |D is a strong Z-embedding, then, for each open cover U of X, there is a strong Z-embedding h : C → X such that h|D = f |D and h is U-close to f .

Here, we introduce one more universality. It is said that X is fine C-universal or fine universal for C if any map f : Y → X from Y ∈ C can be approximated by strong Z-embeddings. Then, the following implications hold: strong -universality

fine -universality

-universality

Recall the following results in Sects. 3.1 and 3.2: • For an ANR X ∈ M1 (τ ), the M1 (τ )-universality and the strong M1 (τ )universality are equivalent (Theorem 3.1.19). • For an ANR X ∈ L(τ ), the M0 -universality, the strong M0 -universality, the proper L(τ )-universality, and the strong proper L(τ )-universality are equivalent (Theorem 3.2.9). In this section, we show several properties of the strong C-universality available to C-absorption bases. In fact, the strong C-universality is related to the C-absorption property as in the following proposition:

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Proposition 5.4.1 Let X be an 2 ()-manifold and M ⊂ X. (1) If M is strongly C(τ )-universal and homotopy dense in X, then M is Cabsorptive in X. (2) If Mfd 0 ⊂ C ⊂ M1 (τ ) and M is C-absorptive in X, then M is strongly Cuniversal. Proof (1): Let C ∈ CZ (X) with D ⊂ M ∩C a closed subset. For each U ∈ cov(X), let V ∈ cov(X) be a star-refinement of U. Since M is homotopy dense in X, we have a map f : C → M such that f |D = id and f is V-close to id. Then, f (D) = D is a strong Z-set in M (cf. Corollary 5.3.6). Because C ∈ C(τ ) and M is strongly C(τ )-universal, f is V-close to a strong Z-embedding h : C → M with h|D = f |D = id. Then, h is U-close to id. Hence, M is C-absorptive in X. (2): Let C ∈ C with D ⊂ C a closed subset, and let f : C → M be a map such that f |D is a strong Z-embedding in M.9 Note that M is an ANR (Corollary 5.3.6). By Proposition 2.8.12, we may assume that f (C \ D) ∩ f (D) = ∅. Since M is homotopy dense in X (Proposition 5.3.5) and f (D) is a Z-set in M, it follows that cl f (D) is a Z-set in X. Then, cl f (D) \ f (D) is Fσ in X, i.e., f (D) is Gδ in X because f (D) ≈ D ∈ M1 (τ ). Hence, cl f (D) \ f (D) is a Zσ -set in X. By the Negligibility Theorem 2.9.9, we have Y = X \ (cl f (D) \ f (D)) ≈ X, where f (D) is a Z-set in Y .10 For each open cover U ∈ cov(M), take V ∈ cov(M) such that st V ≺  U. For  be an open set in Y such that V  ∩ M = V . Then, Y0 =  each V ∈ V, let V V ∈V V 2 is an open set in Y , which is an  ()-manifold. By the strong M1 (τ )-universality of Y0 (Theorem 3.4.2), we have a Z-embedding g : C → Y0 such that g|D = f |D   = {V  | V ∈ V} ∈ cov(Y0 ). and g is V-close to f , where V Observe that Y0 \ g(D) ⊂ Y \ g(D) = X \ cl f (D). Since Y0 is open in Y , Y0 \ g(D) is open in X. Then, g(C \ D) is a Zσ -set in Y0 \ g(D) and g(C \ D) ∈ Cσ .  and W is By Proposition 2.1.17, we can take W ∈ cov(Y0 \ g(D)) so that W ≺ V fitting in Y0 . Applying condition (c) of Theorem 5.3.8, we have a homeomorphism h : Y0 \ g(D) → Y0 \ g(D) such that h is W-close to id and h(g(C \ D)) ⊂ M ∩ (Y0 \ g(D)). Then, h can be extended to a homeomorphism h˜ : Y0 → Y0 by ˜ ˜ : C → M ∩ Y0 that is st Vh|g(D) = id. Thus, we have a closed embedding hg  close to f |C \ D, where (st V)|M ∩ Y0 = st V because M ∩ Y0 is dense in Y0 . Since ˜ is U-close to f . Hence, M is strongly C-universal. st V ≺ U, hg " ! Remark 5.9 In the above proof of (2), if f (D) is closed in X, then the proof can be performed without the Negligibility Theorem 2.9.9, where Y = X. As a combination of Propositions 5.4.1, 5.3.5 and 5.3.12, we have the following characterization of the C-absorption property: Corollary 5.4.2 Let X be an 2 ()-manifold, M ⊂ X, and Mfd 0 ⊂ C ⊂ M1 (τ ). 9 Note 10 It

that f (D) is not a (strong) Z-set in X unless it is closed in X. is unknown whether M is C-absorptive in Y , or not.

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389

(1) M is C-absorptive in X if and only if M is strongly C-universal and homotopy dense in X. (2) M is a C-absorption base for X if and only if M is a strongly C-universal and homotopy dense Zσ -set in X and M ∈ (τ C)σ . " ! The same proof as Proposition 5.4.1(1) is available even when X is a Qmanifold. If Mfd 0 ⊂ C ⊂ M0 , then f (D) is closed in X in the proof of Proposition 5.4.1(2). Then, as mentioned in Remark 5.9, we have the proof without the Negligibility Theorem 2.9.9, which is valid even when X is a Qmanifold. Thus, we have the following Q-manifold versions of Proposition 5.4.1 and Corollary 5.4.2: Proposition 5.4.3 Let X be a Q-manifold and M ⊂ X. (1) If M is strongly C-universal and homotopy dense in X, then M is C-absorptive in X. (2) If Mfd 0 ⊂ C ⊂ M0 and M is C-absorptive in X, then M is strongly C-universal. " ! Corollary 5.4.4 Let X be a Q-manifold, M ⊂ X, and Mfd 0 ⊂ C ⊂ M0 . (1) M is C-absorptive in X if and only if M is strongly C-universal and homotopy dense in X. (2) M is a C-absorption base for X if and only if M is a strongly C-universal and homotopy dense Zσ -set in X and M ∈ Cσ . " ! Combining Corollaries 5.4.2, 5.4.4 with Proposition 5.3.15, we have: Proposition 5.4.5 Let X be an 2 - or Q-manifold and M ⊂ X. Then, M is an (f.d.)cap set for X if and only if M is a (strongly countable-dimensional) σ compact,11 strongly M0 - (or Mfd " ! 0 -)universal homotopy dense Zσ -set in X. It is said that C is I-stable if C ×I ∈ C for every C ∈ C, which implies that C ×In for every C ∈ C and n ∈ N. If C is I-stable, then C contains all cubes In , n ∈ N, which implies that Mfd 0 ⊂ C. A class C said to be multiplicative if C1 × C2 ∈ C for every C1 , C2 ∈ C. Then, we have the following: Fact When C is multiplicative, Mfd 0 ⊂ C ⇔ C is I-stable. We regard  as a discrete space with card  = τ . The following is obvious: Proposition 5.4.6 If In ×  ∈ C for each n ∈ N, then every C-universal space has the τ -discrete cells property. In particular, every τ C-universal space has the τ -discrete cells property if C is I-stable.

11 A strongly countable-dimensional σ -compact metrizable space can be written as a countable union of finite-dimensional compact subsets.

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

We prove the following: Proposition 5.4.7 Suppose that C is I-stable. Let X and Y be ANRs such that every Z-set in X × Y is a strong Z-set. If X is strongly C-universal, then X × Y is strongly C-universal, too. Proof Let f : C → X × Y be a map of C ∈ C such that f |D is a Z-embedding for a closed set D ⊂ C. Then, f (D) is a strong Z-set in X × Y . By Proposition 2.8.12, we may assume that f (C \ D) ∩ f (D) = ∅ and f is closed over f (D). Given d ∈ Metr(X) and d  ∈ Metr(Y ), we define ρ ∈ Metr(X × Y ) as follows: ρ((x, y), (x  , y  )) = d(x, x  ) + d  (y, y  ). We inductively apply the strong C-universality of X to obtain strong Z-embeddings gn : C → X, n ∈ N, such that gn (C) ∩ gm (C) = ∅ for n = m and gn 1/4(n+1) prX f, where prX : X × Y → X is the projection. Then, we have gn 1/2(n+1) gn+1 . Since C × [n, n + 1] ∈ C, we can apply the strong C-universality of X to obtain a map g : C × [1, ∞) → X such that: (1) (2) (3) (4) (5)

g|C × [n, n + 1] is a strong Z-embedding, g(x, n) = gn (x) for x ∈ C, diam g({x} × [n, n + 1]) < 3/4(n + 1) for x ∈ C, g(C × [n − 1, n]) ∩ g(C × [n, n + 1]) = gn (C), g(C × [n, n + 1]) ∩ g(C × [m, m + 1]) = ∅ if |n − m|  2.

Now, we can define a homotopy h : C × I → X × Y as follows: h0 = f and ht (x) = (g(x, t −1 ), prY f (x)) for t > 0, where prY : X × Y → Y is the projection. Then, ρ(f (x), h(x, t))  t for every (x, t) ∈ C × I. Indeed, if 1/(n + 1) < t  1/n, then ρ(f (x), h(x, t))  d(f (x), gn (x)) + diam g({x} × [n, n + 1]) < 1/4(n + 1) + 3/4(n + 1) = 1/(n + 1) < t. For each map α : X × Y → (0, 1], let γ : X × Y → I be the map defined as follows: γ (z) =

1 2

min{α(z), ρ(z, f (D))} for each z ∈ X × Y.

We define a map f  : C → X × Y by f  (x) = (prX h(x, γ (f (x))), prY f (x)) for each x ∈ C.

5.4 Absorption Bases and Strong Universality

391

Then, ρ(f  (x), f (x))  γ (f (x)) < α(f (x)) for each x ∈ C. Since γ −1 (0) = f (D), we have f  |D = f |D. For each x ∈ D, prX f  (x) = g(x, γ (f (x))−1 ), where prX : X × Y → X is the projection. Hence, prX f  |C \ D is injective. Since f  (C \ D) ∩ f  (D) = ∅, it follows that f  is injective. To see that f  is closed, let xn ∈ C, n ∈ N, such that f  (xn ) → z ∈ X × Y (n → ∞). When z ∈ f (D), we have ρ(f (xn ), z)  2ρ(f  (xn ), z). Indeed, ρ(f (xn ), z)  ρ(f (xn ), f  (xn )) + ρ(f  (xn ), z)  γ (f (xn )) + ρ(f  (xn ), z)  12 ρ(f (xn ), f (D)) + ρ(f  (xn ), z)  12 ρ(f (xn ), z) + ρ(f  (xn ), z). Hence, f (xn ) → z, which means xn → f −1 (z) because f is closed over f (D). When z ∈ f (D), if lim infn∈N γ (f (xn )) = 0, replacing (xn )n∈N by a subsequence, we can assume that limn→∞ γ (f (xn )) = 0, which implies that f (xn ) → z. Then, γ (f (xn )) → γ (z) > 0, which is a contradiction. Hence, lim infn∈N γ (f (xn )) > 0, then 1/(k + 1)  lim infn∈N γ (f (xn )) < 1/k for some k ∈ N. Replacing (xn )n∈N by a subsequence, we assume that 1/(k + 1)  γ (f (xn )) < 1/k for every n ∈ N. Then, h(xn , γ (f (xn ))) = g(xn , γ (f (xn ))−1 ) → prY (z) (n → ∞). Since g|C × [k, k + 1] is a closed embedding, (xn )n∈N is convergent. Thus, f  : C → X × Y is a closed embedding. To see that f  (C) is a Z-set in X × Y , let k : Q → X × Y be a map and ε > 0. Since f  (D) = f (D) is a Z-set in X × Y , we have a map k  : Q → X × Y such that k  (Q) ∩ f  (D) = ∅ and ρ(k  , k) < ε/2. Then, there is some n0 ∈ N such that x ∈ C, f  (x) ∈ k  (Q) ⇒ γ (f (x)) > 1/n0 . Otherwise, for each n ∈ N, there is xn ∈ C such that f  (xn ) ∈ k  (Q) and γ (f (xn ))  1/n. Then, (xn )n∈N has a subsequence (xni )i∈N such that f  (xni ) converges to some z0 ∈ k  (Q). Since γ (z0 ) = limi→∞ γ (f (xni )) = 0 and α(z0 ) > 0, it follows that ρ(z0 , f (D)) = 0, which means that z0 ∈ f (D) = f  (D). This contradicts the fact that k  (Q) ∩ f  (D) = ∅. Now, let D  = (γf )−1 ([0, 1/n0 ]). Then, D ⊂ D  and D  is closed in C. In the above, we actually proved that f  (D  ) ∩ k  (Q) = ∅, where dist(k  (Q), f  (D  )) > 0 because k  (Q) is compact. Let δ=

1 2

  min ε, dist(k  (Q), f  (D  )) > 0.

Since g(C × [1, n0 ]) is a strong Z-set in X, we have a map k  : Q → X such that d(k  , prX k  ) < δ and k  (Q) ∩ g(C × [1, n0 ]) = ∅. Thus, we have the map

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

k ∗ : Q → X × Y defined by k ∗ (x) = (k  (x), prY k  (x)). Then, ρ(k ∗ , k)  ρ(k ∗ , k  ) + ρ(k  , k) = d(k  , prX k  ) + ρ(k  , k) < δ + ε/2  ε. For each x ∈ C, if γ (f (x)) > 1/n0 , then prX f  (x) = g(x, γ (f (x))−1 ) ∈ g(C × [1, n0 ]) ⊂ X \ k  (Q), which implies that f  (x) ∈ k ∗ (Q). If γ (f (x))  1/n0 , then x ∈ D  , so it follows that ρ(f  (x), k ∗ (Q))  ρ(f  (x), k  (Q)) − ρ(k ∗ , k  ) = ρ(f  (x), k  (Q)) − d(k  , prX k  )  dist(f  (D  ), k  (Q)) − δ > dist(k  (Q), f  (D  ))/2 > 0, which means that f  (x) ∈ k ∗ (Q). Consequently, k ∗ (Q) ∩ f  (C) = ∅. Hence, f  is a Z-embedding. " ! Concerning the product of an absorption base for an 2 ()-manifold and a completely metrizable ANR, we have the following: Proposition 5.4.8 Suppose that Mfd 0 ⊂ C ⊂ M1 (τ ) and C is multiplicative. Let M ∈ (τ C)σ be a homotopy dense Zσ -set in an 2 ()-manifold X, and let Y ∈ C be an ANR with w(Y )  τ . If one of M and Y is strongly C-universal, then M × Y is a C-absorption base for X × Y . Proof By using the Toru´nczyk Factor Theorem 2.2.14, it is easy to see that X × Y is an 2 ()-manifold. Then, every Z-set in X × Y is a strong Z-set. Since C is multiplicative, Mfd 0 ⊂ C means that C is I-stable, and also we have M × Y ∈ (τ C)σ . Since M is a homotopy dense Zσ -set in X, it follows that M × Y is a homotopy dense Zσ -set in X × Y . Because M or Y is strongly C-universal, M × Y is strongly C-universal by Proposition 5.4.7. Applying Corollary 5.4.2(2), we have the result. " ! For a non-degenerate metrizable space X, let FX be the class of spaces that are homeomorphic to closed sets in X (equivalently they can be embedded into X as closed sets). The following should be remarked: • If C ⊂ FX and X is strongly FX -universal, then X is strongly C-universal. Lemma 5.4.9 For a non-degenerate metrizable space X, the following hold: (1) FX is topological and closed hereditary. (2) If X ≈ X × Y for some non-degenerate path-connected space Y , then FX is I-stable. (3) If X is an AR and X ≈ X × X, then FX is additive and multiplicative.

5.4 Absorption Bases and Strong Universality

393

(4) If w(X) = τ and X ≈ X × Y for some non-compact space Y , then  ∈ FX . Proof Item (1) follows from the definition and (2) follows from the fact that Y contains an arc (i.e., a copy of I) by Proposition 1.1.1. (3): Since X ≈ X × X, it easily follows that FX is multiplicative. To see the additivity, let C = A ∪ B, where A, B ∈ FX are closed in C. Since X is an AR, we have maps f, g : C → X such that f |A and g|B are closed embeddings. Moreover, there exists an embedding ϕ : I → X. Let k : C → I be a map with k −1 (0) = A. Then, we can define a closed embedding h : C → X × X × X by h(x) = (f (x), g(x), ϕ(k(x))). Since X ≈ X × X × X, it follows that FX is additive. (4): We may assume that X = (X, d) is a metric space. Note that X is noncompact. For each n ∈ N, we can use Zorn’s Lemma to find a maximal 2−n -discrete set Xn in X. Since X∗ = n∈N Xn is dense in X, we have supn∈N card Xn = w(X) because w(X) = dens X  card X∗ = sup card Xn  c(X) = w(X). n∈N

On the other hand, since Y is a non-compact metric space, it has a countable discrete set {yn | n ∈ N}, where yn = ym if n = m. Then, Z = n∈N Xn × {yn } is discrete in X × Y and card Z = supn∈N card Xn = w(X). Since  ≈ Z and X × Y ≈ X, it follows that  ∈ FX . " ! Given a base point ∗ ∈ X, we define    XfN = x ∈ XN  x(n) = ∗ except for finitely many n ∈ N , which has the base point ∗ = (∗, ∗, . . . ) ∈ XfN . Then, we have (XfN )N f , which is N N homeomorphic to Xf . If X is an AR, Xf is also an AR. Indeed, by the Arens–Eells Embedding Theorem 1.13.2, X can be regarded as a retract of a normed linear space E, where it can be assumed that ∗ = 0 by translation. Then, XfN is a retract of EfN . Since EfN is an AE (so an AR) by the Dugundji Extension Theorem 1.13.1, it follows that XfN is an AR. Hence, the following is obtained from Lemma 5.4.9 above: Proposition 5.4.10 Let X be a non-degenerate AR with w(X) = τ (and ∗ ∈ X a base point). Then, the classes FXN and FXN are topological, closed hereditary, f

additive, multiplicative, and contain In ×  for every n ∈ N. Proof The last assertion follows from the fact that I and  belong to the classes " ! FXN and FXN . f

Concerning the strong universality of the spaces XN and XfN , we have the following: Proposition 5.4.11 Let X be a non-degenerate AR (with ∗ ∈ X a base point). Then, XN is strongly FXN -universal and XfN is strongly FXN -universal. f

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Proof Because of similarity, we give a proof for the space XfN . We may assume N N that X is non-degenerate. Since XfN ≈ (XfN )N f , it suffices to show that (Xf )f is N strongly FXN -universal. For each k ∈ N, let prk : (XfN )N f → Xf be the projection f

 −k if onto the k-th factor. We use a metric d ∈ Metr((XfN )N f ) such that d(x, x )  2 x(i) = x  (i) for every i < k. Let f : C → (XfN )N f be a map of C ∈ FXfN such that f |D is a Z-embedding for a closed set D ⊂ C. By Theorem 2.8.6, we may assume that f (C \ D) ∩ f (D) = ∅. Observe that

XfN ≈ {∗ } × XfN × {(∗, ∗, . . . )} ⊂ X × XfN × XfN ≈ XfN , hence XfN can be embedded into itself as a Z-set that misses ∗. Then, there is a Z-embedding g : C → XfN with ∗ ∈ g(C). Since XfN is an AR, it has an equiconnecting map λ : XfN × XfN × I → XfN , that is, λ(x, y, 0) = x, λ(x, y, 1) = y and λ(x, x, t) = x for every t ∈ I. We define a homotopy h : C × I → (XfN )N f as follows: h0 = f , and for 2−k  t  2−k+1 , k ∈ N, h(x, t) = (f1 (x), . . . , fk−1 (x), λ(fk (x), g(x), 2k t − 1), g(x), g(x), λ(g(x), ∗, 2k t − 1), ∗, ∗, . . . ), where fk = prk f : C → XfN . Then, d(f (x), h(x, t))  t for every (x, t) ∈ C × I. Indeed, if 2−k < t  2−k+1 then d(f (x), h(x, t))  2−k < t. Observe prk+1 h|C × [2−k−1 , 2−k+1 ] = gprC , where prC : C × I → C is the projection. Moreover, h(C × [2−k , 1]) ∩ h(C × [2−k−2 , 2−k−1 ]) ⊂ (XfN )k+3 × {∗} × {∗} × {(∗, ∗, . . . )} ∩ (XfN )k+3 × g(C) × XfN × {(∗, ∗, . . . )} = ∅. N N For each map α : (XfN )N f → (0, 1], let γ : (Xf )f → I be the map defined by

γ (y) =

1 2

min{α(y), d(y, f (D))} for each y ∈ (XfN )N f.

We define a map f  : C → (XfN )N f by f  (x) = h(x, γ (f (x))) for each ∈ C.

5.4 Absorption Bases and Strong Universality

395

Then, f  |D = f |D and d(f  (x), f (x))  γ (f (x)) < α(f (x)) for every x ∈ C. Since f  (C \D)∩f  (D) = ∅ and f  |C \D is injective, it follows that f  is injective. To see that f  is closed, let xn ∈ C, n ∈ N, such that f  (xn ) → y ∈ (XfN )N f (n → ∞). When y ∈ f (D), we have d(f (xn ), f  (xn ))  γ (f (xn ))  12 d(f (xn ), f (D))  d(f (xn ), f (D)) − d(f (xn ), f  (xn ))  d(f (xn ), f (D)) → 0, hence f (xn ) → y. Since f is closed over f (D), we have xn → f −1 (y). When y ∈ f (D), if lim infn∈N γ (f (xn )) = 0, replacing (xn )n∈N by a subsequence, we assume that limn→∞ γ (f (xn )) = 0, which implies that f (xn ) → y. Then, γ (f (xn )) → γ (y) > 0, which is a contradiction. Hence, lim infn∈N γ (f (xn )) > 0. Choose k ∈ N so that 2−k  lim infn∈N γ (f (xn )) < 2−k+1 . Replacing (xn )n∈N by a subsequence, we assume that 2−k  γ (f (xn )) < 2−k+1 for every n ∈ N. Then, g(xn ) → y(k +1) (n → ∞). Since g is a closed embedding, (xn )n∈N is convergent. Thus, f  : C → (XfN )N f is a closed embedding. Since g(X) is a Z-set in XfN and ∗ ∈ XfN \ g(X), there are maps of (XfN )N f into N N N N  (Xf \ g(X))f arbitrarily close to id. Therefore, f (X) is a Z-set in (Xf )f , that is, f  is a Z-embedding. " ! As a corollary of the above proposition, we have the following: Corollary 5.4.12 Let X be a non-degenerate AR (with ∗ ∈ X a base point). Then, XN (resp. XfN ) is strongly C-universal for any class C ⊂ FXN (resp. C ⊂ FXN ). ! " f

Lemma 5.4.13 Assume that C is multiplicative. (1) If X, Y ∈ (τ C)σ , then X × Y ∈ (τ C)σ . (2) If X = (X, ∗) ∈ (τ C)σ , then XfN ∈ (τ C)σ .   Proof (1): We can write X = n∈N Xn and Y = n∈N Yn  , where each Xn and Yn is closed in X and Y respectively, Xn = α∈A(n) Cα , Yn = β∈B(n) Dβ , Cα , Dβ ∈ C, and card A(n), card B(n)  τ . Then, X×Y = Xn × Ym =



Xn × Ym and

(n,m)∈N2

Cα × Dβ for each (n, m) ∈ N2 ,

(α,β)∈A(n)×B(m)

where Xn × Ym is closed in X × Y , Cα × Dβ ∈ C, and card A(n) × B(m)  τ . Hence, X × Y ∈ (τ C)σ .   (2): Let X = n∈N Xn , where each Xn is closed in X, Xn = α∈A(n) Cα , Cα ∈ C, and card A(n)  τ . Identifying Xk = {x ∈ XN | x(i) = ∗ for i > k }, we

396

have XfN =

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces



k∈N X

k.

Then, as is easily observed, we have XfN =

k   # k∈N ν∈Nk

k # i=1

Xν(i) = α∈

k i=1

k #

Xν(i) and

i=1

Cα(i) for each ν ∈ Nk , k ∈ N,

A(ν(i)) i=1

   where ki=1 Xν(i) is closed in XfN , ki=1 Cα(i) ∈ C, and card ki=1 A(ν(i))  τ . ! " Hence, XfN ∈ (τ C)σ . If M and N are homotopy dense in X and Y respectively, then M×N is homotopy dense in X × Y . If A is a Z-set (resp. a Zσ -set) in an ANR X and B is a closed set (resp. an Fσ set) in Y , then A×B is a Z-set (resp. a Zσ -set) in X ×Y . By combining Corollary 5.4.2(2), Proposition 5.4.7, and Lemma 5.4.13(1), we have the following generalization of Proposition 5.4.8: Proposition 5.4.14 Suppose that Mfd 0 ⊂ C ⊂ M1 (τ ) and C is multiplicative. Let X 2 be an  ()-manifold and Y ∈ M1 (τ ) be an ANR. Let M and N be homotopy dense Fσ sets in X and Y respectively and M, N ∈ (τ C)σ . If M (or N) is a C-absorption base for X (or Y ),12 then M × N is a C-absorption base for X × Y . " ! Concerning simplicial complexes, we have the following: Proposition 5.4.15 For an arbitrary  simplicial complex K with card K  τ , |K|m ∈ (τ Mfd i∈N Di , where each Di is a discrete union of 0 )σ . Actually, |K|m = ni -cells. Proof For each n-simplex σ ∈ K and k ∈ N, let    σk = x ∈ σ  ρK (x, ∂σ )  (n + k)−1 ⊂ rint σ.  Then, σk ∈ Mfd k∈N σk . For each k ∈ N, {σk | σ ∈ K(n)} is 0 and rint σ = discrete in |K|m , where K(n) = K (n) \ K (n−1) is the set of all n-simplexes of K.  (n) Hence, Mn,k = σ ∈K(n) σk ∈ τ M0 = (τ M0 )(n) is closed in |K|m . Thus, we have  |K|m = n,k∈N Mn,k ∈ ((τ M0 )fd )σ . " ! Remark 5.10 Evidently, (τ M0 )fd ⊂ τ Mfd 0 . But the equality does not hold. In  fd n fd fd fact, n∈N I ∈ ℵ0 M0 \ (ℵ0 M0 ) . Nevertheless, (τ Mfd 0 )σ = ((τ M0 ) )σ = fd fd scd 13 fd (τ M0 )σ . Indeed, ((τ M0 ) )σ ⊂ (τ M0 )σ is trivial. Each X ∈ (τ M0 )σ can 12 Or,

M is strongly C-universal and N is a Zσ -set in Y , or otherwise N is strongly C-universal and M is a Zσ -set in X. 13 For the notation Cscd , see p. 377. σ

5.4 Absorption Bases and Strong Universality

397

 fd be written i∈N Xi , where Xi ∈ τ M0 . Each Xi can be written as  as X = Xi = λ∈i Xi,λ , where dim Xi,λ < ∞ and card i  τ . Let i,n = {λ ∈ i |  dim Xi,λ = n}. Then, we have Xi,n = Xi,λ ∈ (τ M0 )(n) by the Locally λ∈i,n Finite Sum Theorem 1.12.9. Hence, X = i,n∈N Xi,n ∈ ((τ M0 )fd )σ . For a locally finite-dimensional simplicial complex K with card K  τ , |K|m is a completely metrizable ANR (cf. Theorems 1.7.2, 1.13.4) and |K|m ∈ (τ Mfd 0 )σ as above. Thus, we have the following corollary: Corollary 5.4.16 Suppose that Mfd 0 ⊂ C ⊂ M1 (τ ) and C is multiplicative. Let M be a C-absorption base for an 2 ()-manifold X. Then, for each locally finitedimensional simplicial complex K with card K  τ , |K|m ×X is an 2 ()-manifold and |K|m × M is a C-absorption base for |K|m × X. " ! Remark 5.11 In the above, it suffices to assume that  C is I-stable instead of assumption that C is multiplicative. Indeed, |K|m = i∈N Di , where each Di is a discrete union of ni -cells. If C is I-stable, then B × C ∈ C for each cell B and C ∈ C, which implies |K|m × M ∈ (τ C)σ . Moreover, we have the following: Proposition 5.4.17 Let X be a completely metrizable AR with ∗ ∈ X a base point and w(X) = τ . Suppose that C is multiplicative. If M is a homotopy dense Fσ set in X, ∗ ∈ M ∈ (τ C)σ , and C ⊂ FM N , then MfN is a C-absorption base for XN f

(≈ 2 ()). Proof Note that XN ≈ 2 () by Theorem 3.5.15. Since M is homotopy dense in X, M = (M, ∗) is also a pointed non-degenerate AR and MfN is homotopy dense in XN (Proposition 1.13.27). Identifying Xn = Xn × {(∗, ∗, · · · )} ⊂ XN , Xn is a Z-set in XN (e.g., it is easy to check condition (b) of Theorem 2.8.6). Hence, XfN is a Zσ -set in XN . Since MfN is Fσ in XfN , it easily follows that MfN is a Zσ -set in XN . By Lemma 5.4.13(2), MfN ∈ (τ C)σ . Moreover, due to Corollary 5.4.12, the strong C-universality of MfN comes from C ⊂ FM N . Hence, MfN is C-absorptive f

by Proposition 5.4.1(1). Thus, it follows from Corollary 5.4.2(2) that MfN is a C! " absorption base for XN . We will show that the strong C-universality is open hereditary, that is, Proposition 5.4.18 Let X be an ANR. If X is strongly C-universal, then every open set in X is strongly C-universal. Proof Let U be an open set in X and f : C → U be a map of C ∈ C such that f |D is a strong Z-embedding for some closed set D in C. Take d ∈ Metr(X) and define Un = {x ∈ U | d(x, X \ U )  2−n } for each n ∈ N. Then, each Un is closed

398

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

in U , Un ⊂ int Un+1 and U = Vn ∈ cov(X), n ∈ N, so that



n∈N Un .

For each V ∈ cov(U ), inductively choose

mesh Vn < 2−n , st Vn ≺ Vn−1 and st(Un+1 , Vn ) ⊂ int Un+2 , where V0 = V ∪ {X \ U1 }. Let An = f −1 (Un ) and Bn = f −1 (U \ int Un+1 ). Then, each An and Bn are disjoint closed sets in C, An ⊂ int An+1 , int An+1 ∪ Bn = C and C =



An .

n∈N

We shall inductively construct maps fn : C → U , n ∈ N, satisfying the following conditions: (1) (2) (3) (4)

fn |An−1 ∪ Bn ∪ D = fn−1 |An−1 ∪ Bn ∪ D, fn |An ∪ D : An ∪ D → U is a Z-embedding, fn is Vn -close to fn−1 , and fn (An+2 ) ⊂ Un+2 ,

where A0 = ∅, B0 = C and f0 = f . Assume that fn−1 has been obtained. Since X is a strongly C-universal ANR, we have a Vn -homotopy h : C × I → X such that h0 = fn−1 , h1 : C → X is a Z-embedding, h1 |An−1 ∪ (D \ int Bn ) = fn−1 |An−1 ∪ (D \ int Bn ) and ht |D \ int Bn = f |D \ int Bn for every t ∈ I. Taking an Urysohn map λ : C → I with λ(Bn ) = 0 and λ(An ) = 1, we define fn : C → X by fn (x) = h(x, λ(x)). Then, fn satisfies the conditions (1)–(3) by definition. To see (4), let x ∈ An+2 . If x ∈ An+1 then fn−1 (x) ∈ Un+1 , hence fn (x) ∈ st(Un+1 , Vn ) ⊂ Un+1 . If x ∈ An+2 \ An+1 ⊂ Bn then fn (x) = f (x) ∈ Un+2 . Thus, fn satisfies (4), and then fn (C) ⊂ U . Since C = n∈N int An , we can define a map g : C → U by g|An = fn |An . Then, g|D = f |D by (1). It is easy to see that g is V-close to f . By (1) and (2), g is injective. For each closed set F in C, observe that g(F ) =

 n∈N

g(F ∩ (An \ int An−1 )) =



fn (F ∩ (An \ int An−1 )),

n∈N

which is a locally finite union of closed sets  in U , hence g(F ) is closed in U . Thus, g is a closed embedding. Since g(C) = n∈N fn (An \ int An−1 ) is a locally finite union of Z-sets in U , it follows that g(C) is a Z-set in U . Hence, g is a Zembedding. " ! Using the above proposition, we can prove the following:

5.5 Four Types of Absorption Bases for 2 ()

399

Theorem 5.4.19 If M is a C-absorption base for an 2 ()-manifold X, then M ∩U is a C-absorption base for every open set U in X. Proof Since M is homotopy dense in X, M is an ANR (Theorem 1.13.26) and M ∩ U is homotopy dense in U (Corollary 1.15.6). Since M is extensively Cuniversal, so is M ∩ U by Proposition 5.4.18.  Hence, M ∩ U is C-absorptive by Proposition 5.4.1(1). We can write M = n∈N Mn , where each Mn is  a discrete union of members of CZ (X), Because U is Fσ in X, we can write U = n∈N An , where each An is closed in X. Since C is closed hereditary, it follows that Mn ∩ Am  is a discrete union of members of CZ (U ), and then M ∩ U = n,m∈N (Mn ∩ Am ). Thus, M ∩ U is a C-absorption base for U . " ! Combining the above theorem with the Open Embedding Theorem 2.5.10, we have the following: Corollary 5.4.20 Suppose that 2 () has a C-absorption base. Then, every 2 ()manifold has a C-absorption base. " ! At the end of this section, we shall show the following: Proposition 5.4.21 Suppose that Mfd 0 ⊂ C ⊂ M1 (τ ). If M is a C-absorption base for an 2 ()-manifold X, then M is strongly τ C-universal. Proof  First, note that every Z-set in M is a strong Z-set (Corollary 5.3.6). Let C = λ∈ Cλ ∈ τ C, where Cλ ∈ C for each λ ∈  and card   τ . Let f : C → M be a map such that f |D is a Z-embedding, where D is a closed set in C. For each U ∈ cov(M), let V ∈ cov(M) be a star-refinement of U. Then, X has   an open collection V  such that V = V|M. Due to Proposition 5.4.19, M is a C Hence, replacing X with the open set V,  we can regard absorption base for V.  ∈ cov(X). V Since C ⊂ M1 (τ ), it follows that τ C ⊂ M1 (τ ). By the strongly M1 (τ )universality of X, we have a strong Z-embedding g : C → X such that g|D = f |D  and g is V-close to f . Then, we have a discrete open collection {Wλ | λ ∈ } in X such that h(Cλ ) ⊂ Wλ for each λ ∈ . Due to Theorem 5.3.8, every M ∩ Wλ is C-absorptive in Wλ . Hence, we have Z-embeddings hλ : Cλ → M ∩ Wλ , λ ∈ ,  such that hλ |Dλ = id and hλ is V-close to id, where Dλ = D ∩ Cλ for each λ ∈ .  We define a Z-embedding h : C → λ∈ Wλ by h|Cλ = hλ , λ ∈ . Then, hg : C → M is a Z-embedding such that hg|D = f |D and hg is U-close to f ,  where st V|M = st V ≺ U. Consequently, M is strongly τ C-universal. " !

5.5 Four Types of Absorption Bases for 2 () In this section, we discuss four types of model spaces being absorption bases for Hilbert space 2 (), that is, an M0 -absorption base, an Mfd 0 -absorption base, an M1 (ℵ0 )-absorption base, and an M1 (τ )-absorption base for 2 ().

400

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Theorem 5.5.1 The subspace 2f () ⊂ 2 () is an Mfd 0 -absorption base for fd 2  (), that is, an M0 -absorptive set which is a σ -discrete union of members of 2 Mfd 0 ( ()). Proof By virtue of Theorem 1.2.3, (2 (), 2f ()) can be replaced with (1 (), 1f ()). This replacement makes calculation of distance easy. fd 1 (The Mfd 0 -Absorption Property) Let C ∈ M0 ( ()) and U be an open set in 1 () with V ∈ cov(U ). We may assume that V is fitting in 1 (). It suffices to construct an injective map h : C ∩ U → U ∩ 1f () that is V-close to id. Indeed, ¯ \ U = id, that is a since V is fitting in 1 (), h extends a map h˜ : C → U by h|C closed embedding because h¯ is injective and C is compact. Since a compact set in ¯ 1 () is a Z-set in 1 () by Proposition 3.2.10 or Corollary 3.3.14, C and h(C) are ¯ Z-sets in 1 (), hence C ∩ U and h(C ∩ U ) = h(C) ∩ Uare Z-sets in U . Now, we shall construct h. First, we write C \ U = i∈N Ci where each Ci is compact and Ci ⊂ intC Ci+1 . We can choose δ1 > δ2 > · · · > 0 so that    B(x, δi )  x ∈ Ci \ int Ci−1 ≺ V for every i ∈ N, where C0 = ∅. Indeed, for each x ∈ Ci \ int Ci−1 , choose δx > 0 so that B(x, δx ) is contained in some Vx ∈ V. Since Ci \ int Ci−1 is compact, Ci \ int Ci−1 ⊂  m j =1 B(xj , δxj /2) for some x1 , . . . , xm ∈ Ci \ int Ci−1 . Then, the desired δi > 0 can be chosen as follows:   0 < δi < min δi−1 , δx1 /2, . . . , δxm /2 .  For each x ∈ Ci \ int Ci−1 , choose Fx ∈  so that γ ∈\Fx |x(γ )| < δi /4. Since Ci \ int C i−1 is compact, we have a finite  set Ai ⊂ Ci \ int Ci−1 such that Ci \ int Ci−1 ⊂ a∈Ai B(a, δi /4). Let Fi = a∈Ai Fa . Then, the following holds: 

|x(γ )| < δi /2 for every x ∈ Ci \ int Ci−1 .

γ ∈\Fi

Indeed, x is contained in B(a, δi /4) for some a ∈ Ai , whence it follows that  γ ∈\Fi

|x(γ )| 





|x(γ ) − a(γ )| +

γ ∈\Fi

 x − a +



γ ∈\Fi

|a(γ )|

γ ∈\Fa

< δi /4 + δi /4 = δi /2.

|a(γ )|

5.5 Four Types of Absorption Bases for 2 ()

401

Let 0 = F1 , 0 = ∅ and take i , i+1 ∈ Fin(), i ∈ N, as follows: i = i−1 ∪ i−1 ∪ Fi+1 , i ∩ i = ∅ and card i = 2 dim C + 2. For each i ∈ , choose any λi ∈ i . By Embedding Theorem 1.12.14, we have embeddings hi : C → Ii such that hi (x)(λi ) = 1 for every x ∈ C. For each i ∈ N, we have a map ϕi : C → I with ϕi−1 (1) = C \ intC Ci and ϕi−1 (0) = Ci−1 . Let εi = δi+1 /6(2 dim C + 2). We define h : C \ U → 1f () as follows: ⎧ ⎪ ⎪ ϕi (x)x(γ ) ⎪ ⎪ ⎪ ⎨ε (1 − ϕ (x))ϕ (x)h (x)(γ ) i i+1 i−1 i h(x)(γ ) = ⎪ ⎪ + ϕi+1 (x)x(γ ) ⎪ ⎪ ⎪ ⎩0

  if γ ∈ i \ i−1 ∪ i−1 , if γ ∈ i , if γ ∈  \



i∈N i .

For each x ∈ Ci \ intC Ci−1 , since ϕj (x) = 0 for j > i and ϕj (x) = 1 for j < i, we have the following: ⎧ ⎪ ⎪ ⎪εi+1 ϕi (x)hi+1 (x)(γ ) ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ε ⎪ i hi (x)(γ ) ⎨ h(x)(γ ) = ϕi (x)x(γ ) ⎪ ⎪ ⎪ ⎪εi−1 (1 − ϕi (x))hi−1 (x)(γ ) + ϕi (x)x(γ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x(γ ) ⎪ ⎪ ⎪ ⎩ 0

if γ ∈ i+1 ,

  if γ ∈ i+1 \ i ∪ i ,

if γ ∈ i ,

  if γ ∈ i \ i−1 ∪ i−1 , if γ ∈ i−1 , if γ ∈ i−1 ,

if γ ∈  \ (i+1 ∪ i+1 ).

Since Fi ⊂ i−1 , it follows that x − h(x) 



|x(γ ) − h(x)(γ )|

γ ∈\i−1





|x(γ )| +

γ ∈\(i−1 ∪i ∪i−1 )

+





εi+1 ϕi (x)|hi+1 (x)(γ )| +

γ ∈i+1

+ 



(1 − ϕi (x))|x(γ )|

γ ∈i ∪i−1





εi |hi (x)(γ )|

γ ∈i

εi−1 (1 − ϕi (x)|hi−1 (x)(γ )|

γ ∈i−1

|x(γ )| + εi+1 card i+1 + εi card i + εi−1 card i−1

\i−1

 δi /2 + 3(2 dim C + 2)εi−1  δi /2 + δi /2 = δi

402

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Therefore, h is V-close to id. To see that h is injective, let x = y ∈ C \ U , where x ∈ Ci \ intC Ci−1 and y ∈ Cj \ intC Cj −1 . We may assume that i  j without loss of generality. When i + 1 < j , we have h(x)(λj ) = 0 < εj = h(y)(λj ), so h(x) = h(y). When i = j , since hi is an embedding, h(x)(γ ) = εi hi (x)(γ ) = εi hi (y)(γ ) = h(y)(γ ) for some γ ∈ i , hence h(x) = h(y). When j = i + 1, if x ∈ Cj \ int Cj −1 then h(x) = h(y) as in the case i = j . If x ∈ int Ci then ϕi (x) < 1, hence h(x)(λi+1 ) = εi+1 ϕi (x) < εi+1 = h(y)(λi+1 ), which implies that h(x) = h(y). (A Cell Complex Structure of 1f ()) For each v ∈ Zf and F ∈ Fin(), we define  Qv,F = v + Ieγ γ ∈F

 = x ∈ 1f ()  x(γ ) = v(γ ) if γ ∈  \ F, 

v(γ )  x(γ )  v(γ ) + 1 if γ ∈ F



⊂ 1f (),

which is an n-cell (an n-cube), where n = card F . Any face of Qv,F can be written as Qv  ,F  , where v = v +



eγ  , F  ⊂ F, F  ⊂ F \ F  .

γ  ∈F 

Moreover, when Qv,F ∩ Qv  ,F  = ∅, we have the following: (1) if γ ∈ F ∪ F  , then v(γ ) = v  (γ ); (2) if γ ∈ F \ F  , then v  (γ ) = v(γ ) or v  (γ ) = v(γ ) + 1, where   [v(γ ), v(γ ) + 1] ∩ {v  (γ )} = max{v(γ ), v  (γ )} ; (3) if γ ∈ F  \ F , then v(γ ) = v  (γ ) or v  (γ ) + 1, where   [v  (γ ), v  (γ ) + 1] ∩ {v(γ )} = max{v(γ ), v  (γ )} ; (4) if γ ∈ F ∩ F  and v(γ ) = v  (γ ), then |v(γ ) − v  (γ )| = 1, where   [v(γ ), v(γ ) + 1] ∩ [v  (γ ), v  (γ ) + 1] = max{v(γ ), v  (γ )} .

5.5 Four Types of Absorption Bases for 2 ()

403

Then, it follows that Qv,F ∩ Qv  ,F  = Qv  ,F  , where v  (γ ) = max{v(γ ), v  (γ )} for γ ∈  and    F  = γ ∈ F ∩ F   v(γ ) = v  (γ ) . Thus, we have the following cell complex:    L = Zf ∪ Qv,F  v ∈ Zf , F ∈ Fin() , where L(0) = Zf . Then, |L| = 1f () as sets. Indeed, for each x ∈ 1f (), F = {γ ∈  | x(γ ) = 0} is finite. Let v ∈ Zf be defined as follows: v(γ ) =

 *x(γ ), if γ ∈ F, 0

otherwise,

where *r, is the largest integer not greater than r. Then, x ∈ v +



γ ∈F

Ieγ = Qv,F .

 1 (1f () is a σ -Discrete Union of Members of Mfd 0 ( ())) For each v ∈ Zf , F ∈ Fin(), and k ∈ N, let

Qv,F,k = v +



[2−k , 1 − 2−k ]eγ .

γ ∈F

 Then, rint Qv,F = k∈N Qv,F,k . We prove that {Qv,F,k | v ∈ Zf , F ∈ Fin()} is discrete in 1f (). Let x ∈ Qv,F,k and x  ∈ Qv  ,F  ,k , where (v, F ) = (v  , F  ). When F = F  , there exists γ ∈ (F \ F  ) ∪ (F  \ F ). Then, v(γ ) + 2−k  x(γ )  v(γ ) + 1 − 2−k and x  (γ ) = v  (γ ) ∈ Z, or v  (γ ) + 2−k  x  (γ )  v  (γ ) + 1 − 2−k and x(γ ) = v(γ ) ∈ Z. In either case, we have x − x    |x(γ ) − x  (γ )|  2−k . When F = F  , it follows that v = v  ∈ Zf , hence |v(γ ) − v  (γ )|  1 for some γ ∈ . When γ ∈ F = F  , since x(γ ) = v(γ ) and x  (γ ) = v  (γ ), it follows that |x(γ ) − x  (γ )| = |v(γ ) − v  (γ )|  1. When γ ∈ F = F  and v(γ ) < v  (γ ), since x(γ )  v(γ ) + 1 − 2−k and v  (γ ) + 2−k  x  (γ ), it follows that x  (γ ) − x(γ )  v  (γ ) + 2−k − (v(γ ) + 1 − 2−k )  2−k+1 .

404

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

When γ ∈ F = F  and v  (γ ) < v(γ ), since x  (γ )  v  (γ ) + 1 − 2−k and v(γ ) + 2−k  x(γ ), it follows that x(γ ) − x  (γ )  v(γ ) + 2−k − (v  (γ ) + 1 − 2−k )  2−k+1 . In either case, we have x − x    |x(γ ) − x  (γ )|  2−k+1 . Consequently, if Qv,F,k = Qv  ,F  ,k , then    dist(Qv,F,k , Qv  ,F  ,k ) = inf x − x    x ∈ Qv,F,k , x  ∈ Qv  ,F  ,k  2−k , which means that {Qv,F,k | v ∈ Zf , F ∈ Fin()} is discrete in 1f (). Observe 1f () = |L| =



rint Qv,F =

v∈Z f F ∈Fin()

 

Qv,F,k ,

k∈N v∈Z f F ∈Fin()

where every Qv,F,k is a Z-set in 1 () because it is compact. Therefore, 1f () is a 1 " ! σ -discrete union of members of Mfd 0 ( (). Now, we can characterize 2f () ⊂ 2 () as an Mfd 0 -absorption base, that is, the following theorem holds: Theorem 5.5.2 For F ⊂ 2 (), the following are equivalent: (a) (2 (), F ) ≈ (2 (), 2f ()); 2 (b) F is an Mfd 0 -absorption base for  (); 2 (c) For each open cover U of  (), there is a homeomorphism f : 2 () → 2 () such that f (F ) = 2f () and f is U-close to id. Proof The implication (c) ⇒ (a) is obvious. Since 2f () is an Mfd 0 -absorption base for 2 () by Theorem 5.5.1, we have (a) ⇒ (b). The implication (b) ⇒ (c) follows from Theorem 5.3.14. " ! Remark 5.12 Due to Theorems 1.2.3 and 5.5.2 above is valid for any other p (p (), f ()), 1  p < ∞. We can show the following M0 -version of Theorem 5.5.2: Theorem 5.5.3 For F ⊂ 2 (), the following are equivalent: (a) (2 (), F ) ≈ (2 () × Q, 2f () × Q); (b) F is an M0 -absorption base for 2 (); (c) For each open cover U of 2 (), there is a homeomorphism f : 2 () × Q → 2 () such that f (2f () × Q) = F and f is U-close to pr1 . Proof Since Q is strongly M0 -universal, we can apply Proposition 5.4.8 and Corollary 5.4.2 to see that 2f () × Q is an M0 -absorption base for 2 () × Q.

5.5 Four Types of Absorption Bases for 2 ()

405

By the Toru´nczyk Factor Theorem 2.2.12 and Corollary 2.3.10, the projection pr1 : 2 () × Q → 2 () is a near-homeomorphism. Since there is a homeomorphism h : 2 () × Q → 2 () arbitrarily close to pr1 , the implication (a) ⇒ (b) is trivial and (b) ⇒ (c) can be obtained by using Theorem 5.3.14. By Theorem 5.3.14, we have the implication (c) ⇒ (a). " ! Note that the pair (2 () × Q, 2f () × Q) is not a pair of linear spaces but it is homeomorphic to a pair of normed linear spaces, that is: Theorem 5.5.4 The pair (2 () × Q, 2f () × Q) is homeomorphic to the pair (2 () × 2 , 2f () × 2Q ). Proof Because card  = card  + ℵ0 , we have 

   2 (), 2f () ≈ 2 () × 2 , 2f () × 2f .

Moreover, (2 × Q, 2f × Q) ≈ (2 , 2Q ) by Corollary 5.1.7(3). Hence, 

   2 () × Q, 2f () × Q ≈ 2 () × 2 × Q, 2f () × 2f × Q   ≈ 2 () × 2 , 2f () × 2Q .

" !

Note Applying Proposition 5.4.14, we can show that 2f ()×2Q is an M0 -absorption base for 2 () × 2 . The above theorem is also a corollary of Theorem 5.5.3.

Next, we will consider the following pairs:     2  () × 2 , 2f () × 2 and 2 () × 2 , 2 () × 2f , where 2 () × 2 ≈ 2 (). First, replacing Q and M0 with 2 and M1 (ℵ0 ) in Theorem 5.5.3 and its proof, we can obtain the following theorem (the details are left to the reader): Theorem 5.5.5 For F ⊂ 2 (), the following are equivalent: (a) (2 (), F ) ≈ (2 () × 2 , 2f () × 2 ); (b) F is an M1 (ℵ0 )-absorption base for 2 (); (c) For each open cover U of 2 (), there is a homeomorphism f : 2 () × 2 → 2 () such that f (2f () × 2 ) = F and f is U-close to pr1 . The following is also obtained: Theorem 5.5.6 The following are homeomorphic to (2 () × 2 , 2f () × 2 ): 

   2 () × 2 × 2 , 2f () × 2Q × 2 and 2 () × Q × 2 , 2f () × Q × 2 .

406

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Proof By Theorem 5.5.4 and the fact Q × 2 ≈ 2 , we have the following: 

   2 () × 2 × 2 , 2f () × 2Q × 2 ≈ 2 () × Q × 2 , 2f () × Q × 2   ≈ 2 () × 2 , 2f () × 2 . " !

For the second pair, since 2 () is strongly M1 (τ )-universal, we can apply Proposition 5.4.8 and Corollary 5.4.2(2) to see that 2 () × 2f is an M1 (τ )absorption base for 2 () × 2 . Because the projection pr1 : 2 () × 2 → 2 () is a near-homeomorphism, the following can be proved in the same way as Theorem 5.5.3 (the details are left to the reader): Theorem 5.5.7 For F ⊂ 2 (), the following are equivalent: (a) (2 (), F ) ≈ (2 () × 2 , 2 () × 2f ); (b) F is an M1 (τ )-absorption base for 2 (); (c) For each open cover U of 2 (), there is a homeomorphism f : 2 () × 2 → 2 () such that f (2 () × 2f ) = F and f is U-close to pr1 . Concerning the pairs (2 () × 2 , 2 () × 2Q ) and (2 ()N , 2 ()N f ), we have the following: Theorem 5.5.8 The pairs (2 () × 2 , 2 () × 2Q ) and (2 ()N , 2 ()N f ) are 2 2 2 2 homeomorphic to ( () ×  ,  () × f ). Proof First, the following is easy: 

   2 () × 2 , 2 () × 2Q ≈ 2 () × 2 × Q, 2 () × 2f × Q   ≈ 2 () × 2 , 2 () × 2f .

 N 2 2 n For the pair (2 ()N , 2 ()N n∈N  () , where each f ), note that  ()f = 2 ()n is a completely metrizable Z-set in 2 ()N with w(2 ()n ) = τ , i.e., a member of (M1 (τ ))(2 ()N ). By Proposition 1.13.27, 2 ()N f is homotopy dense in 2 ()N . Since M1 (τ ) = F2 () ⊂ F2 ()N , it follows from Corollary 5.4.12 that f

2 N 2 ()N f is strongly M1 (τ )-universal. Hence,  ()f is an M1 (τ )-absorption base 2 N 2 N 2 for  () (Corollary 5.4.2(2)). Since  () ≈  () (Corollary 2.2.13), we have



  2  2 2 2 2 ()N , 2 ()N f ≈  () ×  ,  () × f .

" !

Remark 5.13 For   ⊂  with card   = τ  > ℵ0 , we can consider the spaces 2f () × 2 (  ) and 2 () × 2f (  ). The former is an M1 (τ  )-absorption base for 2 () × 2 (  ) but the latter is an M1 (τ )-absorption base for 2 () × 2 (  ), which

5.5 Four Types of Absorption Bases for 2 ()

407

is the same as 2f () × 2 and 2 () × 2f . Then, for the former pair, the same result as Theorem 5.5.5 holds. For the latter pair, we have 

   2 () × 2 (  ), 2 () × 2f (  ) ≈ 2 () × 2 , 2 () × 2f .

Combining Proposition 5.4.17 with the above results, we obtain the following theorem: Theorem 5.5.9 fd 2 N (1) The space 2f ()N f is an M0 -absorption base for  () , hence

   2  2 N 2 2  () , f ()N f ≈  (), f () . 2 2 N (2) The space (2f () × 2Q )N f is an M0 -absorption base for ( () ×  ) , hence

  2   2 2 2 2 ( () × 2 )N , (2f () × 2Q )N f ≈  () ×  , f () × Q . 2 2 N (3) The space (2f () × 2 )N f is an M1 (ℵ0 )-absorption base for ( () ×  ) , hence

  2   2 2 2 2 ( () × 2 )N , (2f () × 2 )N f ≈  () ×  , f () ×  . 2 2 N (4) The space (2 ()×2f )N f is an M1 (τ )-absorption base for ( ()× ) , hence



  2  2 2 2 14 (2 () × 2 )N , (2 () × 2f )N f ≈  () ×  ,  () × f .

" !

By the above theorem, we have 2 2 2 N 2 2 2f ()N f ≈ f (), (f () × Q )f ≈ f () × Q , 2 2 2 2 N 2 2 (2f () × 2 )N f ≈ f () ×  , and ( () × f )f ≈  () × f .

Thus, we can conclude as follows: Theorem 5.5.10 All results on E-manifolds in Chap. 2 are valid for 2f ()-, (2f () × 2Q )-, (2f () × 2 )-, and (2 () × 2f )-manifolds. " ! In particular, the Open Embedding Theorem 2.5.10 and the Classification Theorem 2.6.1 are valid for these manifolds.

 N 2 N 2 N is equivalent to ((2 ()N )N , (2 ()N f )f ≈ ( () ,  ()f ), which is easily seen. In fact,  N N      N×N N N N N×N N (X ) , (Af )f ≈ X , Af ≈ X , Af for an arbitrary pair (X, A) of spaces.

14 This

408

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Remark 5.14 In the separable case, s × s f , s × s Q , and s N f are M1 (ℵ0 )-absorption N bases for s × s, s × s, and s , respectively. Hence, we have  2       × 2 , 2 × 2f ≈ 2 × 2 , 2 × 2Q ≈ (2 )N , (2 )N f   ≈ (s × s, s × s f ) ≈ (s × s, s × s Q ) ≈ s N , s N f . Since every closed set in Q is compact, Q has no M1 (ℵ0 )-absorption bases N homeomorphic to s N f . Although I (Q) × Qf ≈ s × s f (≈ s f ) and I (Q) × Qf is homotopy dense in Q × Q (≈ Q), I (Q) × Qf is not a Zσ -set in Q × Q because it is not σ -compact. Remark 5.15 Although 2f (), 2f () × 2Q , and 2f () × 2 are not homeomorphic to each other, their countable products are homeomorphic to each other. Indeed, since 2f () ≈ 2f () × R, we have the following: 2f ()N ≈ 2f ()N × RN = 2f ()N × s ≈ 2f ()N × s N ≈ (2f () × s)N ≈ (2f () × 2 )N . Moreover, 2 ≈ 2 × Q. Then, it follows that (2f () × 2 )N ≈ (2f () × 2 × Q)N ≈ (2f () × 2 )N × QN ≈ 2f ()N × QN ≈ (2f () × Q)N ≈ (2f () × 2Q )N . Additionally, the countable power (2 () × 2f )N is also homeomorphic to 2f ()N . More generally, let F be a C-absorption base for 2 (), where Mfd 0 ⊂ C ⊂ N 2 N M1 (τ ). Then, we can prove F ≈ f () . This will be done in Sect. 5.10 (see Theorem 5.12.10).

5.6 C-Universality and Cσ -Universality In the previous section, we have considered four types of absorption bases for Hilbert space 2 () and manifolds modeled on those spaces. To consider a more general situation, the strong Cσ -universality is important. In this section, we discuss when the strong C-universality implies the strong Cσ -universality. Evidently, Cσ is closed hereditary, additive, and topological because C is. A class C is said to be open hereditary if C contains any open subspace of a member of C. Proposition 5.6.1 The class Cσ is open hereditary.

5.6 C-Universality and Cσ -Universality

409

 Proof Let X ∈ Cσ , that is, X = n∈N Xn , where Xn ∈ C, n ∈ N. Each open subspace  U of X is Fσ in X, that is, there are closed sets Fi , i ∈ N, in X such that U =  i∈N F i . Since C is closed hereditary, Xn ∩ Fi ∈ C for each n, i ∈ N. Then, U = n∈N i∈N (Xn ∩ Fi ) ∈ Cσ . " ! By virtue of the above proposition, the following two propositions can be applied to the class Cσ even if C is not open hereditary. Proposition 5.6.2 Suppose that C is open hereditary and let X be an ANR. Then, X is strongly C-universal if and only if every open subspace of X is fine C-universal. Proof The “only if” part follows from Proposition 5.4.18. To see the “if” part, let f : C → X be a map of C ∈ C such that f |D is a strong Z-embedding for a closed set D in C. By Proposition 2.8.12, f is arbitrarily close to a map g : C → X such that g|D = f |D, g(C \ D) ∩ f (D) = ∅ and g is closed over g(D) = f (D). Then, C \ D ∈ C because C is open hereditary. By the fine C-universality of X \ g(D), g|C \ D is arbitrarily close to a strong Z-embedding h : C \ D → X \ g(D). If h is sufficiently close to g|C \ D, then the extension h : C → X of h defined by h|D = g|D is continuous and closed over g(D). Thus, we have a strong Zembedding h arbitrarily close to f such that h|D = f |D. This means that X is strongly C-universal. " ! Remark 5.16 As already seen in Proposition 5.4.18, the converse implication in the above proposition also holds even if C is not open hereditary. Applying Michael’s Theorem 1.3.20 on local property, we prove the following proposition: Proposition 5.6.3 Let X be an ANR such that every Z-set is a strong Z-set. If each point of X has a strongly C-universal neighborhood, then X is strongly C-universal. Proof To apply Michael’s Theorem 1.3.20, it is enough to prove the following: (G-1) (G-2) (G-3)

For open sets U ⊃ V in X, if U is strongly C-universal, then V is strongly C-universal; For two open sets U1 and U2 in X, if each Ui is strongly C-universal, then U1 ∪ U2 is strongly C-universal; For a discrete collection {Uλ }λ∈ of open sets in X, if each Uλ is strongly  C-universal, then λ∈ Uλ is strongly C-universal.

Since (G-1) is a direct consequence of Proposition 5.4.18 and (G-3) is trivial, we only have to show (G-2). We write U = U1 ∪ U2 . Let f : A → U be a map from A ∈ C such that f |B is a Z-embedding for a closed set B in A. Choose open sets Wi in U , i = 0, 1, 2 so that U1 \ U2 ⊂ W0 ⊂ clU W0 ⊂ W1 ⊂ clU W1 ⊂ W2 ⊂ clU W2 ⊂ U1 .

410

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

For each open cover U of U , let V be an open star-refinement of U such that st(clU W1 , V) ⊂ W2 , st(U \ W0 , V) ⊂ U2 and st(U \ W1 , st2 V) ⊂ U \ clU W0 . See Fig. 5.2. Since C is closed hereditary, we have C = (B ∩ f −1 (clU W2 )) ∪ f −1 (clU W1 ) ∈ C, which is closed in A. Note that f |B ∩ f −1 (clU W2 ) is a strong Z-embedding into U1 . Since U1 is strongly C-universal, we have a strong Z-embedding g : C → U1 such that g|B ∩ f −1 (clU W2 ) = f |B ∩ f −1 (clU W2 ) and g V f |C rel. B ∩ f −1 (clU W2 ). Since st(clU W1 , V) ⊂ W2 , it follows that g(f −1 (clU W1 )) ⊂ W2 , so g(f −1 (clU W1 )) ∩ f (f −1 (U \ W2 )) = ∅. Using the Homotopy Extension Theorem 1.13.11, we can extend g to a map g˜ : A → U such that g|B ˜ ∪ f −1 (U \ W2 ) = f |B ∪ f −1 (U \ W2 ) and g˜ V f rel. B ∪ f −1 (U \ W2 ).

U1

A B f −1 (U

2)

f −1 (clU W 2 ) f

W1

W0

f −1 (U 1 )

W2 f −1 (U

W0)

Fig. 5.2 Approximating a map by using Z-embeddings

U2

5.6 C-Universality and Cσ -Universality

411

Then, g|B ˜ ∪ C is a strong Z-embedding because g|C ˜ = g|C and g|B ˜ = f |B are strong Z-embeddings and   g(C) ˜ ∩ g(B) ˜ = g(B ∩ f −1 (clU W2 )) ∪ g(f −1 (clU W1 )) ∩ f (B)     = f (B ∩ f −1 (clU W2 )) ∩ f (B) ∪ g(f −1 (clU W1 )) ∩ f (B)   = f (B ∩ f −1 (clU W2 )) ∪ g(f −1 (clU W1 )) ∩ f (B ∩ f −1 (W2 ))   = f (B ∩ f −1 (clU W2 )) ∪ g(f −1 (clU W1 )) ∩ g(B ∩ f −1 (W2 ))  = g(B ∩ f −1 (clU W2 )) ∪ g(B ∩ f −1 (clU W1 )) = g(B ∩ f −1 (clU W2 )) = g(C ˜ ∩ B) Since C is closed hereditary, we have f −1 (U \ W0 ) ∈ C. Observe that g(f ˜ −1 (U \ W0 )) ⊂ st(U \ W0 , V) ⊂ U2 . So, the restriction g|f ˜ −1 (U \ W0 ) is a map into U2 . Let D = (B ∪ C) ∩ f −1 (U \ W0 ) = (B \ f −1 (W0 )) ∪ f −1 (clU W1 \ W0 ). Then, g|D ˜ is a strong Z-embedding into U2 . Since U2 is strongly C-universal, we have a strong Z-embedding h : f −1 (U \ W0 ) → U2 such that ˜ −1 (U \ W0 ) rel. D. h|D = g|D ˜ and h V g|f Since h st V f |f −1 (U \ W0 ), we have h(f −1 (U \ W1 )) ⊂ st(U \ W1 , st V). Moreover, g(f −1 (clU W0 )) ⊂ st(clU W0 , V). Observe that st(U \ W1 , st2 V) ⊂ U \ clU W0 ⇒ st(U \ W1 , st V) ∩ st(clU W0 , V) = ∅. Thus, it follows that h(f −1 (U \ W1 )) ∩ g(f −1 (cl W0 )) = ∅. Since h|f −1 (clU W1 \ W0 ) = g|f −1 (clU W1 \ W0 ), we can extend h to a map ˜ −1 (clU W0 ) = g|f −1 (clU W0 ). Then, h˜ is a strong Z-embedding h˜ : A → U by h|f −1 ˜ ˜ −1 (clU W1 ) = g|f −1 (clU W1 ) because h|f (U \ W1 ) = h|f −1 (U \ W1 ) and h|f are strong Z-embeddings and ˜ −1 (U \ W1 )) ∩ h(f ˜ −1 (clU W1 )) = h(f −1 (U \ W1 )) ∩ g(f −1 (clU W1 )) h(f = h(f −1 (U \ W1 )) ∩ g(f −1 (clU W1 \ W0 )) = h(f −1 (U \ W1 )) ∩ h(f −1 (clU W1 \ W0 ))

412

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

= h(f −1 (U \ W1 ) ∩ f −1 (clU W1 \ W0 )) ˜ −1 (U \ W1 ) ∩ f −1 (clU W1 ). = h(f ˜ = f |B because Since st V ≺ U, h˜ is st V-close to f . Moreover, h|B ˜ ∩ f −1 (U \ W0 ) = g|B ˜ ∩ f −1 (U \ W0 ) = f |B ∩ f −1 (U \ W0 ) and h|B ˜ ∩ f −1 (clU W0 ) = g|B ∩ f −1 (clU W0 ) = f |B ∩ f −1 (clU W0 ). h|B This completes the proof.

" !

A metrizable space X is called a Zσ -space (resp. a strong Zσ -space) is such that X itself is a Zσ -set (resp. a strong Zσ -set) in X. Since every closed set in a strong Zσ -space X is a strong Zσ -set in X, we have the following from Proposition 3.2.11: Lemma 5.6.4 Let X be an ANR with w(X) = τ . If X is a strong Zσ -space with the τ -discrete cells property, then every Z-set in X is a strong Z-set. " ! For a strong Zσ -space being an ANR, a combination of the strong C-universality and the τ -discrete cells property implies the strong C∞ -universality, that is: Proposition 5.6.5 Let X be a strong Zσ -space that is an ANR with the τ -discrete cells property, where w(X) = τ . If X is strongly C-universal, then X is strongly Cσ -universal. Proof Since Cσ is open hereditary, Proposition 5.6.2 can be applied to Cσ . Hence, it suffices to show that each non-empty open set U in X satisfies condition (auCσ ) of Proposition 5.6.2. Observe that U is an ANR with w(U ) = τ ,15 and a strongly Cuniversal strong Zσ -space by Propositions 5.4.18 and 2.8.14. Moreover, U has the τ -discrete cells property by Proposition 3.2.3. Thus, U satisfies the same condition as X. Consequently, it is enough to show that X is fine Cσ -universal. Since X has the τ -discrete cells property, every Z-set in X is a strong Z-set by Lemma 5.6.4. Hence, “strong Z-embedding” can be replaced by “Z-embedding.” Let f : C → X be a map of C ∈ Cσ . For each U ∈ cov(X), by 1.3.22(1), we can choose d ∈ Metr(X) so that {Bd (x, 1) | x ∈ X} ≺ U. Then, it suffices to  construct a Z-embedding f∗ : C → X such that d(f∗ , f ) < 1. We write X = i∈N Xi , where ∅ = X1 ⊂ X2 ⊂ · · · are strong Z-sets in X. (Particular Case) If C is an open set in some member of C, we can write  C = i∈N Ci , where each Ci is closed in C, Ci ∈ C and Ci ⊂ int Ci+1 . We shall inductively construct maps fi : C → X, αi : X → (0, 2−i ], i ∈ N, such that (1) fi |Ci is a Z-embedding,

15 Indeed, w(U )  w(X)  τ and, using the τ -discrete cells property of X, we can easily prove w(U )  τ .

5.6 C-Universality and Cσ -Universality

(2) (3) (4) (5)

413

fi |Ci−1 = fi−1 |Ci−1 , cl fi (C \ int Ci+1 ) ∩ Xi+1 = ∅ and cl fi (C \ int Ci ) ∩ Xi = ∅, fi ∈ Nαi (fi−1 ), αi (x)  12 d(x, Xi ) for each x ∈ cl fi−1 (C \ int Ci ) and αi (x)  12 d(x, Xi−1 ) for each x ∈ cl fi−1 (C \ int Ci−1 ),

where f0 = f and C0 = ∅. Assume that f0 ,. . . ,fi−1 have been obtained. By (3), we can apply the Tietze Extension Theorem 1.3.1 to obtain maps β, β  : X → (0, ∞) such that β(x) = d(x, Xi ) for x ∈ cl fi−1 (C \ int Ci ), 

β (x) = d(x, Xi−1 ) for x ∈ cl fi−1 (C \ int Ci−1 ). We define a map αi : X → (0, 2−i ] by   αi (x) = min 2−i , 12 β(x), 12 β  (x) for each x ∈ X. Then, αi satisfies (5). By Proposition 3.1.2, we have V ∈ cov(X) such that (st V)(fi−1 ) ⊂ Nαi (fi−1 ). By the fine C-universality of X, we have a Z-embedding vi : Ci → X such that vi |Ci−1 = fi−1 |Ci−1 , and vi V fi−1 |Ci . By the Homotopy Extension Theorem 1.13.11, vi extends to a map v˜i : C → X, that is V-homotopic to fi−1 . Since Xi+1 is a strong Z-set in X, we have a V-homotopy h : X × I → X such that clX h1 (X) ∩ Xi+1 = ∅. We define fi : C → X by fi (x) = hλ(x) (v˜i (x)) for each x ∈ C, where λ : X → I is an Urysohn map with λ(Ci ) = 0 and λ(C \ Ci+1 ) = 1. Then, fi satisfies the conditions (1)–(4). By (2) and (4), we have a map f∗ : C → X such that f∗ |Ci = fi |Ci for each i ∈ N and d(f∗ , f )  1, hence f∗ is U-close to f . By (3) and (5), we can see cl f∗ (C \ int Ci ) ∩ Xi = ∅ for each i ∈ N. Thus, f∗ is an embedding and f∗ (C) =

 i∈N

f∗ (Ci \ int Ci−1 ) =



fi (Ci \ int Ci−1 )

i∈N

is a locally finite union of Z-sets in X, hence it is a Z-set in X. Therefore, f∗ is a Z-embedding.  (General Case) We now consider the general case C ∈ Cσ , that is, C = i∈N Ci , where C1 ⊂ C2 ⊂ · · · are closed in C and Ci ∈ C. In this case, it cannot be assumed that Ci ⊂ int Ci+1 . For each k ∈ N, since C(Ik , X) with the sup-metric has the same weight as X, there is a map gk : Ik ×  → X such that {gk,γ | γ ∈ } is dense in C(Ik , X), where gk,γ : Ik → X is defined by gk,γ (x) = gk (x, γ ). Given

414

5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

an open cover U of X, let U0 be an open star-refinement of U. By induction, we shall construct maps fi : C → X, gki : Ik ×  → X (k  i), and open covers Ui of X \ (fi (Ci ) ∪ Xi ), i ∈ N, such that: (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

fi |Ci is a Z-embedding, fi |Ci−1 = fi−1 |Ci−1 , fi (C \ Ci ) ∩ fi (Ci ) = ∅, fi is closed over fi (Ci ) ∪ Xi , fi |C \ Ci−1 is Ui−1  -close to fi−1 |C \ Ci−1 ,  cl fi (C \ Ci−1 ) ∩ Xi \ (fi−1 (Ci−1 ) ∪ Xi−1 ) = ∅, st Ui ≺ Ui−1 , diam U < min{2−(i+1), 12 d(U, fi (Ci ) ∪ Xi )} for each U ∈ Ui , gki (Ik × ) is a Z-set in X,  j fi (C) ∩ kj i gk (Ik × ) = ∅, i | γ ∈ } is 2−i -dense in C(Ik , X), that is, each g ∈ C(Ik , X) is 2−i -close {gk,γ i , to some gk,γ

where f0 = f and C0 = X0 = ∅. Assume that fi−1 , gki−1 (k  i − 1) and Ui−1 have been obtained. By (6), fi−1 (Ci−1 ) is a Z-set in X, so a strong Z-set in X. Using the τ -discrete cells property, we have a map gki : Ik ×  → X (k  i) such that gki (Ik × ) ∩ fi−1 (Ci−1 ) = ∅, and each gki is 2−(i+1) -close to gk , hence it satisfies (16). Since X is a Zσ -space, every closed set is a Zσ -set in X. Because of compactness, gki (Ik × {γ }) is a Z-set in X for each γ ∈  (Proposition 2.8.8). Their discrete union gki (Ik × ) is also a Z-set in X (Corollary 2.8.7). Thus, gki satisfies (14) too. Now, we write W = X \ (fi−1 (Ci−1 ) ∪ Xi−1 ). Then, Ui−1 is an open cover of W . Let V be an open star-refinement of Ui−1 . The open set W in X is a strong Zσ -space and it has τ -discrete cells property (Proposition 3.2.3), hence each Z-set in W is a strong Z-set by Proposition 3.2.11. Note that Xi ∩ W is a strong Z-set in W by Proposition 2.8.14 and W is strongly C-universal by Proposition 5.4.18. We apply the particular case to the open set Ci \ Ci−1 in Ci ∈ C, and use the Homotopy Extension Theorem 1.13.11 to construct a map h : C \ Ci−1 → W such that: (17) h|Ci \ Ci−1 is a Z-embedding, (18) h is V-close to fi−1 |C \ Ci−1 ,    j (19) cl h(C \ Ci−1 ) ∩ W ∩ Xi ∪ kj i gk (Ik × ) = ∅.

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Since h(Ci \ Ci−1 ) ∪ (Xi ∩ W ) is a strong Z-set in W , we apply Proposition 2.8.12 to obtain a map h˜ : C \ Ci−1 → W such that: (20) (21) (22) (23) (24)

h˜ is V-close to h, hence it is Ui−1 -close to fi−1 |C \ Ci−1 by (18),    j k ˜ \ Ci−1 ) ∩ W ∩ Xi ∪ cl h(C kj i gk (I × ) = ∅, ˜ i \ Ci−1 = h|Ci \ Ci−1 , h|C ˜ \ Ci ) ∩ h(C ˜ i \ Ci−1 ) = ∅, h(C ˜ i \ Ci−1 ) ∪ (Xi ∩ W ). h˜ is closed over h(C

For each z ∈ Ci−1 and ε > 0, since fi−1 is continuous, we have a neighborhood V of z in C such that y ∈ V implies d(fi−1 (y), fi−1 (z)) < ε/2. For each ˜ y ∈ V \ Ci−1 , choose U ∈ Ui−1 so that h(y), fi−1 (y) ∈ U , whence we have 1 ˜ d(h(y), fi−1 (y)) < 2 d(fi−1 (y), fi−1 (z)) by (13) for i − 1. Then, we have ˜ ˜ fi−1 (y)) + d(fi−1 (y), fi−1 (z)) d(h(y), fi−1 (z))  d(h(y), < 32 d(fi−1 (y), fi−1 (z)) < ε. ˜ we can obtain the map fi : C → X satisfying (7), Therefore, as an extension of h, that clearly satsfies (8), (10), (11), and (15) (cf. (23), (20), (21)). Since fi |Ci−1 = fi−1 |Ci−1 and fi |Ci \ Ci−1 = h|Ci \ Ci−1 are injective and ˜ i \ Ci−1 ) ∩ fi−1 (Ci−1 ) = ∅, fi (Ci \ Ci−1 ) ∩ fi−1 (Ci−1 ) = h(C it follows that fi |Ci is injective. If fi satisfies (9), that is, fi is closed over fi (Ci ) ∪ Xi , then fi |Ci is an embedding. Suppose that fi is not closed over fi (Ci ) ∪ Xi . Then, there exist a ∈ fi (Ci ) ∪ Xi , a neighborhood U of fi−1 (a) in C (we allow U = fi−1 (a) = ∅) and a sequence (zn )n∈N in C \ U with limn→∞ fi (zn ) = a. Since fi |Ci−1 = fi−1 |Ci−1 is a closed embedding into X by (6) for i − 1, we have zn ∈ C \ Ci−1 for sufficiently large n ∈ N. Since fi |C \ Ci−1 is closed over fi (Ci \ Ci−1 ) ∪ (Xi ∩ W ) by (24), it follows that a ∈ fi (Ci \ Ci−1 ) ∪ (Xi ∩ W ). Recall a ∈ fi (Ci ) ∪ Xi . Then, we have a ∈ fi (Ci−1 ) ∪ (Xi \ W ) = fi−1 (Ci−1 ) ∪ Xi−1 . For sufficiently large n ∈ N, we can choose Un ∈ Ui−1 so that fi (zn ), fi−1 (zn ) ∈ Un by (10), whence d(fi−1 (zn ), a)  d(fi (zn ), fi−1 (zn )) + d(fi (zn ), a) < 32 d(fi (zn ), a). −1 (a) = ∅ by (9) for i − 1. Since Then, limn→∞ fi−1 (zn ) = a, which implies that fi−1 −1 −1 fi−1 (a) ⊂ Ci−1 by (8) for i − 1, it follows from (7) that fi−1 (a) ⊂ fi−1 (a) ⊂ U . −1 (V ) ⊂ U . Again by (9) for i − 1, we have a neighborhood V of a in X such that fi−1 −1 For sufficiently large n ∈ N, fi−1 (zn ) ∈ V , hence zn ∈ fi−1 (V ) ⊂ U . This is a contradiction. Therefore, fi satisfies (9).

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To see (6), it suffices to show that fi (Ci ) is a Z-set in X. Observe that X \ (fi (Ci ) ∪ Xi−1 ) = W \ h(Ci \ Ci−1 ), which is open in W . Then, fi (Ci ) ∪ Xi−1 is closed in X, hence fi (Ci ) ∪ Xi is also closed in X. Since fi (Ci \ Ci−1 ) = h(Ci \ Ci−1 ) is a Z-set in W and X \ W = (fi−1 (Ci−1 ) ∪ Xi ) is a Z-set in X, it follows that fi (Ci ) ∪ Xi is a Z-set in X. By (8) and (9), we can see that fi (Ci ) is closed in fi (Ci ) ∪ Xi . Therefore, fi (Ci ) is a Z-set in X. Finally, by choosing an open cover Ui of X \ (fi (Ci ) ∪ Xi ) so as to satisfy (12) and (13), we can obtain fi , gki (k  i) and Ui that satisfy all conditions (6)–(16). By (7), we can define f∗ : C → X defined by f∗ |Ci = fi |Ci . It follows from (10) and (13) that f∗ is 2−i -close to fi . Thus, f∗ is the uniform limit of (fi )i∈N , so f∗ is continuous. By (6), f∗ is injective. Then, to see that f∗ is a Z-embedding, it remains to be shown that f∗ is a closed map and f∗ (C) is a Z-set in X. Now, assume that f∗ is not closed. Then, we have a sequence (zn )n∈N in C such that (zn )n∈N has no convergent subsequences but (f∗ (zn ))n∈N converges to some a ∈ X. Let a ∈ Xm \ Xm−1 . Then, zn ∈ C \ Cm for sufficiently large n ∈ N. Otherwise, Cm contains a subsequence of (zn )n∈N , that is convergent because f∗ |Cm = fm |Cm is a closed embedding. From (7), (10) and (12), it follows that f∗ |C \ Cm is st Um -close to fm |C \ Cm . By (13), we have xn , yn ∈ X for sufficiently large n ∈ N such that d(f∗ (zn ), xn ) < 12 d(f∗ (zn ), a), d(xn , yn ) < 12 d(xn , a), and d(yn , fm (zn )) < 12 d(yn , a). Then, (fm (zn ))n∈N also converges to a, hence a ∈ cl fm (C \ Cm ) ⊂ cl fm (C \ Cm−1 ), which implies that a ∈ fm−1 (Cm−1 ) by (11). By (6), (7) and (8), there is unique c ∈ Cm−1 such that fm (c) = fm−1 (c) = a, where fm−1 (a) = {c} by (8). Since (zn )n∈N does not converge to c and fm is closed over fm (Cm ) by (9), we have a neighborhood V of a in X such that infinitely many zn are not contained in fm−1 (V ), that is, infinitely many fm (zn ) are not contained in V . This is a contradiction. Therefore, f∗ is a closed map. To see that f∗ (C) is a Z-set in X, let g : Ik → X be a map and ε > 0. Choose j j  k so that 2−j < ε. Then, g is ε-close to some gk,γ by (16), whence fi (Ci ) ∩  j gk,γ (Ik ) = ∅ for every i  j by (15). Since f∗ (C) = ij fi (Ci ), it follows that j

f∗ (C) ∩ gk,γ (Ik ) = ∅. Hence, f∗ (C) is a Z-set in X. As a corollary, we have the following:

" !

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Corollary 5.6.6 Suppose that In ×  ∈ C for every n ∈ N. Let X be an ANR that is a strong Zσ -space. Then, the following are equivalent: (a) (b) (c) (d)

X is strongly C-universal; X is strongly Cσ -universal; Every open set in X is strongly Cσ -universal; Every open set in X is fine Cσ -universal.

Proof By Propositions 5.6.5 and 5.4.18, we have (a) ⇒ (b) ⇒ (c). The implications (b) ⇒ (a) and (c) ⇒ (d) are trivial. Since Cσ is open hereditary, the implication (d) ⇒ (b) follows from Proposition 5.6.2. " ! In Proposition 5.6.5, it is assumed that In ×  ∈ C for every n ∈ N. However, this assumption is not necessary when C is a class of separable metrizable spaces and X is separable. The following are the separable versions of Proposition 5.6.5 and Corollary 5.6.6: Proposition 5.6.7 Let X be a separable ANR and C ⊂ M(ℵ0 ). If X is a strongly C-universal strong Zσ -space, then X is strongly Cσ -universal. Corollary 5.6.8 Let X be a separable ANR that is a strong Zσ -space and C ⊂ M(ℵ0 ). Then, the following are equivalent: (a) (b) (c) (d)

X is strongly C-universal; X is strongly Cσ -universal; Every open set in X is strongly Cσ -universal; Every open set in X is fine Cσ -universal.

Indeed, in the separable case, Lemma 5.6.4 holds without assuming the discrete cells property, that is, the following is obtained from Proposition 3.2.12: Lemma 5.6.9 Let X be a separable ANR. If X is a strong Zσ -space, then every Z-set in X is a strong Z-set. " ! Under the assumption that C is I-stable and  ∈ C, the C-universality of X implies the τ -discrete cells property of X (Proposition 5.4.6). However, when X is separable, the discrete cells property is not necessary because Lemma 5.6.9 above can be used instead of Lemma 5.6.4. Thus, for a separable ANR X, the first half (before the general case) of the proof of Proposition 5.6.5 is valid without assuming that C is I-stable and  ∈ C. In the second half (the general case) of the proof, using the condition that Ik ×  ∈ C, we obtained a dense set {gk,γ | γ ∈ } in C(Ik , X). However, when X is separable, the space C(Ik , X) with the sup-metric is also separable, that is, it has a countable dense set {gk,n | n ∈ N}. In the proof, using gk,n instead of gk,γ , we can i inductively construct maps gk,n : Ik → X, k, n  i, so as to satisfy the following conditions:  j i (Ik ) is a Z-set (hence k,nj i gk,n (Ik ) is a Z-set in X), (14 ) gk,n  j (15 ) fi (C) ∩ k,nj i gk,n (Ik ) = ∅,

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i (16 ) d(gk,n , gk,n ) < 2−(i+1) . i : Ik → In the inductive step, since fi−1 (Ci−1 ) is a Z-set in X, we can find maps gk,n i X \ fi−1 (Ci−1 ), k, n  i, so as to satisfy (16 ). Then, gk,n (Ik ) is a compact (strong) i Zσ -set in X because X is a strong Zσ -space. Hence, gk,n (Ik ) is a Z-set in X by Proposition 2.8.8, so (14 ) is also satisfied. Moreover, the discrete cells property of W (= X\(fi−1 (Ci−1 )∪Xi−1 )) is not necessary for the same reason as X mentioned i above. Thus, for a separable ANR X, we can construct maps fi : C → X, gk,n : k I → X, k, n  i, and open covers Ui of X \ (fi (Ci ) ∪ Xi ), i ∈ N, so as to satisfy conditions (6)–(13) and (14)–(16) without assuming that C is I-stable and  ∈ C. By the same definition as in the proof of Proposition 5.6.5, we define f∗ : C → X. By the same proof, we can show that f∗ is an closed embedding and d(f∗ , f ) < 1. To see that f∗ (C) is a Z-set in X, let g : Ik → X be a map and ε > 0. Since {gk,n | n ∈ N} is dense in C(Ik , X), g is ε/2-close to some gk,n . Choose j  k, n j j so that 2−j < ε. Since d(gk,n , gk,n ) < 2−(j +1) < ε/2, we have d(gk,n , g)
0. Proof For each map f : A → X, let f0 = f prA : A × I → X. Since X is an ANR, we can apply the assumption to obtain a map g : A × I → X such that f0 2−2 g and (g(Aλ × I))λ∈ is locally finite in X. Then, we can easily construct a map f1 : A×I → X such that f1 |A×[0, 1/2] = f0 |A×[0, 1/2], f1 |A×{1} = g|A×{1}, and d(f1 (x, t), f (x)) = d(f1 (x, t), f0 (x, t)) < 2−2 for every (x, t) ∈ A × I. By induction, we can construct maps fn : A × I → X, n ∈ N, so that (fn (Aλ × [2−n+1 , 1]))λ∈ is locally finite in X, fn |A × ([0, 2−n ] ∪ [2−n+2 , 1]) = fn−1 |A × ([0, 2−n ] ∪ [2−n+2 , 1]), and d(fn (x, t), fn−1 (x, t)) < 2−n−1 for every (x, t) ∈ A × I, where [2−n+2 , 1] = ∅ if n = 1. Indeed, assume that fn has been constructed. Since (fn (Aλ × [2−n+1 , 1]))λ∈ is locally finite in X, we have an open cover U such that each U ∈ U meets only finitely many fn (Aλ × [2−n+1 , 1]), λ ∈ . Take V ∈ cov(X) so that st V ≺ U and mesh V < 2−n−2 . Then, each V ∈ V meets only finitely many st(fn (Aλ × [2−n+1 , 1]), V), λ ∈ , which implies that (st(fn (Aλ × [2−n+1 , 1]), V))λ∈ is also locally finite in X. For each λ ∈ , choose an open neighborhood Wλ of Aλ × [2−n+1 , 1] in Aλ × I so that fn (Wλ ) ⊂ st(fn (Aλ × [2−n+1 , 1]), V). Since X is an ANR, we can apply the assumption to obtain a map g : A × I → X such that fn V g and (g(Aλ × I))λ∈ is locally finite in X. Then, we can easily construct a map fn+1 : A × I → X such that fn+1 is V-close to fn , fn+1 |A × ([0, 2−n−1 ] ∪ [2−n+1 , 1]) = fn |A × ([0, 2−n−1 ] ∪ [2−n+1 , 1]), and fn+1 |(Aλ × [2−n , 1]) \ Wλ = g|(Aλ × [2−n , 1]) \ Wλ for each λ ∈ ,

20 This is also called the countable locally finite approximation property—see the footnote on p. 238.

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Since mesh V < 2−n−2 , it follows that d(fn+1 (x, t), fn (x, t)) < 2−n−2 for every (x, t) ∈ A × I. Moreover, for each λ ∈ , fn+1 (Aλ × [2−n , 1]) ⊂ g(Aλ × I) ∪ st(fn (Aλ × [2−n+1 , 1], V), which implies that (fn+1 (Aλ × [2−n , 1]))λ∈ is locally finite in X. Thus, fn+1 satisfies the desired properties. Finally, let f¯ = limn→∞ fn : A × I → X be the uniform limit of (fn )n∈N . For each t > 0, take n ∈ N so that 2−n < t  2−n+1 . Since f¯|A × [2−n , 1] = fn+1 |A × [2−n , 1] and (fn+1 (Aλ × [2−n , 1])λ∈ is locally finite in X. Hence, (f¯(Aλ × [t, 1])λ∈ is also locally finite in X. Moreover, observe fn−1 |A × [0, 2−n+1 ] = · · · = f0 |A × [0, 2−n+1 ] = f prA |A × [0, 2−n+1 ]. Then, it follows that d(f¯(x, t), f (x)) = d(fn+1 (x, t), fn (x, t)) + d(fn (x, t), fn−1 (x, t)) < 2−n−2 + 2−n−1 < 2−n < t. Thus, f¯ satisfies (1) and (2).

" !

Proof (Theorem 5.8.1) (The “only if” Part) Assume that X is a homotopy dense set in an 2 ()-manifold M. Then, w(X) = w(M) = τ and X is an ANR because every homotopy dense set in an ANR is also an ANR (Theorem 1.13.26). We will show that X has the τ -locally finite cells property (resp. the locally finite displacing property). Let f : In ×  → X be a map. For each U  ∈ cov(X), there is an open  in M such that U|X   is an 2 ()-manifold collection U = U. Then, N = U  f is V-close to a as an open set in M. For a star-refinement V ∈ cov(N) of U, n map g : I ×  → X → N (resp. prN : N × N → N is V-close to a map g : N × N → N) such that (gγ (In ))γ ∈ (resp. (g(N × {n}))n∈N ) is locally finite in N. Then, V has an open refinement W ∈ cov(N) such that the star st(W, W) of each W ∈ W meets only finitely many gγ (In ), γ ∈  (resp. g(N × {n}), n ∈ N), which means that each W ∈ W meets only finitely many st(gγ (In ), W), γ ∈  (resp. st(g(N × {n}), W), n ∈ N). Since X is homotopy dense in M, there is a map h : M → X that is W-close to id. Thus, we have a map hg : In ×  → X (resp. hg|X × N : X × N → X) that is V-close to f (resp. prX = prN |X × N). Since each W ∈ W meets only finitely many hgγ (In ), γ ∈  (resp. hg(X × {n}), n ∈ N) (hgγ (In ))γ ∈ (resp. (hg(X × {n}))n∈N ) is locally finite in X. (The “if” Part) Assume that X is an ANR with w(X) = τ and X has both the τ -locally finite cells property and the locally finite τ -skeletal tower property. By combination Theorems 1.3.10, 1.3.17 (2), and Proposition 1.13.36 (or

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Theorems 1.13.37 and 1.13.35), X can be embedded into a completely metrizable  as a homotopy dense set, where w(X)  = w(X) = τ . Let d ∈ Metr(X).  ANR X n For each n ∈ N, let Fn be a dense subset of C(I , X) with card Fn = dens C(In , X) = w(C(In , X) = w(X) = card  (cf. Proposition 1.1.2 (5)). Taking a bijection ηn :  →  × Fn , we define a map fn : In ×  → X by fn (x, γ ) = prFn ηn (γ )(x) for each (x, γ ) ∈ In × . Since X has the τ -locally finite cells property, we can apply Lemma 5.8.2 to obtain a map f¯n : In ×  × I → X such that: (1) d(f¯n (x, γ , t), fn (x, γ )) < t for each (x, γ , t) ∈ In ×  × I, (2) for every t > 0, (f¯n (In × {γ } × [t, 1]))γ ∈ is locally finite in X.  so that Then, for each m ∈ N, we can find an open neighborhood Umn of X in X (3) (f¯n (In × {γ } × [1/m, 1]))γ ∈ is locally finite in Unm . Since X has the locally finite displacing property, we can again apply Lemma 5.8.2 to obtain a map p : X × N × I → X such that: (4) d(p(x, n, t), x) < t for each (x, n, t) ∈ X × N × I, (5) for every t > 0, (p(X × {n} × [t, 1]))n∈N is locally finite in X.  so that Again, for each m ∈ N, we can find an open neighborhood Vm of X in X (6) (p(X × {n} × [1/m, 1]))n∈N is locally finite in Vm .  We will show that M = n,m∈N (Umn ∩ Vm ) is the required 2 ()-manifold.  (Theorem 1.3.17(2)). Since X is First, M is completely metrizable as a Gδ set in X  X is also homotopy dense in M. From Theorem 1.13.26 it homotopy dense in X, follows that M is an ANR. To prove the τ -locally finite cells property of M, let g : In ×  → M be a map. For each γ ∈ , let gγ : In → M be the map defined by gγ (x) = g(x, γ ) for x ∈ In . For each U ∈ cov(M), let V ∈ cov(M) be a star-refinement of U. Since X is homotopy dense in M, we have a map h : M → X that is V-close to idM , which implies that hg is V-close to g. Because Fn is dense in C(In , X), each hgγ is V-close to a map kγ ∈ Fn . Then, hg is V-close to the map k : In ×  → X defined by k(x, γ ) = kγ (x) for each (x, γ ) ∈ In × . Moreover, we define an injection χ :  →  × Fn by χ(γ ) = (γ , kγ ) for each γ ∈ . Then, ξ = ηn−1 χ :  →  is an injection and fn (idIn × ξ ) = k. Indeed, for each (x, γ ) ∈ In × , fn (idIn × ξ )(x, γ ) = fn (x, ξ(γ )) = fn (x, ηn−1 (γ , kγ )) = prFn ηn (ηn−1 (γ , kγ ))(x) = kγ (x) = k(x, γ ).

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By virtue of Lemma 3.1.3, we can take a Lipschitz map α : M → (0, 1) with the Lipschitz constant  1 so that Nα (k) ⊂ V(k). We define a map k ∗ : In ×  → M as follows: k ∗ (x, γ ) = f¯n (x, ξ(γ ), αk(x, γ )) for each (x, γ ) ∈ In × . Then, it follows from (1) that for each (x, γ ) ∈ In × , d(k ∗ (x, γ ), k(x, γ )) = d(f¯n (x, ξ(γ ), αk(x, γ )), fn (x, ξ(γ ))) < αk(x, γ ). Therefore, k ∗ ∈ Nα (k) ⊂ V(k), so k ∗ is V-close to k. Since st V ≺ U, it follows that k ∗ is U-close to g. Moreover, it should be noticed that   αk ∗ (x, γ ) − αk(x, γ )  αk ∗ (x, γ ) − αk(x, γ )  d(k ∗ (x, γ ), k(x, γ )) < αk(x, γ ). Thus, we have αk ∗ (x, γ ) < 2αk(x, γ ). We will show that (k ∗ (In × {γ }))γ ∈ is locally finite in M. For each x ∈ M, Ux = {y ∈ M | α(y) > α(x)/2} is a neighborhood of x in M. For (y, γ ) ∈ In × , if k ∗ (y, γ ) ∈ Ux , then αk ∗ (y, γ ) > α(x)/2, which implies that αk(y, γ ) > α(x)/4 because αk ∗ (x, γ ) < 2αk(x, γ ). Hence, Ux ∩ k ∗ (In × {γ }) ⊂ Ux ∩ f¯n (In × {ξ(γ )} × [α(x)/4, 1]). Choose m ∈ N so that 1/m  α(x)/4. By (3), x has a neighborhood Vx in M which meets only finitely many f¯n (In × {γ } × [1/m, 1], γ ∈ . Since ξ :  →  is injective, Ux ∩ Vx meets only finitely many k ∗ (In × {γ }), γ ∈ . Therefore, (k ∗ (In × {γ }))γ ∈ is locally finite in M. We finally show the locally finite displacing property of M. For each U ∈ cov(M), let α : M → (0, 1) be a map such that Nα (prM ) ⊂ U(prM ), where prM : M × N → M is the projection. We can take a map h : M → X such that d(h(x), x) < α(x)/2 because X is homotopy dense in M. We define g : M × N → M as follows: g(x, n) = p(h(x), ¯ n, α(x)/2) for each (x, n) ∈ M × N. From (4), it follows that for each (x, n) ∈ M × N, d(g(x, n), prM (x, n))  d(p(h(x), n, α(x)/2, h(x)) + d(h(x), x) < α(x)/2 + α(x)/2 = α(x),

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which means that g ∈ Nα (prM ) ⊂ U(prM ), that is, g is U-close to prM . For each x ∈ M, Ux = {y ∈ M | α(y) > α(x)/2} is a neighborhood of x in M. For y ∈ M, if g(y, n) ∈ Ux , then αg(y, n) > α(x)/2. Hence, Ux ∩ g(M × {n}) ⊂ Ux ∩ p(h(M) ¯ × {n} × [α(x)/2, 1]). Choose m ∈ N so that 1/m  α(x)/2. By (6), x has a neighborhood Vx in M which meets only finitely many p(h(M) ¯ × {n} × [1/m, 1]), n ∈ N. Hence, Ux ∩ Vx meets only finitely many g(M × {n}), n ∈ N. Thus, we have shown the locally finite displacing property of M. " ! Since the locally finite cell-tower property is equivalent to the countable locally finite Hilbert cubes property, the following can be proved in the same way as the above. The details of the proof are left to the reader. Theorem 5.8.3 A space X can be embedded into an 2 -manifold as a homotopy dense set if and only if X is a separable ANR and X has the locally finite cell-tower property. In an 2 ()-manifold M, every Z-set is a strong Z-set (Theorem 2.9.3). Due to Proposition 2.9.4, every homotopy dense subset X of M has the same property. Thus, we have the following: Corollary 5.8.4 Let X be an ANR. In the following cases, every Z-set in X is a strong Z-set: (1) w(X) = τ > ℵ0 , and X has both the τ -locally finite cells property and the locally finite displacing property; (2) X is separable (i.e, w(X) = ℵ0 ) and X has the locally finite cell-tower property. " !

5.9 Four Types of Infinite-Dimensional Manifolds In this section, we characterize four types of manifolds considered in Sect. 5.5, that is, manifolds modeled on the following spaces: 2f (), 2f () × 2Q , 2f () × 2 , or 2 () × 2f . These model spaces can be embedded into 2 () as absorption bases for the classes Mfd 0 , M0 , M1 (ℵ0 ), or M1 (τ ). Due to Theorem 5.7.17, these manifolds can be characterized as strongly C-universal ANRs with the τ -discrete cells property which are strong Zσ -spaces in the class (τ C)σ , where C is one of Mfd 0 , M0 , M1 (ℵ0 ), or M1 (τ ). Thus, we have to characterize spaces belonging to the classes (τ Mfd 0 )σ , (τ M0 )σ , (τ M1 (ℵ0 ))σ , and M1 (τ )σ (= (M1 )σ (τ )).

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It is said that X is σ -completely metrizable if X ∈ (M1 )σ , that is, it is a countable union of completely metrizable closed subspaces, where (M1 )σ (τ ) = M1 (τ )σ . A space X is said to be locally separable if each point x ∈ X has a separable neighborhood in X. Evidently, every space in τ (M1 (ℵ0 )) is locally separable. It is said that X is σ -locally separable if it is a countable union of locally separable metrizable closed subspaces. Proposition 5.9.1 Every locally separable metrizable space X is a topological sum (i.e., a discrete union) of separable metrizable spaces. Proof Since X is paracompact and locally separable, X has a locally finite open cover U such that each U ∈ U is separable. Then, U ∈ U has a countable dense subset D. Since each member of U[U ] contains some x ∈ D and each x ∈ D is contained in at most finitely many members of U[U ], it follows that U[U ] is countable. We define an equivalence relation ∼ on X as follows: x ∼ y ⇐⇒∃U0 , U1 , . . . Un ∈ U def

such that x ∈ U0 , y ∈ Un , Ui−1 ∩ Ui = ∅.

This equivalence relation decompose X into disjoint equivalence classes Xλ , λ ∈ . Since each Xλ is open in X, {Xλ | λ ∈ } is a discrete collection of closed sets in X. Thus, we have X = λ∈ Xλ . It is easy to see that each Xλ is a union of countably many members of U. Hence, each Xλ is separable. " ! As a corollary, we have the following: Corollary 5.9.2 In order that X ∈ τ (M1 (ℵ0 ))σ , it is necessary and sufficient that X is a σ -locally separable and σ -completely metrizable space with w(X)  τ . ! " Note that locally compact metrizable spaces are locally separable. Concerning locally compact spaces, we have the following: Proposition 5.9.3 Every locally compact metrizable space X is a countable union of closed sets Xn , n ∈ N, such that each Xn is a discrete union of compacta, that is, X ∈ (⊕M0 )σ , where (⊕M0 )σ (τ ) = (τ M0 )σ . Moreover, if X is strongly countabledimensional, then the condition dim Xn  n can be required, hence X ∈ (⊕Mfd 0 )σ , fd scd . where (⊕Mfd ) (τ ) = (τ M ) = (τ M ) 0 σ 0 σ 0 σ Proof By the same reasoning as Proposition 5.9.1, X has a locally finite open cover U such that cl U is compact for each U ∈ U. As is easily observed, U is starfinite, that is, U[U ] is finite for each U ∈ U (cf. Theorem 4.9.10 in [GAGT]). Then, using this cover U, we can define an equivalence relation ∼ on X as in the proof of Proposition 5.9.1, whichdecomposes X into disjoint equivalence classes Xλ , λ ∈ . Thus, we have X = λ∈ Xλ , where each Xλ is a union of countably many members of U. Then, we can write Xλ = n∈N cl Uλ,n , where Uλ,n ∈ U. For each n ∈ N,{cl Uλ,n | λ ∈ } is a discrete collection  of compacta in X. Then, we have Xn = λ∈ cl Uλ,n ∈ ⊕M0 , n ∈ N, and X = n∈N Xn ∈ (⊕M0 )σ .

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If X is strongly countable-dimensional, then there is a tower A1 ⊂ A2 ⊂ · · · of  closed sets in X such that X = n∈N An and dim An  n (cf. the Countable Sum Theorem 1.12.8 or the Locally Finite Sum Theorem 1.12.9). By replacing each Xn by An ∩ in Xi , the condition dim Xn  n can be satisfied. " ! Proposition 5.9.3 can be stated as follows: L(τ ) ⊂ (τ M0 )σ and L(τ )scd ⊂ (τ M0 )scd σ , where the equalities do not hold. In fact, (M0 )σ ⊂ L because Q, the space of rationals, is σ -compact but nowhere locally compact. A metrizable space said to be σ -locally compact if it is a countable union of locally compact metrizable closed subspaces. Thus, Lσ (Lσ (τ )) is the class of all σ locally compact metrizable spaces (with weight  τ ). The following easily follows from Proposition 5.9.3: scd L(τ )σ ⊂ (τ M0 )σ and L(τ )scd σ ⊂ (τ M0 )σ .

Since every member of ⊕M0 (or τ M0 ) is locally compact, the converse inclusion also holds, so we have scd (τ M0 )σ = L(τ )σ and (τ Mfd 0 )σ = L(τ )σ .

Corollary 5.9.4 The class (⊕M0 )σ (or (τ M0 )σ ) is equal to the class of all σ fd locally compact metrizable spaces (with weight  τ ) and (⊕Mfd 0 )σ (or (τ M0 )σ ) is the class of all strongly countable-dimensional and σ -locally compact metrizable spaces (with weight  τ ). " ! The above results are summarized as follows: scd • (τ Mfd 0 )σ = L(τ )σ —the class of all strongly countable-dimensional σ -locally compact metrizable spaces with weight  τ ; • (τ M0 )σ = L(τ )σ —the class of all σ -locally compact metrizable spaces with weight  τ ; • (τ M1 (ℵ0 ))σ —the class of all σ -locally separable and σ -completely metrizable spaces with weight  τ ; • M1 (τ )σ —the class of all σ -completely metrizable spaces with weight  τ .

Combining the above characterizations of four classes with Theorem 5.7.17, we can characterize manifolds modeled on the following spaces: 2f (), 2f () × 2Q , 2f () × 2 , or 2 () × 2f . Theorem 5.9.5 For a space X, the following are equivalent: (a) X is an 2f ()-manifold;

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437

 such that the pair (X,  X) is an (b) X can be embedded into an 2 ()-manifold X (2 (), 2f ())-manifold pair; (c) X can be embedded into an 2 ()-manifold as an Mfd 0 -absorption base; -universal, strongly countable-dimensional, and σ -locally (d) X is a strongly Mfd 0 compact ANR that is a strong Zσ -space with the τ -discrete cells property. Theorem 5.9.6 For a space X, the following are equivalent: (a) X is an (2f () × 2Q )-manifold;  such that the pair (X,  X) is an (b) X can be embedded into an 2 ()-manifold X 2 2 2 2 ( () ×  , f () × Q )-manifold pair; (c) X can be embedded into an 2 ()-manifold as an M0 -absorption base; (d) X is a strongly M0 -universal σ -locally compact ANR that is a strong Zσ -space with the τ -discrete cells property. Remark 5.18 In condition (d) of Theorem 5.9.6 (5.9.5), X is a strong Zσ -space if and only if every (finite-dimensional) compact set in X is a strong Z-set. Note that every compact set in X is a Z-set in X when X is an ANR with the τ -discrete cells property (Proposition 3.2.10). Theorem 5.9.7 For a space X, the following are equivalent: (a) X is an (2f () × 2 )-manifold;  such that the pair (X,  X) is an (b) X can be embedded into an 2 ()-manifold X 2 2 2 2 ( () ×  , f () ×  )-manifold pair; (c) X can be embedded into an 2 ()-manifold as an M1 (ℵ0 )-absorption base; (d) X is a strongly M1 (ℵ0 )-universal, σ -locally separable, and σ -completely metrizable ANR that is a strong Zσ -space with the τ -discrete cells property. Remark 5.19 In the above (d), when τ > ℵ0 , X is a strong Zσ -space if and only if every separable completely metrizable closed set in X is a strong Z-set. Note that every closed set in X with weight < τ is a Z-set in X when X is an ANR with the τ -discrete cells property (Proposition 3.2.10). Theorem 5.9.8 For a space X, the following are equivalent: (a) X is an (2 () × 2f )-manifold;  such that the pair (X,  X) is an (b) X can be embedded into an 2 ()-manifold X (2 () × 2 , 2 () × 2f )-manifold pair; (c) X can be embedded into an 2 ()-manifold as an M1 (τ )-absorption base; (d) X is a strongly M1 (τ )-universal, σ -completely metrizable ANR that is a strong Zσ -space. Remark 5.20 It should be noted that the τ -discrete cells property is removed from (d) of Theorem 5.9.8 different from the other three theorems. In fact, since In ×  ∈ M1 (τ ) for every n ∈ N, the strong M1 (τ )-universality implies the τ -discrete cells property (cf. Remark 5.17).

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Remark 5.21 As mentioned in Remark 5.13 (p. 406), given an uncountable cardinal τ  < τ , we can consider an M1 (τ  )-absorption base for 2 (). Let   ⊂  with card   = τ  . Then, 2f ()×2 (  ) is an M1 (τ  )-absorption base for 2 ()×2 (  ) (≈ 2 ()). We can obtain the same characterization of 2f () × 2 (  )-manifolds as Theorem 5.9.7. In the separable case (i.e., τ = ℵ0 ), we have the following: scd • (ℵ0 Mfd 0 )σ = (M0 )σ —the class of all strongly countable-dimensional σ -compact metrizable spaces; • (ℵ0 M0 )σ = (M0 )σ —the class of all σ -compact metrizable spaces; • (ℵ0 M1 (ℵ0 ))σ = M1 (ℵ0 )σ —all separable σ -completely metrizable spaces.

Then, we can use Theorem 5.7.18 instead of Theorem 5.7.17. So, the discrete cells property can be removed from condition (d) of each theorem. For the separable version of Theorem 5.9.6, recall that (2 × 2 , 2f × 2Q ) ≈ (2 , 2Q ), hence 2f × 2Q ≈ 2Q . Moreover, Theorems 5.9.7 and 5.9.8 have the same separable version. The following are separable versions of the above theorems: Theorem 5.9.9 For a space X, the following are equivalent: (a) X is an 2f -manifold;  such that the pair (X,  X) is an (b) X can be embedded into an 2 -manifold X (2 , 2f )-manifold pair; (c) X can be embedded into an 2 -manifold as an f.d.cap set; (d) X is a strongly Mfd 0 -universal, strongly countable-dimensional, and σ -compact ANR that is a strong Zσ -space. Theorem 5.9.10 For a space X, the following are equivalent: (a) X is an 2Q -manifold;  such that the pair (X,  X) is an (b) X can be embedded into an 2 -manifold X (2 , 2Q )-manifold pair; (c) X can be embedded into an 2 -manifold as a cap set; (d) X is a strongly M0 -universal σ -compact ANR that is a strong Zσ -space. In condition (d) of Theorem 5.9.10 (5.9.9), “that is a strong Zσ -space” can be replaced by “such that every (finite-dimensional) compact set in X is a strong Zset.” Indeed, a (strongly countable-dimensional) σ -compact metrizable space X is a strong Zσ -space if and only if every (finite-dimensional) compact set in X is a strong Z-set. Thus, we have the following MOGILSKI CHARACTERIZATIONS of 2f - and 2Q -manifolds: Theorem 5.9.11 (MOGILSKI) A space X is an 2Q - (or 2f -)manifold if and only if X is a strongly M0 -(or Mfd 0 -)universal (strongly countable-dimensional) σ -compact ANR and every (finite-dimensional) compact set in X is a strong Z-set. " !

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439

Theorem 5.9.12 For a space X, the following are equivalent: (a) X is an (2f × 2 )-manifold;  such that the pair (X,  X) is an (b) X can be embedded into an 2 -manifold X 2 2 2 2 ( ×  , f ×  )-manifold pair; (c) X can be embedded into an 2 -manifold as an M1 (ℵ0 )-absorption base; (d) X is a strongly M1 (ℵ0 )-universal, separable σ -completely metrizable ANR that is a strong Zσ -space. Concerning a local-compactification of an 2Q - (or 2f -)manifold, we have the following:  is a local-compactification of an 2 - (or 2 -)manifold X. Theorem 5.9.13 Let X Q f  then X  is a Q-manifold and X is an (f.d.)cap set for If X is homotopy dense in X,  X.  and X is an ANR, it follows that X  is Proof Since X is homotopy dense in X fd an ANR (Theorem 1.13.26). Using the strong M0 - (or M0 -)universality of X,  has the disjoint cells property. Hence, X  is a Qwe can easily prove that X manifold by the Toru´nczyk Characterization of Q-manifolds (Theorem 3.8.6). Due  to Proposition 5.4.5, X is an (f.d.)cap set for X. " !  of an 2 - (or 2 -)manifold X is not necessary to Remark 5.22 A completion X Q f  For example, consider the be an 2 -manifold even if X is homotopy dense in X. example defined on p. 162 (see Fig. 2.18), that is,    X = I × {0} ∪ n−1  n ∈ N × I ⊂ R2 and A = {(0, 0)} ⊂ X. Then, X is a separable completely metrizable AR, in which A is a Z-set but not a strong Z-set. As stated in Remark 2.18, (X × 2 )A is not an 2 -manifold. Since dim X = 1 and X is σ -compact, it follows that (X × 2f )A \ A = (X \ A) × 2f ≈ 2f , which is homotopy dense in (X × 2 )A because 2f is a homotopy dense in 2 and A is a Z-set in (X × 2 )A . Thus, (X × 2 )A is a completion of (X × 2f )A \ A such that (X × 2f )A \ A is homotopy dense in (X × 2 )A but (X × 2 )A is not an 2 -manifold. Similarly, (X × 2Q )A \ A ≈ 2Q which is homotopy dense in (X × 2 )A . In the remaining part of this section, we introduce an abosorptively universal tower characterizing 2Q - (or 2f -)manifolds. For a subclass C of M0 , a tower X1 ⊂ X2 ⊂ · · · of subsets of a space X is said to be absorptively universal for C or absorptively C-universal in X if it has the following property:21

21 In

[51], the word “strongly” is used instead of “absorptively.”

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(auC )

For each C ∈ C and each closed set D ⊂ C, if f : C → X is a map such that f |D is an embedding into some Xm , then, for each open cover U of X, there is an embedding h : C → Xn for some n > m such that h|D = f |D and h is U-close to f .

Evidently, the property (auM0 ) (resp. (auMfd )) implies (cap) (resp. (f.d.cap)). 0

When X is strongly M0 - (resp. Mfd 0 -)universal and X1 ⊂ X2 ⊂ · · · is a tower of strong Z-sets, the property (cap) (resp. (f.d.cap)) implies (auM0 ) (resp. (auMfd )). 0 For a subclass C ⊂ M0 , the existence of an absorptively C-universal tower of strong Z-sets implies the strong C-universality, that is: Proposition 5.9.14 Suppose that C ⊂ M0 is closed hereditary and topological. An ANR X is strongly C-universal if X contains an absorptively C-universal tower (Xn )n∈N of strong Z-sets. Proof Let f : C → X be a map of C ∈ C  with a closed set D such that f |D is a strong Z-embedding. We write C \ D = i∈N Ci , where C1 ⊂ C2 ⊂ · · · are closed sets in C. Then, Ci ∈ C for every i ∈ N because C is closed hereditary. For each U ∈ cov(X), let d ∈ Metr(X) such that {Bd (x, 1) | x ∈ X} ≺ U (Proposition 1.3.22 (1)). We will inductively construct maps gi : C → X, i ∈ N, with n(1) < n(2) < · · · so as to satisfy the following: (1) (2) (3) (4)

gi (Ci ) ⊂ Xn(i) , gi |Ci ∪ D is an embedding, gi |Ci−1 ∪ D = gi−1 |Ci−1 ∪ D, d(gi , gi−1 ) < 2−i ,

where g0 = f and C0 = ∅. In the above (2), gi |Ci ∪ D is a strong Z-embedding. Indeed, gi (Ci ) is a strong Z-set in X because gi (Ci ) is a closed subset of a strong Z-set Xn(i) in X. Hence, gi (Ci ∪ D) = gi (Ci ) ∪ f (D) is a strong Z-set in X. Suppose gi−1 has been constructed. We can apply Proposition 2.8.12 to obtain a map g  : C → X such that: (5) (6) (7) (8)

g  is 2−(i+1) -close to gi−1 , g  |Ci−1 ∪ D = gi−1 |Ci−1 ∪ D, g  (C \ (Ci−1 ∪ D)) ⊂ X \ gi−1 (D), g  is closed over gi−1 (D).

Since g  |Ci−1 = gi−1 |Ci−1 by (6), it follows from (8) and (7) that gi−1 (D) ∩ cl g  (Ci ) = gi−1 (D) ∩ g  (Ci ) = (gi−1 (D) ∩ g  (Ci \ Ci−1 )) ∪ (gi−1 (D) ∩ gi−1 (Ci−1 )) = gi−1 (D ∩ Ci−1 ) = ∅. Hence, we can take V ∈ cov(X) so that mesh V < 2−(i+1) and st(g  (Ci ), V) ∩ gi−1 (D) = ∅.

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Using (auC ) for (Xn )n∈N , we can obtain an embedding g  : Ci → Xn(i) for some n(i) > n(i − 1) such that g  |Ci−1 = g  |Ci−1 and g  V g  |Ci . Since g  (Ci ) ⊂ st(g  (Ci ), V), it follows that g  (Ci ) ∩ gi−1 (D) = ∅. Hence, g  extends to an embedding g ∗ : Ci ∪ D → X defined by g ∗ |D = gi−1 |D, where g ∗ V g  |Ci ∪ D. Using the Homotopy Extension Theorem 1.13.11, we can obtain a map gi : C → X such that gi |Ci ∪ D = g ∗ is an embedding, and gi V g  . Then, gi (Ci ) = g  (Ci ) ⊂ Xn(i) and d(gi , gi−1 )  d(gi , g  ) + d(g  , gi−1 ) < 2−i . Observe that gi |Ci−1 = g  |Ci−1 = g  |Ci−1 = gi−1 |Ci−1 and gi |D = g ∗ |D = gi−1 |D. Thus, we have gi |Ci−1 ∪ D = gi−1 |Ci−1 ∪ D. Now, we have obtained maps gi : C → X, i ∈ N, with n(1) < n(2) < · · · satisfying (1)–(4). Then, the required embedding g : C → X can be obtained as the uniform limit g = limi→∞ gi . Indeed, g|D = f |D because gi |D = f |D for every i ∈ N. Moreover,   d(gi , gi−1 ) < 2−i = 1, d(g, f ) = d(g, g0 )  i∈N

i∈N

 which implies that g is U-close to f . Since C = i∈N (Ci ∪D) and each g|Ci ∪D = gi |Ci ∪ D is an embedding, it follows that g is injective, which means that g is an embedding  because C is compact. Because each gi (Ci ∪ D) is a strong Z-set in X, g(C) = i∈N gi (Ci ∪ D) is a strong Zσ -set in X. Due to Corollary 2.8.12, every compact strong Zσ -set is a strong Z-set. Thus, g(C) is a strong Z-embedding. ! " Using the above proposition, we can characterize 2Q - (or 2f -)manifolds as follows: Theorem 5.9.15 A space X is an 2Q - (or 2f -)manifold if and only if X is a (strongly countable-dimensional) σ -compact ANR that is a strong Zσ -space and contains an absorptively M0 - (or Mfd 0 -)universal tower. Proof (The “if” Part) Since X contains an absorptively M0 - (or Mfd 0 -)universal tower, X is strongly M0 - (or Mfd -)universal by Proposition 5.9.14. Hence, X is an 0 2Q - (or 2f -)manifold by Theorem 5.9.10 (or 5.9.9). (The “only if” Part) Due to Theorem 5.9.10 (or 5.9.9), an 2Q - (or 2f -) manifold  as an (f.d.)cap set, where X is homotopy X can be embedded into an 2 -manifold X  dense in X (Propositions 5.3.1 and 5.3.5). Hence, there  is a tower (Xn )n∈N of (finite such that X = n∈N Xn and (Xn )n∈N has the dimensional) compact Z-sets in X property (cap) (or (f.d.cap)). Since every Z-set in an 2 -manifold is a strong Z which implies that each Xn is a strong Z-set set, each Xn is a strong Z-set in X,  in X (Proposition 2.9.4). Since X is strongly M0 -universal, the property (cap) (or  which implies that (Xn )n∈N has the (f.d.cap)) implies (auM0 ) (resp. (auMfd )) in X, 0

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property (auM0 ) (resp. (auMfd )) in X. Hence, (Xn )n∈N is an absorptively M0 - (or Mfd 0 -)universal tower in X.

0

" !

To apply the above theorem to convex sets, we show the following: Lemma 5.9.16 Let C be a separable convex set in a metrizable topological linear space E. (1) If C is infinite-dimensional, then C has an absorptively Mfd 0 -universal tower. (2) If C contains a convex subset affinely homeomorphic to Q, then C has an absorptively M0 -universal tower. Proof Due to Theorem 1.4.15, E has an admissible F -norm  · . Since C is separable, it has a countable dense subset {ai | i ∈ N}, where we may assume that a1 = 0. (1): For each n ∈ N, let Cn = a1 , . . . , an . Since C is infinite-dimensional and {ai | i ∈ N} is dense in C, there is some m > n such that am ∈ C \ fl Cn , and then dim Cm > dim Cn . Thus, supn∈N dim Cn = ∞. To see the absorptive Mfd 0 -universality of (Cn )n∈N , let f : X → C be a map of such that f |A : A → Cm is an embedding for some m ∈ N and a closed X ∈ Mfd 0 set A ⊂ X. For each ε > 0, we will find n  m and construct a map f  : X → Cn so that f  is ε/2-close to f and f  |A = f |A. Since Cm is an AR, f |A extends to a map f  : X → Cm . Then, A has an open neighborhood U such that f (x) − f  (x) < ε/8 for every x ∈ U. Let α : X → I be a map such that α −1 (0) = A and α −1 (1) = X \ U . By Theorem 1.13.21 with Remark 1.13, we have a finite simplicial complex K and maps ϕ : X → |K| and ψ : |K| → C such that f ε/8 ψϕ. Let k = dim K. Then, subdividing K if necessary, we can assume that diam ψ(σ ) < ε/8(k + 1) for every σ ∈ K. For each v ∈ K (0), choose i(v) ∈ N so that ψ(v) − ai(v) < ε/8(k + 1), and let n = max{n, i(v) | v ∈ K (0)}. We can define ψ  : |K| → Cn as follows: ψ  (y) =



βvK (y)ai(v).

v∈K (0)

Then, for each y ∈ |K|, ψ(y) − ψ  (y) 



βvK (y)ψ(y) − βvK (y)ai(v)

v∈cK (y)(0)





ψ(y) − ai(v)

v∈cK (y)(0)





v∈cK

(y)(0)

ψ(y) − ψ(v) +

 v∈cK

ψ(v) − ai(v)

(y)(0)

< (k + 1) diam ψ(cK (y)) + ε/8 < ε/4,

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where cK (y) ∈ K is the carrier of y in K. Thus, we have a map ψ  ϕ : X → Cn that is 3ε/8-close to f . Indeed, for each x ∈ X, f (x) − ψ  ϕ(x)  f (x) − ψϕ(x) + ψϕ(x) − ψ  ϕ(x) < ε/8 + ε/4 = 3ε/8. The desired map f  : X → Cn can be defined as follows: f  (x) = (1 − α(x))f  (x) + α(x)ψ  ϕ(x) for each x ∈ X. Indeed, f  |A = f  |A = f |A and f  |X \ U = ψ  ϕ|X \ U , so f  |X \ U is ε/2-close to f |X \ U . For each x ∈ U , f  (x) − f (x) = (1 − α(x))f  (x) + α(x)ψ  ϕ(x) − f (x)  (1 − α(x))(f  (x) − f (x)) + α(x)(ψ  ϕ(x) − f (x))  f  (x) − f (x) + ψ  ϕ(x) − f (x) < ε/8 + 3ε/8 = ε/2. Now, by Theorem 1.12.14, we have an embedding h : X → I2 dim X+1 . Taking a map λ : X → I with α −1 (0) = A, we define a map g : X → Cn × I2 dim X+2 as follows: g(x) = (f  (x), λ(x), λ(x)h(x)) for each x ∈ X. Then, it is easy to see that g(x) = (f  (x), 0, 0) for any x ∈ A and g is injective, which means that g is an embedding. Choose n > n so that dim Cn  dim Cn + 2 dim X + 2. Since the boundary of each cell is collared in the cell, there exists an embedding e : Cn × I2 dim X+2 → Cn such that e(x, 0) = x and diam e({x} × I2 dim X+2 ) < ε/2 for each x ∈ Cm . Then, eg : X → Cn is an embedding, eg|A = f  |A = f |A, and eg is ε/2-close to f  , so ε-close to f . (2): By the assumption, we have an affine embedding h : Q → C, where we can assume that h(0) = 0 ∈ C. Let    C0 = h(x)  x ∈ Q and x(2i) = 0 for every i ∈ N .

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Then, {h(e2n ) | n ∈ N} is a linearly independent subset of E \ fl C0 , where fl C0 is a linear subspace of E because 0 ∈ C0 . Hence, we can inductively choose vn , n ∈ N, so that vn ∈ E \ fl(C0 ∪ {a1 , . . . , an , v1 , . . . , vn−1 }). For each n ∈ N, let 9 : Dn = C0 ∪ {a1 , . . . , an , v1 , . . . , vn−1 } and 9 : Cn = C0 ∪ {a1 , . . . , an , v1 , . . . , vn } . Then, vn ∈ E \ fl Dn−1 and Cn is the cone over Dn with vn the vertex. Since Cn−1 ⊂ Dn and Dn is collared in Cn , it follows that Cn−1 is a Z-set in Cn . We will show that every Cn is affinely homeomorphic to a convex set in a normed linear space, which implies that Cn is an AR. Then, applying Keller’s Theorem 3.9.2, we will conclude that Cn is homeomorphic to Q. Assume that Cn−1 is affinely homeomorphic to a convex set in a normed linear space L. Since Cn is a cone over Dn , it suffices to show that Dn is affinely homeomorphic to a convex set in a normed linear space. Let ϕ : Cn−1 → L be an affinely embedding of Cn−1 into L. Then, ϕ extends to a linear isomorphism ϕ˜ : fl Cn−1 → L, where ϕ˜ is not a homeomorphism in general. When an ∈ fl Cn−1 , we have Dn ⊂ fl Cn−1 . There exist maps q : Cn−1 ×I → Dn and p : Cn−1 ×I → L defined as follows: q(x, t) = (1 − t)an + tx and p(x, t) = (1 − t)ϕ(a ˜ n ) + tϕ(x), ˜ = p, ϕ|D ˜ n : Dn → where q is a quotient map because Cn−1 is compact. Since ϕq ϕ(D ˜ n ) is a continuous bijection, which is a homeomorphism because Dn is compact. Thus, Dn is affinely homeomorphic to a convex set in L. When an ∈ fl Cn−1 , Dn is the cone over Cn−1 with an the vertex. Hence, every x ∈ Dn \ {an } is uniquely written as x = (1 − t)an + ty, y ∈ Cn−1 , 0 < t  1. We define an affine map ϕ ∗ : Dn → L × R by ϕ ∗ ((1 − t)an + ty) = (tϕ(y), 1 − t) for each y ∈ Cn−1 and t ∈ I. Then, ϕ ∗ is a continuous injection, which is an embedding because Dn is compact. Hence, Dn is affinely homeomorphic to a convex set in L × R. Consequently, Cn is affinely homeomorphic to a convex set in a normed linear space. We have just shown that each Cn is a Z-set in Cn+1 and homeomorphic to Q. Now, using this fact, we will show that the tower (Cn )n∈N is absorptively M0 universal. Let f : X → C be a map of X ∈ M0 such that f |A : A → Cm is an embedding for some m ∈ N and a closed set A ⊂ X. By the same argument as (1), for each ε > 0, we can find n > m and construct a map f  : X → Cn so that f  is ε/2-close to f and f  |A = f |A. Since Cm is a Z-set in Cn , it follows that f  |A = f |A : A → Cn ≈ Q is a Z-embedding. Applying the Strong Universality

5.9 Four Types of Infinite-Dimensional Manifolds

445

Theorem 2.3.16, we can obtain an embedding f  : X → Cn such that f  is ε/2close to f  and f  |A = f  |A, so f  is ε-close to f and f  |A = f |A. " ! Lemma 5.9.17 Let C be a separable convex set in a metrizable topological linear space E. If C is an AR and some point of C has no totally bounded neighborhood in C, then every compact set in C is a strong Z-set in C. Proof Since C has the discrete cell-tower property by Lemma 3.5.8, it follows from Corollary 5.8.4 that every compact set in C is a strong Z-set in C. " ! A convex set C in a metric linear space E = (E, d) is said to be locally complete at x ∈ C if x has a complete neighborhood in C. Proposition 5.9.18 Let C be an infinite-dimensional convex set in a metric linear space E = (E, d). If C is locally complete at some point, then C contains a convex set A such that A is affinely homeomorphic to the Hilbert cube Q. Proof We may assume that d is induced from an F -norm  ·  and C is locally complete at 0 ∈ C, that is, there exists ε > 0 such that every Cauchy sequence in C ∩ B(0, ε) converges in C. Since C is infinite-dimensional, C has a linearly independent sequence (xi )i∈N such that xi  < ε for each i ∈ N. We can inductively choose 0 < ai < 1, i ∈ N, so that ai xi  < 2−(i−k+1) δk , k = 0, . . . , i − 1, where δ0 = 2ε and for k > 0,  $  k k  #   k  (si )k , (ti )k ∈  si xi − ti xi  [0, ai ], δk = inf  i=1 i=1  i=1

i=1

i=1

% ∃i = 1, . . . , k such that |si − ti |  min{1/k, ai } .

 Indeed, we have an embedding ϕk : ki=1 [0, ai ] → C defined by ϕ((ti )ki=1 ) = k i=1 ti xi . Then, it follows that δk > 0. Hence, we can proceed the inductive choice of ai .   For each k ∈ N, let prk : i∈N [0, ai ] → ki=1 [0, ai ] be the projection. For each  (ti )ki=1 ∈ ki=1 [0, ai ], ϕ((ti )ki=1 )

  k k k k           ti xi   ai xi  <  = ti xi  2−(i+1) δ0 < ε.  i=1

Then, we have maps ϕk prk :  i∈N [0, ai ],

i=1



i∈N [0, ai ]

i=1

i=1

→ C ∩ B(0, ε), k ∈ N. For each (ti )i∈N ∈

  ϕk+1 pr ((ti )i∈N ) − ϕk pr ((ti )i∈N ) = tk+1 xk+1  k+1 k  ak+1 xk+1  < 2−(k+2) δ0 = 2−(k+1)ε,

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

which means that (ϕk prk  )k∈N is uniformly Cauchy. Hence, (ϕk prk )k∈N uniformly converges to the map ϕ : i∈N [0, ai ] → C ∩ B(0, ε). Then, ϕ((ti )i∈N ) =



ti xi for each (ti )i∈N ∈

i∈N

#

[0, ai ].

i∈N

Hence, ϕ is an affine map. Moreover, for each k ∈ N and (ti )i∈N ∈



i∈N [0, ai ],

       ϕ((ti )i∈N ) − ϕk prk ((ti )i∈N ) =  ti xi  ti xi    



i>k

ai xi 
k



i>k

2−(i−k+1) δk = δk /2.

i>k

 To see that ϕ is injective, let (ti )i∈N = (si )i∈N ∈ i∈N [0, ai ]. Then, ti = si for some i ∈ N. Choose k  i so that 1/k  |si − ti | ( ai ). It follows that     ϕ((si )i∈N ) − ϕ((ti )i∈N )  ϕk pr ((si )i∈N ) − ϕk pr ((ti )i∈N ) k k   − ϕ((si )i∈N ) − ϕk prk ((si )i∈N )   − ϕk prk ((ti )i∈N ) − ϕ((ti )i∈N )  k  k     > si xi − ti xi   − δk /2 − δk /2 i=1

i=1

 δk − δk = 0, which means that ϕ((si )i∈N ) = ϕ((ti )i∈N ). Thus, ϕ is injective, so it is an affine embedding. Therefore, the image of ϕ is a convex set in E, which is affinely  homeomorphic to i∈N [0, ai ] and so to Q. " ! Now, we can prove the following: Theorem 5.9.19 Let C be a σ -compact convex set in a metrizable topological linear space E such that some point of C has no totally bounded neighborhood in C. (1) C ≈ 2f if C is strongly countable-dimensional. (2) C ≈ 2Q if C is an AR and contains an infinite-dimensional compact convex set. Proof In Lemma 5.9.17, we have already shown that every compact set in C is a strong Z-set in C. Hence, C is a strong Zσ -space. (1): As a convex set in a topological linear space, C is contractible and locally contractible. Since C is countable-dimensional, it follows that C is an AR (Haver’s Theorem 1.13.38 and Proposition 1.13.3(2)). By virtue of Lemma 5.9.16(1), C

5.10 Absorbing Sets

447

contains an absorptively Mfd 0 -universal tower. Thus, applying Theorem 5.9.15, we have (1). (2): As we saw in Proposition 5.9.18, C contains a convex subset affinely homeomorphic to the Hilbert cube Q. By Lemma 5.9.16(2), C has an absorptively M0 -universal tower. Again, applying Theorem 5.9.15, we have (2). " ! Recall that a topological linear space E is finite-dimensional if and only if 0 ∈ E has a totally bounded neighborhood in E (Theorem 1.4.12). As a special case of Theorem 5.9.19 above, we have the following: Theorem 5.9.20 Let E be a metrizable topological linear space. (1) E ≈ 2f if E is ℵ0 -dimensional, that is, E has a countable Hamel basis. (2) E ≈ 2Q if E is a σ -compact AR and contains an infinite-dimensional compact convex set. " !

5.10 Absorbing Sets In this chapter, as before, •

let C be a closed hereditary, additive topological class of metrizable spaces.

A strongly C-universal homotopy dense set X in M is called an absorbing set in M for C or simply a C-absorbing set if X is a Zσ -space and X ∈ Cσ , equivalently X is a countable union of members of CZ (X), where CZ (X) is the collection of Z-sets in X that is a member of C.   Indeed, let X = i∈N Ai = j ∈N Cj , where each Ai is a Z-set in X, each Cj is closed  in X and Cj ∈ C. Then, X = i,j ∈N Ai ∩ Cj . Since C is closed hereditary and any closed subset of a Z-set in X is a Z-set in X, it follows that Ai ∩ Cj ∈ CZ (X) for every i, j ∈ N.

The following is a direct consequence of Proposition 5.4.1(1): Lemma 5.10.1 Every C-absorbing set in an 2 ()-manifold M is C-absorptive in M. " ! 2 When Mfd 0 ⊂ C ⊂ M1 (τ ), we can characterize C-absorbing sets in an  ()manifold as follows:

Proposition 5.10.2 Let M be an 2 ()-manifold with X ⊂ M. When Mfd 0 ⊂ C ⊂ M1 (τ ), X is a C-absorbing set in M if and only if X is a C-absorptive Zσ -space and X ∈ Cσ . Proof The “only if” part follows from Lemma 5.10.1. By Propositions 5.3.5 and 5.4.1(2), a C-absorptive X in M is a strongly C-universal homotopy dense set in M, which means the “if” part. " ! The following differences should be noted:

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

• A C-absorption base for M is a Zσ -set in M but a C-absorbing set in M is a Zσ -set in itself. • Every C-absorption base for an 2 ()-manifold M is σ -completely metrizable (i.e., it belongs to M1 (τ )σ ), but a C-absorbing set in M is not necessary σ completely metrizable. We have to keep in mind the following: • It is not required for a C-absorbing set to be Fσ in M. In fact, we have the following: Proposition 5.10.3 Let M be an 2 ()-manifold. (1) A C-absorbing (or τ C-absorbing) set X in M is a C-absorption base for M if and only if X is Fσ in M. (2) When Mfd 0 ⊂ C ⊂ M1 (τ ), every C-absorption base X for M is a τ C-absorbing Fσ set in M, which is a C-absorbing set in M if (τ C)σ = Cσ . Proof (1): The “only if” part is trivial. To see the “if” part, note that X is Cabsorptive in M (Lemma 5.10.1).Since X is Fσ in M, there are closed sets An , n ∈ N, n∈N An . As a C-absorbing set, we can write  in M such that X = X = n∈N Xn , where Xn ∈ CZ (X) (or Xn ∈ τ CZ (X)) for each n ∈ N. Then, every Ai ∩ Xn is closed in M. Since X is homotopy dense in X and each Xn is a Z-set in X, it is easy to see that every Ai ∩ Xn is a Z-set in M. Because C is closed hereditary, Ai ∩ Xn ∈ CZ (M) (or Ai ∩ Xn ∈ τ CZ (M)) for each i, n ∈ N. Thus, X is a C-absorption base for M. (2): Combining Corollaries 5.3.9 and 5.4.2(2), we can see that X is a strongly τ C-universal homotopy dense Zσ -set in M and X ∈ (τ C)σ . Since X is homotopy dense in M, if A ⊂ X is a Z-set in M, then A is a Z-set in X. Hence, X is a Zσ -space. Thus, X is a τ C-absorbing set in M. " ! 2 In particular, an M0 - (or Mfd 0 -)absorption base for an  ()-manifold M is an fd fd τ M0 - (or τ M0 -)absorbing set in M, but an M0 - (or M0 -)absorption base for an 2 -manifold M (i.e., an (f.d.)cap set in M) is an M0 - (or Mfd 0 -) absorbing set in M. Moreover, an M1 (τ )- (or M1 (ℵ0 )-)absorption base for an 2 ()-manifold M is an M1 (τ )- (or τ (M1 (ℵ0 ))-)absorbing set in M. An M1 (ℵ0 )-absorption base for an 2 -manifold M is an M1 (ℵ0 )-absorbing set in M. Due to Theorem 3.4.9, if N is a homotopy dense Gδ set in an 2 ()-manifold M, then the inclusion N → M is a near-homeomorphism. Hence, the following lemma can be easily obtained, which will be frequently used:

 is a Gδ Lemma 5.10.4 Let X be a C-absorbing set in an 2 ()-manifold M. If X  then X  is an 2 ()-manifold, X is a C-absorbing set in X,  set in M with X ⊂ X,  → M is a near-homeomorphism. and the inclusion X " ! We have the following uniqueness of absorbing sets in an 2 ()-manifold, which is weaker than the uniqueness of absorption bases for an 2 ()-manifold (Theorem 5.3.14):

5.10 Absorbing Sets

449

Theorem 5.10.5 (UNIQUENESS OF ABSORBING SETS) Let X and Y be two Cabsorbing sets in an 2 ()-manifold M. Then, for each open cover U of M, there is a homeomorphism f : X → Y that is U-close to the inclusion X ⊂ M.   Proof We can write X = n∈N Xn , Y = n∈N Yn , where Xn ∈ CZ (X) and Yn ∈ CZ (Y ) for each n ∈ N. Take open covers of M as follows: ∗

∗

∗

U  U1  U2  · · · and mesh Un < 2−n . We shall inductively construct homeomorphisms fn : Gn → Hn , gn : Hn → Gn between Gδ -sets in M such that X ⊂ Gn ∩ Gn , Y ⊂ Hn ∩ Hn and the following are satisfied: (1) (2) (3) (4)

fn |X is Un -close to fn−1 |X and gn |Y is Un -close to gn−1 |Y ; fn |Xn−1 = fn−1 |Xn−1 and gn |Yn−1 = gn−1 |Yn−1 ; fn (Xn ) is a Z-set in Y and gn (Yn ) is a Z-set in X; gn fn |Xn = id and fn gn |Yn = id.

Then, we have the maps f = limn→∞ fn |X : X → Y and g = limn→∞ gn |Y : Y → X, where fg = idY and gf = idX , that is, f is a homeomorphism with f −1 = g. Let G0 = G0 = H0 = H0 = M and f0 = g0 = idM . Assume that fi : Gi → Hi , gi : Hi → Gi have been constructed for i < n so as to satisfy (1)–(4). By using the strong C-universality of Y , we have a Z-embedding h : Xn ∪ gn−1 (Yn−1 ) → Y such that h|Xn−1 ∪ gn−1 (Yn−1 ) = fn−1 |Xn−1 ∪ gn−1 (Yn−1 ) and h Un+5 fn−1 |Xn ∪ gn−1 (Yn−1 ). By Lavrentieff’s Homeomorphism Extension Theorem 1.3.19, h extends to a homeomorphism h˜ : A → B between Gδ -sets A and B of Gn−1 and Hn−1 , respectively, such that   Xn−1 ∪ gn−1 (Yn−1 ) ⊂ A ⊂ clGn−1 Xn−1 ∪ gn−1 (Yn−1 ) and h(Xn−1 ∪ gn−1 (Yn−1 )) ⊂ B ⊂ clHn−1 h(Xn−1 ∪ gn−1 (Yn−1 )). By using Lavrentieff’s Gδ Extension Theorem 1.3.18, we may assume that h˜ Un+5 fn−1 |A in Hn−1 . We can define Gδ -sets Gn and Hn in M as follows:   Gn = Gn−1 \ clGn−1 (Xn−1 ∪ gn−1 (Yn−1 )) \ A ,   Hn = Hn−1 \ clHn−1 h(Xn−1 ∪ gn−1 (Yn−1 )) \ B .

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Then, by Lemma 5.10.4, Gn , Gn−1 , Hn and Hn−1 are homeomorphic to M and also the inclusions Gn → Gn−1 and Hn → Hn−1 are near-homeomorphisms. So, we have a homeomorphism α : Gn−1 → Gn so close to idGn−1 that fn−1 α −1 Un+6 fn−1 |Gn in Hn−1 . We have also a homeomorphism β : Hn−1 → Hn such that β Un+6 idHn−1 in Hn−1 . Observe that A and B are Z-sets in Gn and Hn , respectively, hence βfn−1 α −1 |A and h˜ are Z-embedding into Hn . Since βfn−1 α −1 |A Un+5 fn−1 |A in Hn−1 and Hn is homotopy dense in Hn−1 , it follows that βfn−1 α −1 |A Un+5 fn−1 |A in Hn . Similarly, h˜ Un+5 fn−1 |A in Hn . Hence, βfn−1 α −1 |A Un+4 h˜ in Hn . By the Z-Set Unknotting Theorem 2.9.7, we have a homeomorphism γ : Hn → Hn such that γβfn−1 α −1 |A = h˜ and γ Un+5 id. The homeomorphism fn = γβfn−1 α −1 : Gn → Hn satisfies (1)–(3). The homeomorphism gn : Hn → Gn can be similarly constructed so as to satify (1)–(4). Thus, the proof is completed. " ! Remark 5.23 In the above, the homeomorphism f : X → Y can be extended  and  → Y  between Gδ sets in M with X ⊂ X to a homeomorphism f˜ : X    Y ⊂ Y . However, X = Y in general. Thus, Theorem 5.10.5 above is weaker than Theorem 5.3.14. Corollary 5.10.6 Let X and Y be C-absorbing sets in 2 ()-manifolds M and N, respectively. If X and Y have the same homotopy type (equiv. M and Y have the same homotopy type), then they are homeomorphic, i.e., X  Y (or M  N) ⇒ X ≈ Y. Proof By the Classification Theorem 2.6.1, we have M ≈ N, that is, there is a homeomorphism f : M → N. Then, f (X) is a C-absorbing set in N. By Theorem 5.10.5 above, we have f (X) ≈ Y . Therefore, X ≈ Y . " ! Theorem 5.10.7 Let X and Y be C-absorbing sets in 2 ()-manifolds M and N, respectively. Then, every fine homotopy equivalence f : X → Y is a nearhomeomorphism. Proof For each U ∈ cov(Y ), let V ∈ cov(Y ) be a star-refinement of U. We  ∈ cov(Y  = V. By  in N and V ) such that Y ⊂ Y  and V|Y have an open set Y  → Y  Lavrentieff’s Gδ Extension Theorem 1.3.18, f extends to a map f˜ : X  in M. Then, X  and Y  are 2 ()-manifolds by Lemma 5.10.4. over a Gδ set X Moreover, f˜ is a fine homotopy equivalence. Indeed, by Theorem 1.15.1, we may show that f˜ is a hereditary weak homotopy equivalence. For each open set U in , since X and Y are homotopy dense in X  and Y  respectively, the inclusions Y f −1 (U ∩ Y ) = f˜−1 (U ) ∩ X → f˜−1 (U ) and U ∩ Y → Y are (weak) homotopy equivalences. Since f |f −1 (U ∩ Y ) : f −1 (U ∩ Y ) → U ∩ Y is a (weak) homotopy equivalence, so is f˜|f˜−1 (U ) : f˜−1 (U ) → U :

5.10 Absorbing Sets

f −1 (U ∩ Y )

451 =

f˜−1 (U ) ∩ X

⊂

f˜−1 (U ) f˜|

f|

U ∩Y

⊂

U

 → Y  is a nearBy the Homeomorphism Approximation Theorem 2.6.3, f˜ : X    homeomorphism. Hence, we have a homeomorphism g : X → Y that is V-close ˜  to f . Since g(X) and Y are C-absorbing sets in Y , we have a homeomorphism  h : g(X) → Y that is V-close to id by Theorem 5.10.5. Then, hg : X → Y is a homeomorphism that is U-close to f . " ! Concerning strong Zσ -spaces, we have the following: Lemma 5.10.8 Let X be an ANR that is a strong Zσ -space. Every open set in X is a strong Zσ -space.  Proof Let X = n∈N An , where each An is a strong Z-set  in X. Due to Proposition 2.8.14, each An ∩ U is a strong Z-set in U . So, U = n∈N (An ∩ U ) is a strong Zσ -space. " ! Concerning open subsets of a C-absorbing set, the following holds: Proposition 5.10.9 Let X be a C-absorbing set in an 2 ()-manifold M. For each open set U in M, X ∩ U is a C-absorbing set in U . Proof From Propositions 5.4.18 and 1.13.6(3), X ∩ U is also a strongly C-universal ANR. Due to Corollary 1.15.6, X ∩ U is homotopy dense in U . Since X is a strong Zσ -space, X ∩ U is a strong Zσ -space by Lemma 5.10.8. Because Cσ is open hereditary, it follows that X ∩ U ∈ Cσ . Thus, X ∩ U is a C-absorbing set in U . ! " Combining the above proposition with the Open Embedding Theorem 2.5.10, we have the following: Corollary 5.10.10 If 2 () has a C-absorbing set !, then every 2 ()-manifold has a C-absorbing set. " ! We can consider manifolds modeled on a C-absorbing set in 2 (). Corollary 5.10.11 Let ! be a C-absorbing set in 2 (). If X is a C-absorbing set in some 2 ()-manifold M, then X is an !-manifold. Proof Each point x ∈ X has an open neighborhood U in M that is homeomorphic to an open ball in 2 (), which means U ≈ 2 (). Then, X ∩ U is a C-absorbing set in U . By Corollary 5.10.6, we have X ∩ U ≈ !. Therefore, X is an !-manifold. ! " We prove that an ANR is a strong Zσ -space if it is a local strong Zσ -space, that is: Proposition 5.10.12 Let X be an ANR. If each point of X has an open neighborhood being a strong Zσ -space, then X is a strong Zσ -space.

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5 Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

Proof Applying Michael’s Theorem 1.3.20, we can show the proposition, where it suffices to prove the following: (G-1) (G-2) (G-3)

For open sets U ⊃ V in X, if U is a strong Zσ -space, then V is a strong Zσ -space; For two open sets U and V in X, if both U and V are strong Zσ -spaces, then U ∪ V is a strong Zσ -space; For a discrete collection {Uλ }λ∈ of open sets in X, if each Uλ is a strong  Zσ -space, then λ∈ Uλ is a strong Zσ -space.

Since (G-1) is a direct consequence of Lemma 5.10.8, we have to show (G-2) and (G-3), but we only prove (G-2) because of similarity. For (G-3), the proof is left to the reader.   Let U = n∈N An and V = n∈N Bn , where each An and Bn is a strong Zset in U and V , respectively. Since U and V are Fσ in X, we can assume that each An and Bn is closed in X. Then, each An and Bn is a strong Z-set in X by Proposition 2.8.9(2). Hence, An ∪ Bn is a strong Z-set in X by Proposition 2.8.9(1). Due  to Proposition 2.8.14, An ∪ Bn is a strong Z-set in U ∪ V . Thus, U ∪ V = " ! n∈N (An ∪ Bn ) is a strong Zσ -space. For the class (τ C)σ , we show the following: Proposition 5.10.13 Let X be a metrizable space with w(X)  τ . If each point of X has an open neighborhood U ∈ (τ C)σ , then X ∈ (τ C)σ . Proof Since the result can be obtained by using Michael’s Theorem 1.3.20, it suffices to prove the following: (G-1) (G-2) (G-3)

For open sets U ⊃ V in X, if U ∈ (τ C)σ , then V ∈ (τ C)σ ; For two open sets U and V in X, if U, V ∈ (τ C)σ , then U ∪ V ∈ (τ C)σ ; For a discretecollection (Uλ )λ∈ of open sets in X, if Uλ ∈ (τ C)σ for each λ ∈ , then λ∈ Uλ ∈ (τ C)σ .

Since (τ C)σ is open hereditary (Proposition5.6.1), (G-1) holds.  For (G-2), let U = n∈N An and V = n∈N Bn , where An , Bn ∈ τ C and each An and Bn is closed in U and V , respectively. Since U and V are Fσ in X, we can assume that each An and Bn is closed in X. Then, An ∪ Bn is closed  in X, hence in U ∪ V . Since τ C is additive, An ∪ Bn ∈ τ C. Hence, U ∪ V = n∈N (An ∪ Bn ) ∈ (τ C)σ . For (G-3), since  (Uλ )λ∈ is discrete in X, it follows that card   w(X)  τ . We can write U = and each Cλ,n is closed in Uλ . Then, n∈N Cλ,n , where Cλ,n ∈ τ C   λ Cn = λ∈ C ∈ τ C and C is closed in n λ∈ Uλ because Cn ∩ Uλ = Cλ,n .  λ,n  Therefore, λ∈ Uλ = n∈N Cn ∈ (τ C)σ . " ! Using the above two propositions, we can prove the following:. Proposition 5.10.14 Suppose that τ C = C. Let ! be a C-absorbing set in 2 (). Then, every !-manifold X is a strongly C-universal ANR that is a strong Zσ -space, X ∈ Cσ , and every Z-set in X is a strong Z-set.

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Proof Each point of X has an open neighborhood U that is homeomorphic to an open set in !, that is, U ≈ W ∩ ! for some open set W in 2 (). Then, W ∩ ! is a C-absorbing set in W by Proposition 5.10.9. Hence, U is a C-absorbing set in , which implies that U is a strongly C-universal ANR that some 2 ()-manifold U is a strong Zσ -space and U ∈ Cσ . Hence, X is an ANR by the Hanner Theorem (Proposition 1.13.6(4)). Form Proposition 5.10.12, it follows that X is a strong Zσ space. We show that every Z-set A in X is a strong Z-set. As we saw above, each a ∈ A has an open neighborhood U in X that is a C-absorbing set in some 2 (). Then, we have a closed set AU in U  such that AU ∩ U = A ∩ U . manifold U  Since U is homotopy dense in U and A ∩ U is a Z-set in U (Corollary 2.8.7(3)), it , so a strong Z-set in U . Hence, follows that AU is a Z-set in an 2 ()-manifold U A ∩ U = AU ∩ U is a strong Z-set in U . Applying Proposition 2.8.15, we can conclude that A is a strong Z-set in X. We can now apply Proposition 5.6.3 to prove that X is strongly C-universal. Moreover, because τ C = C, we have X ∈ Cσ by Proposition 5.10.13. The proof is complete. " ! Now, we can obtain the following characterization: Theorem 5.10.15 (CHARACTERIZATION) Suppose that τ C = C and C is Istable.22 Let ! be a C-absorbing set in 2 (). For a space X, the following are equivalent: (a) X is an !-manifold; (b) X can be embedded into an 2 ()-manifold as a C-absorbing set; (c) X ∈ Cσ is a strongly C-universal ANR with w(X) = τ that is a strong Zσ space. Proof (a) ⇒ (c): There exists U ∈ cov(X) such that each U ∈ U is homeomorphic to an open set in !. Since ! is a strong Zσ -space, each U is also a strong Zσ space by Lemma 5.10.8. By virtue of Proposition 5.10.12, X is a strong Zσ -space. Since ! is homotopy dense in 2 (), it is an AR, hence each U ∈ U is an ANR (Proposition 1.13.6(3)). By Hanner Theorem (Proposition 1.13.6(4)), X is also an ANR. Since Cσ is open hereditary and ! ∈ Cσ , it follows that U ∈ Cσ . Applying Proposition 5.10.13, we have X ∈ (τ C)σ = Cσ . Because of the definition of manifolds, w(X) = w(!) = τ .  in 2 () such that U ≈ U  ∩ !. Since For each U ∈ U, take an open set U    U ∩ ! is a C-absorbing set in U (Proposition 5.10.9), U ∩ ! is a strongly C. Hence, each U ∈ U is strongly C-universal. universal homotopy dense set in U Due to Proposition 5.10.14, each Z-set in X is a strong Z-set. Hence, we can apply Proposition 5.6.3 to see that X is strongly C-universal.

that τ C = C implies  ∈ C. Since C is I-stable, In ×  ∈ C for every n ∈ N. Then, the strong C-universality implies the τ -discrete cells property.

22 Note

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(c) ⇒ (b): Due to the definition of a C-absorbing set, it suffices to show that X can be embedded into an 2 ()-manifold as a homotopy dense set. This can be done as in the proof of Theorem 5.7.17 (d) ⇒ (c). For readers’ convenience, we repeat the proof. First, take d ∈ Metr(X) such that (X, d) is  of (X, d) is an ANR a uniform ANR (Theorem 1.13.37). Then, the metric completion X  by Theorem 1.13.35. Because X has the τ -discrete cells and X is homotopy dense in X  has the τ -discrete cells property. Since In ×  ∈ C for every n ∈ N, it follows property, X fd that M0 ⊂ C. So, the strong C-universality of X implies the strong Mfd 0 -universality. By Corollary 5.7.15 X has the discrete τ -polyhedral sequence property of X, which implies the  because X is homotopy dense in X.  By virtue discrete τ -polyhedral sequence property of X  is an 2 ()-manifold. of the Toru´nczyk Characterization (Theorem 3.4.2), X

(b) ⇒ (a): Let M be an 2 ()-manifold in which X is a C-absorbing set. By the Open Embedding Theorem 2.5.10, M can be regarded as an open set in 2 (). Then, ! ∩ M is a C-absorbing set in M (Proposition 5.10.9). By the Uniqueness of Absorbing Sets (Theorem 5.10.5), X is homeomorphic to the open subset ! ∩ M of !. Therefore, X is an !-manifold. " ! The following is a separable version of Theorem 5.10.15 above: Theorem 5.10.16 (CHARACTERIZATION) Let ! be a C-absorbing set in 2 .23 For a space X, the following are equivalent: (a) X is an !-manifold; (b) X can be embedded into an 2 -manifold as a C-absorbing set; (c) X ∈ Cσ is a strongly C-universal ANR that is a strong Zσ -space.24 For manifolds modeled on a C-absorbing set in 2 (), the Classification and the Open Embedding Theorems hold: Theorem 5.10.17 Suppose that τ C = C. Let ! be a C-absorbing set in 2 (). (1) (CLASSIFICATION) For two !-manifolds X and Y , X ≈ Y if and only if X  Y . Sketch of Proof. This is just combination of Theorem 5.10.15 with Corollary 5.10.6.

(2) (OPEN EMBEDDING) Every !-manifold can be embedded into ! as an open set.  as a CSketch of Proof. Embed each !-manifold M into an 2 ()-manifold M  is homeomorphic to an open set U in absorbing set (Theorem 5.10.15). Then, M 2 () (the Open Embedding Theorem 2.5.10), where ! ∩ U is a C-absorbing set in U (Proposition 5.10.9) Since the image of M is also a C-absorbing set in U , we can apply Theorem 5.10.5 to obtain the result.

(3) For two !-manifolds X and Y , every fine homotopy equivalence f : X → Y is a near-homeomorphism. Sketch of Proof. This is just combination of Theorem 5.10.15 with Theorem 5.10.7.

23 This

implies C ⊂ M(ℵ0 ). Cσ ⊂ M(ℵ0 ), it follows that X is separable.

24 Since

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To prove the Triangulation Theorem, we need the following: Theorem 5.10.18 Suppose that τ C = C and C is I-stable. Let ! be a C-absorbing set in 2 (). For an arbitrary simplicial complex K with card K (0)  τ , |K|m × ! is an !-manifold. Proof First, note that |K|m × ! is an ANR and w(|K|m × !) = w(!)w(|K|m )  τ ℵ0 card K (0)  τ 2 = τ. Since ! is strongly C-universal, |K|m × ! is strongly C-universal (Proposi tion 5.4.7). By Theorem 5.4.15, we can write |K|m = i∈N Di , where each Di is a discrete union of at most τ many ni -cells. Let ! = j ∈N Cj , where Cj ∈ CZ (!) for each j ∈  N. Since C is I-stable, Di × Cj ∈ τ C = C for each i, j ∈ N. Thus, |K|m × ! = i,j ∈N Di × Cj ∈ Cσ . It is easy to see that each |K|m × Cj is a strong Z-set in |K|m × !. Hence, |K|m × ! is a strong Zσ -space. Therefore, |K|m × ! is an !-manifold by Theorem 5.10.15. " ! Remark 5.24 In the above theorem, it is not assumed that K is locally finitedimensional. Then, |K|m is not necessarily completely metrizable (cf. Theorem 1.7.2), that is, |K|m × 2 () is not necessarily an 2 ()-manifold.25 Now, we prove the Triangulation Theorem: Theorem 5.10.19 (TRIANGULATION) Suppose that τ C = C and C is I-stable. Let ! be a C-absorbing set in 2 (). For each !-manifold M, there exists a locally finite-dimensional simplicial complex K such that M ≈ |K|m × !. Proof Since M is an ANR (Theorem 5.10.15), there exists a locally finitedimensional simplicial complex K such that card K (0)  w(M) = w(!) = τ and M  |K|m (Theorem 1.13.22). Due to Theorem 5.10.18, |K|m × ! is an !-manifold. Then, applying the Classification Theorem 5.10.17, we have the result. " ! When C is multiplicative, Theorem 5.10.18 can be generalized as follows: Theorem 5.10.20 (ANR FACTOR) Suppose that τ C = C, I ∈ C, and C is multiplicative.26 Let ! be a C-absorbing set in 2 (). For each ANR X ∈ Cσ ,  ∈ M1 (τ ), X × ! is an !-manifold. Moreover, if X is homotopy dense in an ANR X 2 2  then X × ! is a C-absorbing set in an  ()-manifold X ×  (). Proof First note that X × ! is an ANR. Since ! is strongly C-universal, X × ! is strongly C-universal (Proposition 5.4.7). Since C is multiplicative and X, ! ∈ Cσ , in [120] that the completion of (|K|, ρK ) is an ANR in which |K| is homotopy dense, where ρK is the metric defined in Sect. 1.7. 26 Hence, C is I-stable. 25 It is proved

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it is easy to see X × ! ∈ Cσ . Since ! is a strong Zσ -space, it follows that X × ! is also a strong Zσ -space. Thus, X × ! is an !-manifold by Theorem 5.10.15. The additional statement can be obtained in combination with the Toru´nczyk Factor Theorem 2.2.14. " ! We have the following corollary (cf. Theorem 5.10.17(1)): Corollary 5.10.21 Suppose that τ C = C, I ∈ C, and C is multiplicative. Let ! be a C-absorbing set in 2 (). For each AR X ∈ Cσ , X × ! ≈ !. Moreover, if  ∈ M1 (τ ), then X × ! is a C-absorbing set in X is homotopy dense in an AR X 2 2  X ×  () ≈  (). " ! Combining the ANR Factor Theorems 5.10.20 and 5.10.17(3), we have the following: Theorem 5.10.22 (STABILITY) Suppose that τ C = C, I ∈ C, and C is multiplicative. Let ! be a C-absorbing set in 2 (). For each !-manifold M, the projection prM : M × ! → M is a near-homeomorphism. " ! We prove the Z-Set Unknotting Theorem: Theorem 5.10.23 (Z-SET UNKNOTTING) Suppose that In ×  ∈ C for every n ∈ N. Let X be a C-absorbing set in an 2 ()-manifold M and U, V open covers of X. If f : A → B is a homeomorphism between Z-sets in X and f is U-homotopic to the inclusion A ⊂ X, then f extends to a homeomorphism h : X → X that is st(U, V)-close to id. Proof First, assume that A ∩ B = ∅. Then, removing clM A ∩ clM B from M, we may assume that clM A ∩ clM B = ∅. Let W ∈ cov(X) be a star-refinement of V. " in M such that U|X " = W. Let  and W  = U and W|X Take open collections U M =



∩ U



 " \ (clM A ∩ clM B). W

Then, X ⊂ M  and M  is Gδ in M. By Lavrentieff’s Theorem 1.3.19, f extends to  → B  between Gδ -sets in M  such that f˜ is U-homotopic a homeomorphism f˜ : A   ⊂ clM  A and B ⊂ B  ⊂ clM  B. Let to id in M , where A ⊂ A    = M  \ (clM  A \ A)  ∪ (clM  B \ B)  . X  and X  is a homotopy dense Gδ -set in M, hence X  is an 2 ()Then, X ⊂ X "  ˜  ˜ manifold. By pushing the homotopy from f to id into X, f is st(U, W)-homotopic     Applying to id in X, Moreover, A = clX  A and B = clX  B, which are Z-sets in X. → the Z-Set Unknotting Theorem 2.9.7, we can obtain a homeomorphism g : X " "    ˜ X such that g|A = f and g is st(st(U, W), W)-close to id. Then, g(X) is a C By Propositions 5.4.18 and 5.6.2, X \ B  and g(X) \ B  are absorbing set in X.     C-absorbing sets in X \ B, where X \ B = X \ B and g(X) \ B = g(X) \ B. Let \B  such that W ≺ W and diam W < dist(W, X  \ B)  W be an open cover of X    for each W ∈ W , where X is given some d ∈ Metr(X). By Theorem 5.10.5, we

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have a homeomorphism h : g(X) \ B → X \ B that is W -close to id. We define h˜ : g(X) → X as follows: ˜ h(x) =

 h(x) x

if x ∈ g(X) \ B, if x ∈ B.

" Then, h˜ is a homeomorphism that is W-close to id. Thus, we have a homeomor"  st W)-close ˜ phism hg|X : X → X that is st(U, to id, hence it is st(U, V)-close to id. Now, applying the above case, we show the general case. Choose W, W ∈ cov(X) so that V  st2 W and W  W . h

Since In ×  ∈ C for every n ∈ N and X is a strong Zσ -space, the strong Cuniversality of X implies the strong Cσ -universality (Proposition 5.6.5). Since A∪B is a Z-set in X and A ∈ Cσ as a closed set in X ∈ Cσ , we have a Z-embedding g : A → X \ (A ∪ B) that is W -close to id, so g and g −1 are W-homotopic to id. Applying the above, we have homeomorphisms h1 : X → X such that h1 |A = g and h1 is st W-close to id. Observe that fg −1 : g(A) → B is st(U, W)-homotopic to id. Again applying the above we have homeomorphisms h2 : X → X such that h2 |g(A) = fg −1 and h2 is st(st(U, W), W)-close to id. Then, h2 h1 : X → X is the desired homeomorphism. " ! The following two lemmas are trivial. Lemma 5.10.24 If X is homotopy dense in 2 (), then XN and XfN are homotopy " ! dense in 2 ()N . In the above, XfN is homotopy dense in X N (cf. Proposition 1.13.27).

Lemma 5.10.25 Let X be a strong Zσ -space. Then, X × Y is a strong Zσ -space for any Y . In particular, XN = X × XN\{1} is a strong Zσ -space. If X = (X, ∗) is N\{1} pointed, then XfN = X × Xf is a strong Zσ -space. " ! Combining Proposition 5.4.11 with the above two lemmas, we can obtain the following: Proposition 5.10.26 Let X be a homotopy dense set in 2 () which is a strong Zσ space (and 0 ∈ X). Then, XN (resp. XfN ) is an absorbing set in 2 ()N for the class FXN (resp. FXN ). " ! f

In the above, note that X N and XfN are Zσ -sets in itself. Every Z-set in X N and XfN belongs to the classes FX N and FX N , respectively. f

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5.11 Absolute Borel Classes For a metrizable space X, let A0 (X) and M0 (X) be the collections consisting of all open sets and of all closed sets in X, respectively. For each n ∈ N, we inductively define An (X) and Mn (X) as the collections consisting of all countable unions of sets in Mn−1 (X) and of all countable intersections of sets in An−1 (X), respectively. Fσ

Gδσ

Fσ δσ

0 (X)

1 (X)

2 (X)

3 (X)

···

0 (X)

1 (X)

2 (X)

3 (X)

···

Gδ

Fσ δ

open

closed

Gδσ δ

Then, M1 (X) and A1 (X) are the collections of all Gδ sets and of all Fσ sets in X, respectively. Sets contained in A2 (X), M2 (X), . . . are said to be Gδσ , Fσ δ , . . . in X, respectively. We can determine the notation of a set of Mn (X) or An (X) as follows: First write the alternate sequence of letters δ and σ with n length and the last letter is δ or σ according to Mn (X) or An (X). It is decided by the first letter (δ or σ ) whether it is Gδσ ··· or Fσ δ··· .

Since X is a metrizable space, open sets and closed sets in X are Fσ and Gδ respectively, that is, A0 (X) ⊂ A1 (X) and M0 (X) ⊂ M1 (X). By induction, we have An−1 (X) ⊂ An (X) and Mn−1 (X) ⊂ Mn (X) for every n ∈ N. Hence, it follows that n−1 

Ai (X) ∪ Mi (X) ⊂ An (X) ∩ Mn (X).

i=1

For infinite ordinals α, by transfinite induction, we define Aα (X) (resp. Mα (X)) as the collectionconsisting of all countable unions (resp.countable intersec tions) of sets in M (X) (resp. A (X)), where β 0, X ∈ Mα (resp. X ∈ Aα ) if and only if X ∈ Mα (Y ) (resp. X ∈ Aα (Y )) for some compactum Y containing X as a subspace. " ! Concerning absolutely Borel spaces, we have the following: Corollary 5.11.4 Let X be a metrizable space with w(X)  τ . For α > 1 (resp. (τ ) (resp. X ∈ M (τ )) if and only if X = α > 2), X ∈ A α α i∈N Xi (resp. X =    X ), where X ∈ M (τ ) (resp. X ∈ A (τ )) for each i ∈ N. i β i i∈N i β 0, we can take v1 , . . . , vn ∈ Vx such that 1−

n  i=1

x(vi ) = x −

n  i=1

x(vi ) < ε/2.

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Then, σ = x, . . . , xn  ∈ (Vx ) ⊂ K. We define y ∈ σ ⊂ |K| as follows:

y(v) =

⎧ n ⎪ ⎪ ⎨x(v1 ) + 1 − i=1 x(vi ) x(vi ) ⎪ ⎪ ⎩0

if v = v1 , if v = v2 , . . . , vn , otherwise.

Observe that x − y =



|x(v) − y(v)| =

n 

  n  |x(vi ) − y(vi )| + 1 − x(vi )

i=1

v∈K (0)

i=1

  n  x(vi ) < ε. =2 1− i=1 1

Therefore, x ∈ |K| .

" ! 1

As we saw in the above lemma, |K| is a subspace of the unit sphere S1 K (0) 1 (0) of 1 (K (0) ). But we also remark that |K| ⊂ IK . Due to Proposition 1.2.2 (0) (Remark 1.2), the unit sphere S1 (K (0) ) is also a subspace of the product space RK . 1

(0)

Hence, |K| can also be regarded as a subspace of the product space IK . Then, 1 |K| ⊂ clIK (0) |K|. Let K be a countable infinite simplicial complex with K (0) = {vi | i ∈ N}, where vi = vj if i = j . Identifying each vertex vi ∈K (0) with the unit vector ei ∈ RN , we regard |K| ⊂ IN (⊂ RN ), that is, each x = i∈N x(vi )vi ∈ |K| is regarded as the point x ∈ IN , where x(i) = x(vi ), i ∈ N. The topology |K|m is clearly the subspace topology of |K| inherited from IN . Thus, |K|m can be regarded as a subspace of IN . Then, clIN |K| is a compactification of |K|m . Let Q0 = IN \ {0}. The following is a local-compactification of |K|m : |K|Q0 = clQ0 |K| = clIN |K| \ {0}, 1

where 0 ∈ clIN |K (0)| ⊂ clIN |K|. Besides, note that |K| ⊂ |K|Q0 . For each x ∈ Q, let Nx = {i ∈ N | x(i) = 0}. We use the metric dp for IN defined as follows:   dp (x, y) = sup min |x(i) − y(i)|, 1/i . i∈N

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Lemma 6.13.7 Let K be a countable simplicial complex without principal simplexes20 andlet K (0) = {vi | i ∈ N} (vi = vj if i = j ). For x ∈ IN , x ∈ clIN |K| if and only if i∈N x(i)  1, and vi1 , . . . , vin  ∈ K if i1 , . . . , in ∈ Nx . Proof The “only if” part is trivial, which is valid for any countable infinite  simplicial complex K. To see the “if” part, suppose that x ∈ IN , i∈N x(i)  1, and vi1 , . . . , vin  ∈ K if i1 , . . . , in ∈ Vx . For each n ∈ N, we write {i ∈ Nx | i < n} = {i1 , . . . , im }, where i1 < · · · < im . Then, vi1 , . . . , vim  ∈ K. Since K has no principal  simplexes, we have im+1 > im such that vi1 , . . . , vim , vim+1  ∈ K. Note that m j =1 x(ij )  1. We can define y ∈ |K| as follows: ⎧ ⎪ ⎪x(ij ) ⎨  y(i) = 1 − m j =1 x(ij ) ⎪ ⎪ ⎩0

i = ij , j = 1, . . . , m, i = im+1 , otherwise.

Then, dp (x, y) < 1/(in + 1) < 1/n. Hence, x ∈ clIN |K|.

" !

Combining the above two lemmas, we have the following: Lemma 6.13.8 For each countable simplicial complex K without principal simplexes,    1 1 |K|Q0 = (0, 1] · |K| = tx  x ∈ |K| , 0 < t  1 .

" !

Recall that X is locally equi-connected (LEC) if there exists a neighborhood U of the diagonal X in X2 and a map λ : U × I → X called an equi-connecting map such that λ(x, y, 0) = x, λ(x, y, 1) = y for all (x, y) ∈ U, λ(x, x, t) = x for all x ∈ X and t ∈ I, where X is equi-connected (EC) if U = X2 . Then, A ⊂ X is λ-convex if A2 ⊂ U and λ(A2 × I) ⊂ A. Due to Corollary 1.14.4, a metrizable LEC (resp. EC) space X is an ANR (resp. AR) if X has an equi-connecting map λ such that each point of X has arbitrarily small λ-convex neighborhoods. Theorem 6.13.9 Let K be a countable simplicial complex without principal simplexes. Then, clIN |K| is an AR; hence |K|Q0 is an ANR. Proof Let μ : (IN )2 → IN be a map defined by   μ(x, y)(i) = min x(i), y(i) , i ∈ N,

20 A

principal simplex of K means a simplex of K that is not a face of any other simplex of K.

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and define a map λ : (IN )2 × I → IN as follows:  λ(x, y, t) =

(1 − 2t)x + 2tμ(x, y)

if 0  t  1/2,

(2t − 1)y + (2 − 2t)μ(x, y) if 1/2  t  1.

Then, λ(x, y, 0) = x, λ(x, y, 1) = y, λ(x, x, t) = x for any x, y  ∈ IN and N N t ∈ I, that is, λ is an equi-connecting map for I . For x, y ∈ I with i∈N x(i),  i∈N y(i) < ∞, Nμ(x,y) = Nx ∩ Ny and



μ(x, y)(i) 

i∈N



x(i),

i∈N



y(i).

i∈N

Then, it follows that Nλ(x,y,t ) ⊂ Nx if 0  t  1/2; Nλ(x,y,t ) ⊂ Ny if 1/2  t  1;    μ(x, y)(i)  x(i), y(i). i∈N

i∈N

i∈N

Hence, λ((clIN |K|)2 × I) ⊂ clIN |K| by Lemma 6.13.7, Thus, the restriction λ|(clIN |K|)2 × I is an equi-connecting map for clIN |K|. Let x, y, z ∈ clIN |K| with dp (x, z), dp (y, z) < 1/n and let t ∈ I. Then, for each i = 1, . . . , n, |x(i) − z(i)|, |y(i) − z(i)| < 1/n ( 1/i), which implies that |μ(x, y)(i) − z(i)| < 1/n. Hence, |λ(x, y, t)(i) − z(i)| < 1/n for each i = 1, . . . , n. Thus, we have   dp (λ(x, y, t), z) = sup min |λ(x, y, t)(i) − z(i)|, 1/i < 1/n, i∈N

which means that the 1/n-neighborhood Bdp (z, 1/n) of z in clIN |K| is λ-convex. By Corollary 1.14.4, clIN |K| is an AR, and its open subset |K|Q0 = clIN |K| \ {0} is an ANR. " ! Theorem 6.13.10 For each combinatorial ∞-manifold K, |K|Q0 is an R+ -stable 1 Q-manifold, for which |K|m is an f.d.cap set and |K|Q0 \ |K| is a cap set. Proof Since |K|m is an 2f -manifold (Theorem 6.13.1), if |K|m is homotopy dense in |K|Q0 , then the result follows from Theorem 5.9.13. Since |K|Q0 is an ANR

6.13 Metric Combinatorial ∞-Manifolds

561

(Theorem 6.13.9), we can apply Proposition 1.15.8 to show that |K|m is homotopy dense in |K|Q0 . Let C be a compact set in |K|Q0 and ε > 0. Since C is closed in IN and 0 ∈ C, we can choose n ∈ N so that 1/n < min{ε/3, dp (0, C)}. For each x ∈ C,   dp (0, x) = sup min |x(i)|, 1/i > 1/n, i∈N

which means that x(i) > 1/n for some i = 1, . . . , n. Let L be the maximal subcomplex of K with L(0) = {v1 , . . . , vn }, where K (0) = {vi | i ∈ N} and the maximality of L means that L is full in K. Let    |L|∗ = tx  x ∈ |L|, t ∈ [1/n, 1] ⊂ |K|Q0 and define a map f : C → |L|∗ as follows: f (x)(i) =

 x(i) if i  n, 0

if i > n.

Then, dp (f, id) < 1/n < ε/3. Since |L| is a Z-set in the 2f -manifold |K|m because of compactness, we have a map g : |K|m → |K|m \ |L| with dp (g, id) < ε/3. We can define a map g ∗ : |L|∗ → |K|Q0 by g(tx) = tg(x) because |L| × [1/n, 1] ≈ |L|∗ by the following correspondence: |L| × [1/n, 1]  (x, t) → tx ∈ |L|∗ . Then, dp (g ∗ , id)  dp (g, id) < ε/3. Observe that    g ∗ (|L|∗ ) ∩ tx  x ∈ |L|, t ∈ [0, 1] = ∅, which means that for each y ∈ g ∗ (|L|∗ ), there is some i(y) > n such that pri(y) (y) = y(i(y)) > 0. Because of the compactness of g ∗ (|L|∗ ), we have  y1 , . . . , y ∈ g ∗ (|L|∗ ) such that g ∗ (|L|∗ ) ⊂ j =1 pri(yj ) ((0, ∞)). Let m = ∗ ∗ max{i(y1), . . . , i(y )}. Then, each y ∈ g (|L| ) has a positive coordinate y(i) > 0 for some i = n + 1, . . . , m. We can define a map k : g ∗ (|L|∗ ) → |K|m as follows: ⎧ ⎪ y(i) ⎪ ⎪ ⎪ ⎨ 1 − n y(j )y(i) j =1 m k(y)(i) = ⎪ ⎪ j =n+1 y(j ) ⎪ ⎪ ⎩0

if i  n, if n < i  m, if i > m.

6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds

562

Then, dp (k, id) < 1/n < ε/3. Thus, we have a map kg ∗ f : C → |K|m , where dp (kg ∗ f, id)  dp (k, id) + dp (g ∗ , id) + dp (f, id) < ε. From Theorem 5.9.13, it follows that |K|Q0 is a Q-manifold and |K|m is an f.d.cap set for |K|Q0 . Next, we show that |K|Q0 is R+ -stable, where we use Theorem 2.11.13. It can be assume that K is connected. Since |K|m is connected, so is |K|Q0 . It suffices to show that |K|Q0 is properly contractible to infinity, that is, for each compact set C in |K|Q0 , there is a proper homotopy h : |K|Q0 × I → |K|Q0 with h0 = id and h1 (|K|Q0 ) ⊂ |K|Q0 \ C. For each compact set C in |K|Q0 , choose n ∈ N so that 1/n < dp (0, C). Then,    Bdp (0, 1/n) = x ∈ IN  x(i)  1/n if i  n ⊂ IN \ C. We define a homotopy h : IN × I → IN as follows: h(x, t) = ((1 − t) + t/n)x for each (x, t) ∈ IN × I, where h0 = id and h1 (IN ) = [0, 1/n]N ⊂ Bdp (0, 1/n) ⊂ IN \ C. For each x ∈ |K|Q0 and t ∈ I, 

ht (x)(i) = ((1 − t) + t/n)

i∈N

 i∈N

x(i) 



x(i)  1.

i∈N

Since (1 − t) + t/n > 0 for each t ∈ I, we have x(i) = 0 ⇔ ht (x)(i) = ((1 − t) + t/n)x(i) = 0, which implies Vx = Vht (x) . By Lemma 6.13.7, we have h(|K|Q0 × I) ⊂ |K|Q0 . By restricting h, the desired proper homotopy can be obtained. 1 Finally, we prove that |K|Q0 \ |K| is a cap set for |K|Q0 . From Lemma 6.13.8, it follows that   1 Q0 1 · |K|Q0 . 1− |K| \ |K| = n n∈N

Each (1−1/n)·|K|Q0 is a Q-manifold because it is homeomorphic to |K|Q0 , where (1 − 1/n) · |K|Q0 is a Z-set in (1 − 1/(n + 1)) · |K|Q0 . More generally, for each 0 < t < 1, t · |K|Q0 is a Z-set in |K|Q0 . This can be shown by using Theorem 2.8.6 as follows: For each map f : Q → |K|Q0 and ε > 0, letting C = f (Q) in the above, we can obtain a map h : f (Q) → |K|m with dp (h, id) < ε. Then, hf : Q → |K|m ⊂ |K|Q0 \ t · |K|Q0 and dp (hf, f ) < ε.

6.13 Metric Combinatorial ∞-Manifolds

563

  It remains to show that the tower (1 − 1/n) · |K|Q0 n∈N has the cap. Let C be a compact set in |K|Q0 and m ∈ N. For each ε > 0, choose n > m so that 1/n < ε/6. As we saw above, we can obtain a map f : C → |K|m such that f ε/6 id. Then, we have a map f  : C → (1 − 1/n) · |K|m ⊂ (1 − 1/n) · |K|Q0 defined by f  (x) = (1 − 1/n)f (x) for each x ∈ C, where f  ε/6 f . Since (1 − 1/n) · |K|Q0 is a Q-manifold, f  is ε/6-homotopic to a Z-embedding g : C → (1 − 1/n) · |K|Q0 (Strong Universality Theorem 2.10.10). Then, g ε/2 id, C∩(1−1/m)·|K|Q0 is a Z-set in (1−1/n)·|K|Q0 , and g|C∩(1−1/m)·|K|Q0 is a Zembedding into (1 − 1/n) · |K|Q0 Applying the Z-Set Unknotting Theorem 2.11.6, we have a homeomorphism h : (1 − 1/n) · |K|Q0 → (1 − 1/n) · |K|Q0 such that h ε/2 id and h|C ∩ (1 − 1/m) · |K|Q0 = g|C ∩ (1 − 1/m) · |K|Q0 . Then, h−1 g : C → (1 − 1/n) · |K|Q0 is a Z-embedding such that h−1 g is ε-close to id and h−1 g|C ∩ (1 − 1/m) · |K|Q0 = id. " ! Below, it will be proved in Theorem 6.13.13 that |K|m is homotopy dense in 1 |K| for an arbitrary simplicial complex K. Combining this with Theorem 6.13.10, we can easily obtain the following: Corollary 6.13.11 For each combinatorial ∞-manifold K, |K| manifold, for which |K|m is an f.d.cap set.

1

is an 2 " !

Corollary 6.13.12 For each contractible combinatorial ∞-manifold K, (clIN |K|, |K|m ) ≈ (Q, Qf ), 1

(clIN |K|, clIN |K| \ |K| ) ≈ (Q, B(Q)), and 1

(clIN |K|, |K| ) ≈ (Q, I (Q)). Proof Combining Theorems 6.13.10 and 2.11.11, we have |K|Q0 ≈ Q × R+ ≈ Q \ {0}. Since clIN |K| is the one-point compactification of |K|Q0 , it follows that clIN |K| ≈ Q. Since the one-point set in Q is a Z-set in Q, |K|Q0 is homotopy dense in clIN |K|. Hence, |K|m is homotopy dense in clIN |K|. Since |K|m ≈ 2f , it follows that |K|m is an f.d.cap set for clIN |K|, that is, (clIN |K|, |K|m ) ≈ (Q, Qf ). 1

As in Lemma 6.13.8, we have clIN |K| = I · |K| . By the same proof as the last part 1 of Theorem 6.13.10, it can be proved that clIN |K| \ |K| is a cap set for clIN |K|, that is, 1

(clIN |K|, clIN |K| \ |K| ) ≈ (Q, B(Q)) and 1

(clIN |K|, |K| ) ≈ (Q, I (Q)). Here, we give a proof of Theorem 6.13.5:

" !

6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds

564

Proof (Theorem 6.13.5) The “only if” part is Theorem 6.13.10 (cf. Corol1 lary 6.13.11), To see the “if” part, assume that |K| is an 2 -manifold. Since 2 every compact set in an  -manifold is a Z-set (Proposition 3.2.10), each simplex 1 σ ∈ K is a Z-set in |K| . As mentioned above, it will be proved that |K|m is 1 homotopy dense in |K| (Theorem 6.13.13). Then, it follows that each simplex σ ∈ K is a Z-set in |K|m . Hence, we can conclude that K is a combinatorial ∞manifold by the Characterization Theorem 6.11.1 for combinatorial ∞-manifolds. " ! 1

Theorem 6.13.13 For every simplicial complex K, the 1 -completion |K| of |K| 1 is an ANR and |K|m is homotopy dense in |K| . 1

Proof To prove that |K| is an ANR, it suffices to construct an equi-connecting 1 map λ such that each point of |K| has arbitrarily small λ-convex neighborhoods. We first define μ : 1 (K (0) )2 → 1 (K (0) ) by   μ(x, y)(v) = min |x(v)|, |y(v)| , v ∈ K (0). Then, μ is continuous. Indeed,       min |x(v)|, |y(v)| − min |x  (v)|, |y  (v)|       max |x(v)| − |x  (v)|, |y(v)| − |y  (v)|      max x(v) − x  (v), y(v) − y  (v)      x(v) − x  (v) + y(v) − y  (v), which implies that μ(x, y) − μ(x  , y  )  x − x   + y − y  . From the definition, it follows that μ(x, y) = 0 if and only if x(v) = 0 or y(v) = 0 for every v ∈ K (0) , which implies that x − y = x + y. Hence, x − y < x + y implies μ(x, y) = 0. Observe that Vμ(x,y) = Vx ∩ Vy for each (x, y) ∈ 1 (K (0) )2 . 1 We define an open neighborhood U of the diagonal |K| in 1 (K (0) )2 as follows:    1 2   x − y < 2 . U = (x, y) ∈ |K| By the above observation, μ(x, y) = 0 for each (x, y) ∈ U . It is easy to see that x,

μ(x, y) μ(x, y) 1 1 ∈ |(Vx )| and y, ∈ |(Vy )| , μ(x, y) μ(x, y)

6.13 Metric Combinatorial ∞-Manifolds 1

565

1

where |(Vx )| and |(Vy )| are convex sets in 1 (K (0) ) and they are contained 1 1 in |K| . Hence, we can define an equi-connecting map λ : U × I → |K| as follows:

λ(x, y, t) =

⎧ 2tμ(x, y) ⎪ (1 − 2t)x + ⎪ ⎪ ⎨ μ(x, y)

if 0  t  1/2,

⎪ ⎪ (2 − 2t)μ(x, y) ⎪ ⎩(2t − 1)y + μ(x, y)

if 1/2  t  1.

1

Now, we show that each point z ∈ |K| has arbitrarily small λ-convex neighborhoods. For each ε > 0, choose v1 , . . . , vn ∈ Vz so that 1−

n 

z(vi ) = z −

i=1

n 

z(vi ) < ε/2.

i=1

For each i = 1, . . . , n, choose 0 < αi < z(vi ) so that 1 −

n

i=1 αi

< ε/2 and define

  1  W = x ∈ |K|  x(vi ) > αi for all i = 1, . . . , n . 1

Then, W is an open neighborhood of z in |K| with diam W < ε. Indeed, for each x, y ∈ W .     n n n        x − y  x(vi ) + 1 − y(vi ) x(vi ) − y(vi ) + 1 − i=1



n 

i=1

  x(vi ) − αi +

i=1

2−2

n 



i=1

 y(vi ) − αi + 2 −

i=1 n 

n 

x(vi ) −

i=1

n 

y(vi )

i=1

αi < ε.

i=1

To see the λ-convexity of W , let x, y ∈ W and t ∈ I. Then, μ(x, y)(vi ) = min{x(vi ), y(vi )} and μ(x, y)  1. When t  1/2, for each i = 1, . . . , n, λ(x, y, t)(vi ) = (1 − 2t)x(vi ) +

2t min{x(vi ), y(vi )} μ(x, y)

 min{x(vi ), y(vi )} > αi , which means λ(x, y, t) ∈ W . Similarly, when t  1/2, we have λ(x, y, t) ∈ W . 1 Therefore, W is λ-convex. Thus, it follows from Theorem 1.14.4 that |K| is an ANR.

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6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds 1

Next, applying Corollary 1.15.5, we show that |K|m is homotopy dense in |K| . 1 For each z ∈ |K| and ε > 0, choose v1 , . . . , vn ∈ Vy (⊂ K (0)) so that 1−

n 

z(vi ) = y −

i=1

n 

z(vi ) < ε/2.

i=1

Then, σ0 = v1 , . . . , vn  ∈ (Vz ) ⊂ K by Lemma 6.13.6. Take δ > 0 so that δ < ε, z(vi )/2, i = 1, . . . , n. Let α : Sk−1 → |K|m ∩ B(z, δ) be a map. Then, α(Sk−1 ) ⊂ | St(σ0 , K)|. Otherwise, σ0  cK (x) for some x ∈ α(Sk−1 ), that is, x(vi ) = 0 for some i = 1, . . . , n. Then, x − z  |x(vi ) − z(vi )| = z(vi ) > δ, which is a contradiction. We define y ∈ rint σ0 as follows: ⎧ ⎪ z(vi )  ⎪ ⎪  if v = vi for some i < n, ⎨ n if v = vn , y(v) = z(vn ) + 1 − i=1 z(vi ) ⎪ ⎪ ⎪ ⎩ 0 otherwise. Then, it follows that y − z = |y(vn ) − z(vn )| + 1 −

n 

  n  z(vi ) = 2 1 − z(vi ) < ε.

i=1

i=1

We can now define a homotopy h : Sk−1 × I → |K|m as follows: h(x, t) = (1 − t)α(x) + ty ∈ cK (x) ⊂ | St(σ0 , K)| ⊂ |K|, where h0 = α and h1 (Sk−1 ) = {y}. Observe that h(x, t) − z  (1 − t)α(x) − z + ty − z  (1 − t)δ + tε < ε, which means that h(Sk−1 × I) ⊂ |K|m ∩ B(z, ε). Hence, α is null-homotopic in |K|m ∩ B(z, ε). Thus, we can apply Corollary 1.15.5 to conclude that |K|m is 1 " ! homotopy dense in |K| . Summing up Theorems 6.11.1, 6.11.12, 6.12.2, 6.13.1, 6.13.4, and 6.13.5, the following are equivalent for a countable simplicial complex K: (a) K is a combinatorial ∞-manifold; (b) |K| is a PL ∞-manifold; (c) |K| is an R∞ -manifold;

6.14 Bi-topological Infinite-Dimensional Manifolds

(c’) (d) (e) (f) (g) (g’)

567

|K| × Q is a Q∞ -manifold; Every simplex of K is a Z-set in |K|; Every simplex of K is a Z-set in |K|m ; Lk(σ, K) is non-empty and contractible for each simplex σ ∈ K; |K|m is an 2f -manifold; |K|m × Q is an 2Q -manifold; 1

(h) |K| is an 2 -manifold. It is unknown whether the converse of Theorem 6.13.10 is true or not, that is:

•

? Open Problem

For a countable simplicial complex K, if |K|Q0 is an R+ -stable Q-manifold, then is K a combinatorial ∞-manifold?

Remark 6.15 Let K be a simplicial complex satisfying the following condition: (Lk-τ ) card Lk(σ, K)(0) = τ and | Lk(σ, K)|  0 for every σ ∈ K. Due to Theorem 6.12.1, the second condition of the above (Lk-τ ) is equivalent to the following: • Every σ ∈ K is a Z-set in |K|.

•

? Open Problems

For an uncountable simplicial complex K, the following questions occur: 1. Does the condition (Lk-τ ) imply that St(v, K) ∼ = F () for each v ∈ K (0) ? 2 2. Is |K|m an f ()-manifold? 1

3. Is |K| an 2 ()-manifold? 1 4. Is |K|m an Mfd 0 -absorption base for |K| ? 5. Let K and L be simplicial complexes satisfying (Lk-τ ). Does |K| ≈ |L| imply K∼ = L?

6.14 Bi-topological Infinite-Dimensional Manifolds A bi-topological space is a space X = (X, T1 , T2 ) given two topologies T1 and T2 , or a couple (X1 , X2 ) of spaces Xi = (X, Ti ), i = 1, 2. For example, the polyhedron |K| of a simplicial complex K has two natural topologies, namely the weak (or Whitehead) topology Tw and the metric topology Tm . Then, (|K|w , |K|m )

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6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds

(or (|K|, Tw , Tm )) is a bi-topological space. The underlying sets of R∞ = lim Rn − → and Q∞ = lim Qn are the same as the subspaces RN (= s f ) and QN of the f f − → countable product spaces RN (= s) and QN , respectively. Thus, (R∞ , RN f ) and ∞ N (Q , Qf ) are also bi-topological spaces, where 2 RN f ≈ f

2 and QN f ≈ Q .

Let X = (X, T1 , T2 ) and Y = (Y, T1 , T2 ) be bi-topological spaces. A function f : X → Y is called a bi-topological map, a bi-topological embedding, or a bitopological homeomorphism if both f : (X, T1 ) → (Y, T1 ) and f : (X, T2 ) → (Y, T2 ) are continuous, embeddings, or homeomorphisms, respectively. A bitopological open (resp. closed) embedding means a bi-topological embedding f : X → Y such that h(X) is open (resp. closed) with respect to both topologies T1 and T2 . A bi-topological space (M, Tw , Tm ) is called a bi-topological manifold mod∞ N N ∞ eled on (R∞ , RN f ) (resp. (Q , Qf )) or simply an (R , Rf )-manifold (resp. a (Q∞ , QN f )-manifold) if the topology Tm is metrizable, Tm ⊂ Tw and each point of M has a Tm -open neighborhood U with a bi-topological open embedding ϕ : (U, Tw , Tm ) → (R∞ , RN f)

  resp. ϕ : (U, Tw , Tm ) → (Q∞ , QN f) .

Then, (M, Tw ) is an R∞ -manifold (resp. Q∞ -manifold) and (M, Tn ) is an RN fN manifold (resp. Qf -manifold). In this section, we show fundamental properties of bi-topological manifolds ∞ N modeled on (R∞ , RN f ) and (Q , Qf ). The following theorem is their source: Theorem 6.14.1 Let M = (M, Tw , Tm ) and N = (N, Tw , Tm ) be bi-topological spaces satisfying the following conditions: (i) (M, Tw ) and (N, Tw ) are R∞ -manifolds (resp. Q∞ -manifolds), N (ii) (M, Tm ) and (N, Tm ) are RN f -manifolds (resp. Qf -manifolds), (iii) idM : (M, Tw ) → (M, Tm ) and idN : (N, Tw ) → (N, Tm ) are fine homotopy equivalences (so Tm ⊂ Tw ). Suppose that (M, Tm ) ≈ (N, Tm ). Then, for each Tm -open cover U of N, each homeomorphism f : (M, Tm ) → (N, Tm ) is U-close to a bi-topological homeomorphism h : M → N. Remark 6.16 The following example shows that the condition (iii) in Theo2 ∞ rem 6.14.1 is essential. Let (M, Tm ) = (N, Tm ) = I2 × RN f , (M, Tw ) = I × R ,  ∞ n and (N, Tw ) = lim Xn × R = lim(Xn × R ), where X1 ⊂ X2 ⊂ · · · are defined − → − → as follows: Xn = {0} × I ∪ I × ∂I ∪ [1/n, 1] × I ⊂ I2 , n ∈ N.

6.14 Bi-topological Infinite-Dimensional Manifolds

569

I

1 n

I

Fig. 6.8 Xn = {0} × I ∪ I × ∂I ∪ [1/n, 1] × I

Then, (N, Tw ) ≈ S1 × R∞ which is not homotopic to (M, Tw ). See Fig. 6.8. Before proving Theorem 6.14.1, we will derive fundamental properties of ∞ N (R∞ , RN f )- and (Q , Qf )-manifolds from Theorem 6.14.1. To this end, the following proposition is necessary: ∞ N Proposition 6.14.2 For each (R∞ , RN f )- or (Q , Qf )-manifold (M, Tw , Tm ), the identity id : (M, Tw ) → (M, Tm ) is a fine homotopy equivalence.

Proof Recall that an R∞ -manifold is an ANE (Proposition 6.1.8). Due to Theorem 1.15.1 with Remark 1.15, it suffices to show that id : (M, Tw ) → (M, Tm ) is a local ∗-connection. Any Tm -neighborhood of each x ∈ M contains a Tm -open neighborhood U of x with a bi-topological open embedding ϕ : (U, Tw , Tm ) → (R∞ , RN f)

  or ϕ : (U, Tw , Tm ) → (Q∞ , QN f)

such that ϕ(U ) is convex, where the following operation is continuous with respect to the topology inherited from R∞ (resp. Q∞ ): ϕ(U )2 × I  (y, z, t) → (1 − t)y + tz ∈ ϕ(U ). Then, it follows that ϕ(U ) is contractible as a subspace of R∞ (resp. Q∞ ). Hence, U is contractible with respect to Tw , which means that id is a local ∗-connection. " ! ∞ N For (R∞ , RN f )- and (Q , Qf )-manifolds, the Characterization, Open Embedding, Classification, and Triangulation Theorems are easily obtained:

Theorem 6.14.3 (CHARACTERIZATION) A bi-topological space (M, Tw , Tm ) ∞ N is an (R∞ , RN f )-manifold (resp. (Q , Qf )-manifold) if and only if (M, Tw ) is N an R∞ -manifold (resp. Q∞ -manifold), (M, Tm ) is an RN f -manifold (resp. Qf manifold), and idM : (M, Tw ) → (M, Tm ) is a fine homotopy equivalence.

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6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds

Proof The “only if” part follows from Proposition 6.14.2. We show the “if” N part. Since (M, Tm ) is an RN f -manifold (resp. a Qf -manifold), there exists an N open embedding g : (M, Tm ) → RN f (resp. g : (M, Tm ) → Qf ) by the Open Embedding Theorem 2.5.10 with Theorem 5.5.10 (or combination of N Theorems 5.2.3, 5.2.5, 5.2.6). Then, g(M) is not only open in RN f (resp. Qf ) but ∞ ∞ also open in R (resp. Q ), where we write the former by Nm and the latter by ∞ N Nw . Thus, we have an (R∞ , RN f )-manifold (resp. (Q , Qf )-manifold) (Nw , Nm ). Note that id : Nw → Nm is a fine homotopy equivalence. By Theorem 6.14.1, we have a bi-topological homeomorphism h : (M, Tw , Tm ) → (Nw , Nm ). Hence, ∞ N (M, Tw , Tm ) is an (R∞ , RN " ! f )-manifold (resp. (Q , Qf )-manifold). In the above proof, h is a bi-topological open embedding into (R∞ , RN f ) (resp. ∞ N (Q , Qf )). Thus, we have also the following Open Embedding Theorem: Theorem 6.14.4 (OPEN EMBEDDING) For each (R∞ , RN f )-manifold (resp. ∞ N (Q , Qf )-manifold) (M, Tw , Tm ), there exists a bi-topological open embedding   ∞ N h : (M, Tw , Tm ) → (R∞ , RN f ) resp. (Q , Qf ) . ∞ N Namely, every (R∞ , RN f )-manifold (resp. (Q , Qf )-manifold) can be bi-topo∞ N " ! logically embedded into (R∞ , RN f ) (resp. (Q , Qf ) as an open set. N By combining the Classification Theorem 2.6.1 for RN f - and Qf -manifolds (cf. Theorem 5.1.10) and Theorem 6.14.1, the following Classification Theorem can be obtained:

Theorem 6.14.5 (CLASSIFICATION) Suppose that both (M, Tw , Tm ) and ∞ N  (N, Tw , Tm ) are (R∞ , RN f )- or (Q , Qf )-manifolds. If (M, Tm )  (N, Tm ), then M and N are bi-topologically homeomorphic, which means that there exists a bi-topological homeomorphism h : M → N. " ! In the above, (M, Tm )  (N, Tm ) can be replaced with (M, Tw )  (N, Tw ). ∞ N Theorem 6.14.6 Any R∞ - or RN f -manifold has the unique (R , Rf )-manifold ∞ N structure and any Q∞ - or QN f -manifold has the unique (Q , Qf )-manifold structure.

Proof By the Classification Theorem 6.14.5, it suffices to show that any R∞ - or ∞ N ∞ N RN f -manifold has an (R , Rf )-manifold structure and any Q - or Qf -manifold has a (Q∞ , QN f )-manifold structure. As is seen by easy observation, the proof of N Theorem 6.14.3 shows that every RN f -manifold (resp. every Qf -manifold) has an ∞ N (R∞ , RN f )-manifold structure (resp. a (Q , Qf )-manifold structure). ∞ By Theorem 6.11.8, any R -manifold M is homeomorphic to |K| for some combinatorial ∞-manifold K. Then, |K|m is an RN f -manifold (Theorem 6.13.1) and id : |K| → |K|m is a fine homotopy equivalence (Theorem 1.7.6). From

6.14 Bi-topological Infinite-Dimensional Manifolds

571

the Characterization Theorem 6.14.3, it follows that (|K|, |K|m ) is an (R∞ , RN f )N ∞ manifold. Thus, M has an (R , Rf )-manifold structure. For any Q∞ -manifold M, there is some combinatorial ∞-manifold K such that M ≈ |K| × Q (Theorem 6.11.12). Then, |K|m × Q is an QN f -manifold (Theorem 6.13.4). Moreover, id : |K| × Q → |K|m × Q is a fine homotopy equivalence (Theorem 1.7.6 and Corollary 1.15.4 with Remark 1.16). Hence, by the Characterization Theorem 6.14.3, (|K| × Q, |K|m × Q) is a (Q∞ , QN f )-manifold. ∞ N " ! Thus, M has a (Q , Qf )-manifold structure. The above proof of the R∞ - or Q∞ -manifold case can be easily modified to a proof of the following Triangulation Theorem: Theorem 6.14.7 (TRIANGULATION) For each (R∞ , RN f )-manifold (resp. ∞ N (Q , Qf )-manifold) (M, Tw , Tm ), there exists a simplicial complex K with a bi-topological homeomorphism   h : (M, Tw , Tm ) → (|K|, |K|m ) resp. (|K| × Q, |K|m × Q) , where K must be a combinatorial ∞-manifold.

" !

The following are also easily obtained: ∞ N Theorem 6.14.8 (STABILITY) For each (R∞ , RN f )-manifold (resp. (Q , Qf )manifold) (M, Tw , Tm ), there exists a bi-topological homeomorphism

h : (Mw × R∞ , Mm × RN f ) → (M, Tw , Tm )

  resp. h : (Mw × Q∞ , Mm × QN f ) → (M, Tw , Tm ) . Given a Tm -open cover U of M, h can be taken U-close to the projection. Sketch of Proof. By Corollary 1.15.4 with Remark 1.16, id : (M, Tw ) × R∞ → (M, Tm ) × ∞ → (M, Tm ) × QN RN f ) is a fine homotopy equivalence. Then, f (resp. id : (M, Tw ) × Q the result can be obtained by combining the Stability Theorem 2.3.7 for 2f -manifolds (resp. 2Q -manifolds) (cf. Theorem 5.1.10) and Theorem 6.14.1.

Theorem 6.14.9 (NEGLIGIBILITY) Let (M, Tw , Tm ) be an (R∞ , RN f )- or ∞ N (Q , Qf )-manifold. If A is a Z-set in (M, Tm ), then there exists a bi-topological homeomorphism h : M \ A → M. Given a Tm -open cover U of M, h can be taken U-close to the identity. Sketch of Proof. Let Tw = Tw |M \ A and Tm = Tm |M \ A. Then, (M \ A, Tw , Tm ) is also ∞ N an (R∞ , RN f )-manifold (resp. a (Q , Qf )-manifold). Combining the Negligibility Theo2 2 rem 2.9.8 for f -manifolds (resp. Q -manifolds) (cf. Theorem 5.1.10) and Theorem 6.14.1, we can obtain the result.

Theorem 6.14.10 (Tw -COMPACT SET UNKNOTTING) Let (M, Tw , Tm ) be an ∞ N (R∞ , RN f )- or (Q , Qf )-manifold and let f : A → B be a homeomorphism

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6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds

between Tw -compact sets in M. If f is homotopic to id in (M, Tm ), then f extends to a bi-topological homeomorphism f˜ : M → M. Given a Tm -open cover U of M, if f U id in (M, Tm ), then f˜ can be taken U-close to id. Sketch of Proof. Note that A and B are Z-sets in (M, Tw ). Combining Z-Set Unknotting Theorem 2.9.7 for 2f -manifolds (resp. 2Q -manifolds) (cf. Theorem 5.1.10) and Theorem 6.14.1, we can obtain the result.

The following is a key lemma to prove Theorem 6.14.1: Lemma 6.14.11 Let ϕ : X → Y is a bijective fine homotopy equivalence from N an R∞ -manifold (resp. a Q∞ -manifold) X onto an RN f -manifold (resp. a Qf manifold) Y . Given finite-dimensional compacta A ⊂ X and B ⊂ Y (there is no dimensional assumption in the (Q∞ , QN f )-case) so that ϕ(A) ⊂ B, for each U ∈ cov(Y ), there exists a U-homotopy inverse h : Y → X of ϕ such that hϕ|A = id (i.e., h|ϕ(A) = ϕ −1 |ϕ(A)) and h|B is an embedding. A ϕ|A

ϕ(A)

⊂

B

⊂ h|B

⊂

X ϕ

h

Y

Proof Choose V ∈ cov(Y ) so that st V ≺ U and let f : Y → X be a Vhomotopy inverse of ϕ. Then, ϕ −1 |ϕ(A) is continuous and ϕ −1 |ϕ(A) ϕ −1 (V) f |ϕ(A) because f ϕ ϕ −1 (V) idY . Since X is an ANE (Theorem 6.1.8), we can apply the Homotopy Extension Theorem 1.13.11 to obtain a map g : Y → X such that g|ϕ(A) = ϕ −1 |ϕ(A) and g ϕ −1 (V) f . Since X has property (Af ) (resp. (A)) (Theorem 6.4.10), there exists an embedding j : B → X such that j |ϕ(A) = g|ϕ(A) = ϕ −1 |ϕ(A) and j ϕ −1 (V) g|B. Again using the Homotopy Extension Theorem 1.13.11, we can extend j to a map h : Y → X such that h ϕ −1 (V) g. Then, h|ϕ(A) = j |ϕ(A) = ϕ −1 |ϕ(A) and h|B = j is an embedding. Moreover, ϕh V ϕg V ϕf V idX

and

hϕ ϕ −1 (V) gϕ ϕ −1 (V) f ϕ ϕ −1 (V) idX , which implies that h is a U-homotopy inverse of ϕ.

" !

Proof (Theorem 6.14.1) For simplicity, let X = (M, Tm ), X∞ = (M, Tw ), Y = (N, Tm ), Y∞ = (N, Tw ), ϕ = idM : X∞ → X, and ψ = idN : Y∞ → Y, where X∞ and Y∞ are R∞ -manifolds (resp. Q∞ -manifolds), X and Y are RN fN manifolds (resp. Qf -manifolds), and, ϕ and ψ are fine homotopy equivalences. Then, a homeomorphism f : X → Y and U ∈ cov(Y ) are given. Due to

6.14 Bi-topological Infinite-Dimensional Manifolds

573

Proposition 1.3.22(1), there is ρ ∈ Metr(Y ) such that {Bρ (y, 1) | y ∈ Y } ≺ U. We will construct a homeomorphism f∞ : X → Y that is also a homeomorphism from X∞ onto Y∞ and ρ(f∞ , f ) < 1. We can write X∞ = lim Xn and Y∞ = lim Yn , where X1 ⊂ X2 ⊂ · · · and Y1 ⊂ − → − → Y2 ⊂ · · · of finite-dimensional compacta (resp. compacta) in X and Y , respectively. Then, we can assume that X1 is a singleton. In the following, let d ∈ Metr(X). N Recall that any compact set in an RN f -manifold (resp. a Qf -manifold) is a Z-set. Put m(1) = 1 and f1 = f . Since Xm(1) is a singleton, f1 (Xm(1) ) ⊂ Yn(1) for some n(1) ∈ N. Choose U1 ∈ cov(Y ) so that mesh st U1 < 2−2 and mesh f1−1 (st U1 ) < 2−2 . Since f1 ϕ is a bijective fine homotopy equivalence, we can apply Lemma 6.14.11 to obtain a U1 -homotopy inverse j1 of f1 ϕ such that j1 f1 |Xm(1) = j1 f1 ϕ|Xm(1) = id and j1 |Yn(1) is an embedding. Then, j1 (Yn(1) ) ⊂ Xm(2) for some m(2) > m(1) (Proposition 6.1.2 (3)). Xm(1) f1 |

Xm(2)

⊂ j1 |

f1 ϕ

emb

Yn(1)

X∞

⊂

j1 |

⊂

j1

Y

Since ϕ(f1 ϕ)−1 (U1 ) = f1−1 (U1 ), it follows that j1 f1 |f1−1 (Yn(1) ) = ϕj1 f1 ϕ|f1−1 (Yn(1) ) f −1 (U1 ) ϕ|f1−1 (Yn(1) ) = id, 1

where f1−1 (Yn(1) ) is a Z-set in X and j1 f1 |f1−1 (Yn(1) ) is a Z-embedding into X. We can apply the Z-Set Unknotting Theorem 2.9.7 for 2f -manifolds (resp. 2Q manifolds) (cf. Theorem 5.1.10) to obtain a homeomorphism h1 : X → X such that h1 |f1−1 (Yn(1) ) = j1 f1 |f1−1 (Yn(1) ) and h1 f −1 (st U1 ) id. Thus, we have a 1

homeomorphism g1 = h1 f1−1 : Y → X such that

g1 |Yn(1) = h1 f1−1 |Yn(1) = j1 |Yn(1) and g1 f −1 (st U1 ) f1−1 . 1

Then, g1 (Yn(1) ) = j1 (Yn(1) ) ⊂ Xm(2) . Because f1 (Xm(1) ) ⊂ Yn(1) , we have g1 f1 |Xm(1) = j1 f1 |Xm(1) = id. Xm(1) f1 |

Yn(1)

⊂

Xm(2)

⊂

X f1

g1 |

⊂

Y

g1

6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds

574

Moreover, it follows that d(g1 , f1−1 )  mesh f1−1 (st U1 ) < 2−2 and ρ(f1 , g1−1 ) = ρ(f1 g1 , id)  mesh st U1 < 2−2 . Next, choose V1 ∈ cov(X) so that mesh st V1 < 2−2 and mesh g1−1 (st V1 ) < 2−2 . In the same way as the previous paragraph, we can apply Lemma 6.14.11 to obtain a V1 -homotopy inverse k1 of g1 ψ such that k1 g1 |Yn(1) = k1 g1 ψ|Yn(1) = id and k1 |Xm(2) is an embedding. Then, k1 (Xm(2) ) ⊂ Yn(2) for some n(2) > n(1). ⊂

Xm(2) g1 |

Yn(1)

X emb

⊂

Yn(2)

k1

g1 ψ

k1 |

k1 |

⊂

Y∞

By the same argument as the previous paragraph, we can apply the Z-Set Unknotting Theorem 2.9.7 to obtain a homeomorphism h2 : Y → Y such that h2 |g1−1 (Xm(2) ) = ψk1 g1 |g1−1 (Xm(2) ) and h2 g −1 (st V1 ) id. Then, f2 = h1 g1−1 : 1 X → Y is a homeomorphism and f2 |Xm(2) = h1 g1−1 |Xm(2) = k1 |Xm(2) and f2 g −1 (st V1 ) g1−1 . 1

Then, f2 (Xm(2) ) = k1 (Xm(2) ) ⊂ Yn(2) and f2 g1 |Yn(1) = k1 g1 |Yn(1) = id. ⊂

Xm(2) g1 |

Yn(1)

X g1

f2 | ⊂

Yn(2)

⊂

f2

Y

Moreover, it follows that ρ(f2 , g1−1 )  mesh g1−1 (st V1 ) < 2−2 and d(g1 , f2−1 ) = d(g1 f2 , id)  mesh st V1 < 2−2 . Therefore, we have ρ(f1 , f2 )  ρ(f1 , g1−1 ) + ρ(f2 , g1−1 ) < 2−2 + 2−2 = 2−1 .

6.14 Bi-topological Infinite-Dimensional Manifolds

575

Now, in the construction of g1 , replacing f1 , m(1), n(1) with f2 , m(2), n(2), and choosing U2 ∈ cov(Y ) instead of U1 so that mesh st U2 < 2−3 and mesh f2−1 (st U2 ) < 2−3 , we can obtain a homeomorphism g2 : Y → X and m(3) > m(2) such that g2 f2 |Xm(2) = id, g2 (Yn(2) ) ⊂ Xm(3) , d(g2 , f2−1 ) < 2−3 , and ρ(f2 , g2−1 ) < 2−3 . Then, it follows that we have d(g1 , g2 )  d(g1 , f2−1 ) + d(g2 , f2−1 ) < 2−2 + 2−3 < 2−1 . By repeating these constructions, we can obtain homeomorphisms fi : X → Y , gi : Y → X, i ∈ N, and increasing sequences m(1) < m(2) < · · · , n(1) < n(2) < · · · such that fi (Xm(i) ) ⊂ Yn(i) , gi (Yn(i) ) ⊂ Xm(i+1) , gi fi |Xm(i) = id, fi+1 gi |Yn(i) = id, d(gi , fi−1 ) < 2−(i+1) , ρ(fi , gi−1 ) < 2−(i+1) , −1 ρ(fi+1 , gi−1 ) < 2−(i+1) , and d(gi , fi+1 ) < 2−(i+1) .

Xm(1) f1 |

Yn(1)

⊂ g1 | ⊂

Xm(2) f2 |

Yn(2)

⊂ g2 | ⊂

Xm(3)

⊂

···

f3 |

Yn(3)

X gi , i∈N

fi ⊂

···

Y

Observe that fi+1 |Xm(i) = fi+1 gi fi |Xm(i) = (fi+1 gi |Yn(i) )fi |Xm(i) = fi |Xm(i) and gi+1 |Yn(i) = gi+1 fi+1 gi |Yn(i) = (gi+1 fi+1 |Xm(i+1) )gi |Yn(i) = gi |Yn(i) . Hence, we can define f∞ : X → Y and g∞ : Y → X by f∞ |Xm(i) = fi |Xm(i) and g∞ |Yn(i) = gi |Yn(i) for each i ∈ N. Due to Proposition 6.1.1, f∞ : X∞ = −1 = g . Moreover, lim Xi → Y∞ = lim Yi is a homeomorphism and f∞ ∞ − → − → ρ(fi , fi+1 )  ρ(fi , gi−1 ) + ρ(fi+1 , gi−1 ) < 2−(i+1) + 2−(i+1) = 2−i and −1 −1 ) + d(gi+1 , fi+1 ) < 2−(i+1) + 2−(i+2) < 2−i . d(gi , gi+1 )  d(gi , fi+1

576

6 Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds

Then, it follows that ρ(fn , f∞ ) 

 in

d(gn , g∞ ) 

ρ(fi , fi+1 )
0, each k-simplex σ can be written as σ = vτ , where v ∈ σ (0) and τ is the opposite face of σ to v, i.e., τ (0) = σ (0) \ {v}. Then, as is easily observed, Lk(σ, K) = Lk(τ, Lk(v, K)). Due to Proposition A.1.1 and Corollary A.1.7, (c) implies that Lk(v, K) is a combinatorial (n − 1)-manifold. Since τ is an (k − 1)-simplex, it follows from the inductive assumption that Lk(σ, K) = Lk(τ, Lk(v, K)) ∼ = F (∂n−k ) or F (n−k−1 ). Hence, (c) ⇒ (d) is valid for n. Thus, we have the implication (c) ⇒ (d) by induction. ! " As a corollary, we have the following: Corollary A.1.11 Let K be a combinatorial n-manifold. (1) Every σ ∈ K is a face of an n-simplex of K. Equivalently, every k-simplex σ ∈ K is joinable to some (n − k − 1)-simplex of K in K. (2) Each (n − 1)-simplex σ ∈ K is a common face of distinct two n-simplexes of K or a face of exactly one n-simplex of K. Equivalently, each (n − 1)-simplex σ ∈ K is joinable to one or two vertices in K. 

590

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PL n-Manifolds and Combinatorial n-Manifolds

A.2 The Boundary of a Combinatorial n-Manifold Every PL n-manifold M is an n-manifold. The interior Int M and the boundary ∂M of M can be defined similarly to those of an n-manifold. Actually, x ∈ M is called an interior point of M if it has a neighborhood N with a PL homeomorphism h : N → In such that h(x) ∈ (0, 1)n . The set Int M of all interior points of M is called the interior of M, which is open in M. We call x ∈ M a boundary point of M if it is not an interior point, that is, it has a neighborhood N with a PL homeomorphic h : N → In such that h(x) ∈ ∂In . We define the boundary ∂M of M as the set of boundary points. A PL n-manifold M with ∂M = ∅ is called a PL n-manifold without boundary. A PL n-manifold without boundary is called a closed PL n-manifold if it is compact. For a combinatorial n-manifold K, we define    ∂K = σ ∈ K  Lk(σ, K) ∼ = F (n−dim σ −1 ) , which is called the boundary of K. For every n-simplex σ ∈ K, Lk(σ, K) = ∅ because σ is principal. Hence, ∂K ⊂ K (n−1) . In fact, dim ∂K = n − 1 if ∂K = ∅. ∼ F (n−1 ), it Indeed, assume ∂K = ∅. Then, we have v ∈ (∂K)(0) . Since Lk(v, K) = is easily seen that Lk(v, K) contains an (n − 2)-simplex σ which is a face of only one (n − 1)-simplex of Lk(v, K). Consider the (n − 1)-simplex vσ ∈ K. Then, it follows that Lk(vσ, K) = Lk(σ, Lk(v, K)) ∼ = F (0 ) = F (n−(n−1)−1 ), which means that vσ ∈ ∂K. Therefore, dim ∂K = n − 1.

Due to Corollary A.1.11(2), the following fact is trivial: Fact For a combinatorial n-manifold K, an (n − 1)-simplex σ ∈ K belongs to the boundary ∂K if and only if σ is a face of only one n-simplex of K, equivalently σ is joinable to only one vertex in K. We will show that ∂|K| = |∂K| for every combinatorial n-manifold. For the simplest case, the following holds: Lemma A.2.1 For every n ∈ N, ∂F (n ) = F (∂n ) and ∂F (∂n ) = ∅. Consequently, |∂F (n )| = ∂n . Proof Obviously, Lk(n , F (n )) = ∅, which means that n ∈ ∂F (n ). For each σ ∈ F (∂n ), let τ be the opposite face of n to σ , that is, n = σ τ . Then, Lk(σ, F (n )) = F (τ ), which means σ ∈ ∂F (n ). Therefore, ∂F (n ) = F (∂n ). Additionally, Lk(σ, F (∂n )) = F (∂τ ), which means σ ∈ ∂F (∂n ). Consequently, ∂F (∂n ) = ∅. " !

A PL n-Manifolds and Combinatorial n-Manifolds

591

A combinatorial n-manifold K with ∂K = ∅ is called a combinatorial n-manifold without boundary. Concerning the boundary of a combinatorial nmanifold, we have the following: Proposition A.2.2 For a combinatorial n-manifold K with ∂K = ∅, ∂K is a combinatorial (n − 1)-manifold without boundary. Proof First, we have to show that ∂K is a subcomplex of K, that is, if σ ∈ ∂K and τ < σ , then τ ∈ ∂K. For τ < σ ∈ K, it suffices to prove that τ ∈ ∂K implies σ ∈ ∂K, that is, Lk(τ, K) ∼ = F (∂n−dim σ ) = F (∂n−dim τ ) implies Lk(σ, K) ∼ (cf. Theorem A.1.10 (c)). We may show the case when dim σ = dim τ + 1, that is, σ = vτ , v ∈ K (0). Then, by induction, the general case can be obtained. When σ = vτ , we have Lk(σ, K) = Lk(v, Lk(τ, K)), where Lk(τ, K) ∼ = F (∂n−dim τ ) by the assumption. Due to Proposition A.1.1 and Lemma A.2.1, F (∂n−dim τ ) is a combinatorial (n − dim τ − 1)-manifold without boundary. Hence, Lk(σ, K) = Lk(v, Lk(τ, K)) ∼ = F (∂n−dim τ −1 ) = F (∂n−dim σ ). Next, we will show that ∂K is a combinatorial (n − 1)-manifold without boundary. For each k-simplex σ ∈ ∂K, obviously Lk(σ, ∂K) ⊂ Lk(σ, K). Due to Lemma A.2.1, Lk(σ, K) ∼ = F (n−k−1 ) is a combinatorial (n − k − 1)-manifold with the following boundary: ∂ Lk(σ, K) ∼ = ∂F (n−k−1 ) = F (∂n−k−1 ). For an (n − k − 2)-simplex τ ∈ Lk(σ, K), τ ∈ ∂ Lk(σ, K) if and only if τ is joinable to only one vertex v ∈ Lk(σ, K)(0) in Lk(σ, K). Then, τ σ ∈ K is an (n − 1)-simplex which is joinable to v in K but not joinable to any other vertex of K in K, which means that τ σ ∈ ∂K. Thus, for an (n − k − 2)-simplex τ ∈ Lk(σ, K), τ ∈ ∂ Lk(σ, K) if and only if τ ∈ Lk(σ, ∂K). For i < n − k − 2, every i-simplex of Lk(σ, K) is a face of an (n − k − 2)-simplex of Lk(σ, K) because Lk(σ, K) is a combinatorial (n − k − 1)-manifold. Therefore, Lk(σ, ∂K) = ∂ Lk(σ, K) ∼ = F (∂n−k−1 ). Consequently, ∂K is a combinatorial (n − 1)-manifold without boundary.

" !

For a combinatorial n-manifold K, |K| is a PL n-manifold (Theorem A.1.6). Concerning boundaries ∂K and ∂|K|, we have the following: Theorem A.2.3 For every combinatorial n-manifold K, |∂K| = ∂|K|. Proof For x ∈ |K|, we may show that x ∈ |∂K| if and only if x ∈ ∂|K|. (⊂): For each x ∈ |∂K|, cK (x) = c∂K (x) ∈ ∂K, which means Lk(cK (x), K) ∼ = F (n−k−1 ), k = dim cK (x).

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Then, by Lemma A.1.8, it follows that 

   St(cK (x), K), F (cK (x)) = F (cK (x)) ∗ Lk(cK (x), K), F (cK (x))   ∼ = F (k ) ∗ F (n−k−1 ), F (k ) .

Moreover, due to Proposition A.1.9(i),   F (k ) ∗ F (n−k−1 ), F (∂k ) ∗ F (n−k−1) ∪ F (k ) ∗ F (∂n−k−1 )   ∼ = F (n ), F (∂n ) . Hence, there is a PL homeomorphism h : | St(cK (x), K)| → n such that h(x) ∈ h(cK (x)) ⊂ ∂n . Since | St(cK (x), K)| is a neighborhood of x in |K|, we have x ∈ ∂|K|. (⊃): For each x ∈ |K| \ |∂K|, cK (x) ∈ ∂K, that is, Lk(cK (x), K) ∼ = F (∂n−k ), k = dim cK (x). By Lemma A.1.8, we have a PL homeomorphism       f :  St(cK (x), K) = F (cK (x)) ∗ Lk(cK (x), K) → F (k ) ∗ F (∂n−k ) such that f (cK (x)) = k and f (∂cK (x)) = ∂k . Since f (x) ∈ f (rint cK (x)) = rint k , it follows that f (x) ∈ |F (∂k ) ∗ F (∂n−k )|. By the way, due to Proposition A.1.9(ii), 

   F (k ) ∗ F (∂n−k ), F (∂k ) ∗ F (∂n−k ) ∼ = F (n ), F (∂n )   ∼ = F (In ), F (∂In ) .

Thus, we have a PL homeomorphism h : | St(cK (x), K)| → In such that h(x) ∈ h(rint cK (x)) = rint In = (0, 1)n . Since | St(cK (x), K)| is a neighborhood of x in |K|, we have x ∈ Int |K| = |K| \ ∂|K|. " !

A.3 Regular Neighborhoods in PL n-Manifolds In this section, we will prove Proposition 6.10.6. Let K be a simplicial complex and L a subcomplex of K. First, we recall the ε-neighborhood of L in K. In Sect. 1.9, we defined a simplicial map ϕL : K → I = {0, 1, I} such that ϕL (L(0) ) = 0 and ϕL (K (0) \ L(0) ) = 1. Due to Proposition 1.9.1, L is full in K if and only if |L| = ϕL−1 (0). Then, for 0 < ε < 1, we define the following cell complexes:

A PL n-Manifolds and Combinatorial n-Manifolds

593

   Nε (L, K) = σ ∩ ϕL−1 (τ )  σ ∈ K[|L|], τ  [0, ε] ,    Cε (L, K) = σ ∩ ϕL−1 (τ )  σ ∈ K \ L, τ  [ε, 1] , KLε = Nε (L, K) ∪ Cε (L, K), where Nε (L, K) is the ε-neighborhood of L in K, |Nε (L, K)| = ϕL−1 ([0, ε]), |Cε (L, K)| = ϕL−1 ([ε, 1]), and bd |Nε (L, K)| = ϕL−1 (ε). Applying Proposition 1.6.8, we can obtain a derived subdivision K  relative to L ∪ C(L, K) such that N(L, K  )  Nε (L, K) (cf. p. 47). Theorem A.3.1 Let M be a PL n-manifold and P a subpolyhedron of M. Then, a regular neighborhood N of P in M is a PL n-manifold with boundary. If P ⊂ Int M = M \ ∂M, then ∂N = bdM N. Proof Let K be a triangulation of M (i.e., M = |K|) such that P = |L| for some full subpolyhedron L of K and N = N(L, K  ) for some derived subdivision K  of K relative to L ∪ C(L, K). Then, K is a combinatorial n-manifold by Theorem A.1.6. By virtue of the uniqueness of regular neighborhoods (Theorem 1.9.5), we may assume that N(L, K  )  Nε (L, K), 0 < ε < 1. As is easily observed, N = |Nε (L, K)| =



OK (v) ∩ |Nε (L, K)|,

v∈L(0)

where OK (v) is the open star at v. For each x ∈ N, choose v ∈ L(0) so that x ∈ OK (v). Then, | St(v, K)| ∩ N is a neighborhood of x in N and          St(v, K) ∩ N =  St(v, K) ∩ Nε (L, K) =  St(v, K ε ). L It should be remarked that KLε is not a simplicial subdivision of K nor | St(v, K  )| = | St(v, KLε )|. However, due to Theorem 1.6.1, giving an order on (KLε )(0) so that C (0) has the maximum vC for each C ∈ KLε , we can define a simplicial subdivision of KLε with the same set of vertices as KLε . Giving such an order on (KLε )(0) satisfying the additional condition that v is the maximum of C (0) for every C ∈ KLε [v] (⊂ St(v, KLε )), we apply Theorem 1.6.1 to define a simplicial subdivision K v  KLε . Then, it follows that | St(v, K v )| = | St(v, KLε )|. Since St(v, K v ) ∼ = St(v, K) ∼ = ε n n F ( ) by Lemma A.1.3, | St(v, KL )| is PL homeomorphic to I . Therefore, N = |N(L, K  )| is a PL n-manifold. In the above, assume that P ⊂ Int M. For each v ∈ L(0) (⊂ P ), it follows from Theorem A.1.10 that OK (v) = | St(v, K)| \ | Lk(v, K)| ≈ n \ ∂n = rint n .

594

A

PL n-Manifolds and Combinatorial n-Manifolds

 Then, we have OK (v) ⊂ Int M. Hence, N ⊂ v∈L(0) OK (v) ⊂ Int M. Each x ∈ N has a polyhedral neighborhood B in M with a PL homeomorphism h : B → In such that h(x) ∈ (0, 1)n . If x ∈ intM N, then the above B can be taken so small that B ⊂ intM N, which implies x ∈ Int N. Conversely, if x ∈ Int N, then there is a polyhedral neighborhood D in N with a PL homeomorphism g : D → In such that g(x) ∈ (0, 1)n , where D can be taken so small that D ⊂ B. By virtue of the Invariance of Domain (Corollary 1.12.5), fg −1 ((0, 1)n ) is open in (0, 1)n , which implies that D is open in B. Therefore, x ∈ intM N. Thus, we have intM N = Int N, which means ∂N = N \ Int N ⊂ N \ intM N = bdM N.

" !

The setting at the beginning of the above proof shows that Theorem A.3.1 is a restatement of Proposition 6.10.6. Thus, we have Proposition 6.10.6.

A.4 PL Embedding Approximation Theorem In this section, we give the PL manifold version of Theorem 1.8.11, in which Q×In is replaced with a PL n-manifold. Namely, we prove the following PL EMBEDDING APPROXIMATION THEOREM: Theorem A.4.1 (PL EMBEDDING APPROXIMATION FOR PL MANIFOLDS) Let P be a compact polyhedron with a subpolyhedron P0 ⊂ P and let M be a PL n-manifold. Suppose that n  2 dim P + 1 and f : P → M is a map such that f |P0 is a PL embedding. Then, for each open cover U of M, there exists a PL embedding h : P → M such that h|P0 = f |P0 and h U f rel. P0 .  Proof Take subpolyhedra C1 , . . . , Ck ⊂ M so that f (P ) ⊂ ki=1 int Ci and each Ci is PL homeomorphic to In . Let ϕi : In → Ci , i = 1, . . . , k, be PL embeddings. There is a sequence of subpolyhedra P0 ⊂ P1 ⊂ · · · ⊂ Pk = P such that f (cl(Pi \ Pi−1 )) ⊂ int Ci for each i = 1, . . . , k. Due to Proposition 1.3.22 (1), M has an admissible metric d such that {Bd (x, 1) | x ∈ X} ≺ U. By induction, we will construct maps fi : P → M, i = 1, . . . , k, such that fi |Pi is a PL embedding, fi |Pi−1 = fi−1 |Pi−1 , fi 1/ k fi−1 rel. Pi−1 , and fi (cl(Pj \ Pj −1 )) ⊂ int Cj for each j > i, where f0 = f . Suppose fi−1 have been constructed. Choose ε > 0 so that ε < 1/k and N(fi−1 (cl(Pj \ Pj −1 )), ε) ⊂ int Cj for each j > i. By the uniform continuity of ϕi , we have δ > 0 such that x, y ∈ In , x − y < δ ⇒ d(ϕi (x), ϕi (y)) < ε.

A PL n-Manifolds and Combinatorial n-Manifolds

595

Since fi−1 (Pi−1 ) is a polyhedron and fi−1 (Pi−1 ) \ int Ci is its subpolyhedron, −1 −1 cl(Pi \ fi−1 (Ci )) = Pi−1 \ int fi−1 (Ci ) is a subpolyhedron of P . It follows that −1 −1 (Ci ) is its subpolyhedron. Pi ∩fi−1 (Ci ) is also a subpolyhedron of P and Pi−1 ∩fi−1 Consider the following map: −1 −1 ϕi−1 f |Pi ∩ fi−1 (Ci ) : Pi ∩ fi−1 (Ci ) → In , −1 (Ci ) is a PL embedding. Applying the PL Embedding where ϕi−1 fi−1 |Pi−1 ∩ fi−1 Approximation Theorem 1.8.11 (the case Q is a singleton), we can obtain a PL −1 embedding g : Pi ∩ fi−1 (Ci ) → In such that −1 −1 g|Pi−1 ∩ fi−1 (Ci ) = ϕi−1 fi−1 |Pi−1 ∩ fi−1 (Ci ) and −1 g δ ϕi−1 fi−1 rel. Pi−1 ∩ fi−1 (Ci ).

Then, a PL embedding f  : Pi → M can be defined by f  |Pi−1 = fi−1 |Pi−1 −1 and f  |Pi ∩ fi−1 (Ci ) = ϕi g. Observe that f  ε fi−1 |Pi rel. Pi−1 . Applying the Homotopy Extension Theorem 1.5.7, we can extend f  to a map fi : P → M such that fi ε fi−1 rel. Pi−1 . Then, fi |Pi = f  is a PL embedding, fi |Pi−1 = fi−1 |Pi−1 , fi 1/ k fi−1 rel. Pi−1 , and fi (cl(Pj \ Pj −1 )) ⊂ int Cj for each j > i. Refer to Fig. A.1. The map h = fk : P = Pk → M is the desired PL embedding. Indeed, h|P0 = fk |P0 = f0 |P0 = f |P0 and h = fk 1/ k fk−1 1/ k · · · 1/ k f0 = f rel. P0 , " !

which implies that h U f rel. P0 .

M P

fi −1 −1 (C i )

fi −1 (P ) Ci

Pi fi −1 P i −1

P i −1

fi −1 −1 (C i ) i

g In

−1 Fig. A.1 Pi \ int fi−1 (Ci ) and fi−1 (Pi−1 ) \ int Ci

fi −1 (P i −1 )

Epilogue

In 1964, answering Klee’s question, R.D. Anderson showed in [3] that the product of a triod, T , and the Hilbert cube Q is homeomorphic to Q. This is the starting point of Infinite-Dimensional Topology.1 And in 1966, answering the longstanding Fréchet’s question, he proved in [4] that Hilbert space 2 is homeomorphic to RN . Since then, Infinite-Dimensional Topology has been developed rapidly and remarkable results have been obtained in a short period (cf. Anderson’s essay [9]). The first book [(1)] on Infinite-Dimensional Topology was prepared by C. Bessaga and A. Pełczynski in 1975. From that book, one could learn basic properties of Q and fundamental results on manifolds modeled on infinite-dimensional metric linear spaces. For instance, Keller’s Theorem, the topological homogeneity of Q, the Classification and Open Embedding Theorems for Hilbert manifolds are covered in that book. In the same year, T.A. Chapman gave us his lecture notes [(2)] on Hilbert cube manifolds, by which we could learn fundamental results on Q-manifolds containing the Classification and the Triangulation Theorems for Q-manifolds. In 1980 and 1981, two celebrated papers [143, 145] by H. Toru´nczyk were published, in which Hilbert cube manifolds and Hilbert manifolds are characterized topologically. With these results, the theories of Hilbert cube manifolds and Hilbert manifolds were almost completed. However, Toru´nczyk’s characterizations can be expected to apply to various spaces appearing in many branches of Mathematics. On the other hand, manifolds modeled on incomplete metric linear spaces have also been studied and given characterizations. Thus, in order to learn those infinitedimensional manifolds, some self-contained textbook has been long-expected.2 In 1989, J. van Mill prepared the excellent self-contained textbook [(3)], which saves us a lot of time in learning Toru´nczyk’s characterization of compact Qmanifolds. However, it is not sufficient to cover the non-compact case. In his

1 Cf.

Footnote 4 in Preface (p. vii). the history of Infinite-Dimensional Topology, refer to the article of T. Koetsier and J. van Mill [96, Sect. 4]. 2 For

© Springer Nature Singapore Pte Ltd. 2020 K. Sakai, Topology of Infinite-Dimensional Manifolds, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-15-7575-4

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Epilogue

paper [148], J.J. Walsh provided an alternative proof of the characterization of Q-manifolds but that paper is not self-contained. The paper [30] coauthored by P. Bowers, M. Bestvina, J. Mogilski, and J.J. Walsh provided an alternative proof of the characterization of 2 -manifolds but is not so easy for a graduate student to read. Concerning Toru´nczyk’s characterization of non-separable Hilbert manifolds, we have to read his paper together with references. Since van Mill’s book was published, over thirty years have passed but there are no books containing these results. In 1996, two books [(4)] and [(5)] were published. The second chapter of Chigogidze’s book [(4)] is devoted to infinite-dimensional manifolds but it is a survey article and not suitable for graduate students to study them. The book [(5)] by Banakh, Radul, and Zarichnyi is a good reference for absorbing sets in infinite-dimensional manifolds but the reader is required to have background knowledge of infinite-dimensional manifolds, that is, fundamental results containing characterization of 2 - and Q-manifolds. Thus, for graduate students, it is hard to study infinite-dimensional manifolds. This book was designed for graduate students to study fundamental results on infinite-dimensional manifolds and related topics. The origin of this project goes back to the time when van Mill’s book published. In every year, for a graduate course in the University of Tsukuba, I had given lectures on various topics, which were (1) basic properties of paracompact spaces, simplicial complexes, and dimensions, (2) basic properties of ANRs, Michael’s Selection Theorem, and Toru´nczyk’s Factor Theorem, and (3) fundamental results on infinite-dimensional manifolds and the characterization of Hilbert manifolds. In 1991, combining these notes, I had written up my personal lecture notes entitled Lectures on ANR’s and Infinite-Dimensional Manifolds consisting of three parts: Part I: Paracompact Spaces, Simplicial Complexes, and Dimensions, Part II: Absolute Neighborhood Retracts, Part III: Infinite-Dimensional Manifolds, where Q-manifolds were not contained in Part III because students can refer to Chapman’s lecture notes [(2)] and van Mill’s book [(3)]. Since then, adding new sections and rearranging contents, Parts I and II were finally divided into five chapters: Chap. 1: Chap. 2: Chap. 3: Chap. 4: Chap. 5:

Metrization and Paracompact Spaces, Topology of Linear Spaces and Convex Sets, Simplicial Complexes and Polyhedra, Dimensions of Spaces, Retracts and Extensors.

Each of these chapters had been used as not only a text for a class of the graduate course but also a text of seminars of graduate students or advanced college seniors. In 2007, these chapters were almost completed and the title was changed to Lectures on Topology toward Infinite-Dimensional Spaces.

Epilogue

599

At that time, the lecture notes had been growing into a thick book because the first five chapters covered not only almost all material to learn about infinite-dimensional manifolds but also basic materials that will be necessary to study related topics. Besides, there were many materials that should be contained in the part on infinitedimensional manifolds. With the joint work with Yaguchi [129] and Mine’s work [108], Bestvina–Mogilski’s results in [29] on absorbing sets have been generalized to the non-separable case, which should also be added to the book. Since van Mill’s book is not enough for the characterization of non-compact Q-manifolds, that should be included. Thus, it seemed better to divide the book into two volumes, the first one treating subjects corresponding to the above five chapters and the second volume treating infinite-dimensional manifolds directly. Actually, the contents of the first volume looked to be useful for students to study any other subject in Topology. By the way, in 1988, A.N. Dranishnikov [56] solved the Alexandroff Problem affirmatively, that is, he showed the existence of an infinite-dimensional compactum with finite cohomological dimension. That means that the CE Problem was answered affirmatively, that is, there exists a cell-like map of a finite-dimensional compactum onto an infinite-dimensional compactum. In 1994, using that result, R. Cauty [38] constructed a linear metric space which is not an AR. To make the book more publishable, the chapter on cell-like maps was added as the last chapter, in which these distinguished results were contained. But, unfortunately this part is not self-contained. Required definitions and basic results on homotopy groups were added in the chapter on simplicial complexes. Since cohomological dimension was contained in the last chapter, it became possible to give a proof of dim X × I = dim X + 1 for every metrizable space X. Thus, the first volume became more useful for students in other branches in Topology and publishable independently from the second volume. Finally, the title of this first volume was changed to Geometric Aspects of General Topology and was published by Springer in 2013. That book is now a textbook on Dimension Theory and ANR Theory, and emphasizes the relation between them. That is also a good textbook on non-locally finite simplicial complexes, and a useful reference. After publication of the first book, the remaining part, the second volume, needed to be reformed as a book independent from the first, and was entitled Topology of Infinite-Dimensional Manifolds. At that time, writing the chapter on absorbing sets was in progress. Concerning the Triangulation Theorem for Q-manifolds, I was considering that it is not necessary to add it because Chapman’s lecture notes can be referred to. I had not yet started to write the chapter on R∞ - and Q∞ -manifolds. Moreover, preliminaries and background results needed to be added. Lots of work was required. Because of personal reasons, I could not continue writing for a couple of years. In the autumn of 2015, I resumed writing. In the first half of the last chapter, basic properties of R∞ - and Q∞ -manifolds, was written in 2016. In 2017, changing my mind, I added Chapman’s proof of the Triangulation Theorem for Q-manifolds based on his lecture notes together with necessary preliminaries. Except for some algebraic preliminary result, each proof was written in detail as much as possible. Then, the proof of Borsuk’s conjecture and the topological invariance of Whitehead

600

Epilogue

torsion was included. By reforming the chapter on absorbing sets, West’s absorption bases were added in 2018. Finally, infinite-dimensional combinatorial manifolds were added to the last chapter. Then, I was able to fix gaps and to give details of the original proof of the Simplicial Approximation Theorem for PL Embeddings. While my colleagues and graduate students were expecting earlier publication of this book, it has taken a long time to get this book ready. Over 30 years have passed since I wrote up the first version of the lecture notes. Time flies! Tsukuba, Japan May 2020

Katsuro Sakai

Bibliography

Books and Texts Infinite-Dimensional Topology (1) C. Bessaga and A. Pełczy´nski, Selected Topics in Infinite-Dimensional Topology, MM 58 (Polish Scientific, Warsaw, 1975) (2) T.A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28, (Amer. Math. Soc., Providence, 1975) (3) J. van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library 43 (Elsevier Sci. Publ. B.V., Amsterdam, 1989) (4) A. Chigogidze, Inverse Spectra, North-Holland Math. Library 53 (Elsevier Sci. Publ. B.V., Amsterdam, 1996) (5) T. Banakh, T. Radul and M. Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds, Math. Studies, Monog. Ser. 1 (VNTL Publishers, Lviv, 1996) (6) J. van Mill, The Infinite-Dimensional Topology of Function Spaces, NorthHolland Math. Library 64 (Elsevier Sci. Publ. B.V., Amsterdam, 2002) General Topology (Containing Dimension and ANR Theories) Topological Spaces (7) J.R. Munkres, Topology, 2nd edn. (Prentice Hall, Upper Saddle River, 2000) (8) J. Dugundji, Topology (Allyn and Bacon, Boston, 1966) (9) R. Engelking, General Topology, Revised and Complete Edition, SSPM 6 (Heldermann Verlag, Berlin, 1989) Dimension Theory (10) W. Hurewicz, H. Wallman, Dimension Theory (Princeton University Press, Princeton, 1941) (11) K. Nagami, Dimension Theory (Academic Press, New York, 1970)

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(12) R. Engelking, Theory of Dimensions, Finite and Infinite SSPM 10 (Heldermann Verlag, Berlin, 1995) ANR Theory (13) K. Borsuk, Theory of Retracts, MM 44 (Polish Scientific, Warsaw, 1967) (14) S.-T. Hu, Theory of Retracts (Wayne State University Press, Detroit, 1965) Dimension and ANR Theories (15) K. Sakai, Geometric Aspects of General Topology, SMM (Springer, Tokyo, 2013) Descriptive Set Theory (16) A.S. Kechris, Classical Descriptive Set Theory, GTM 156 (Springer, New York, 1995) Piecewise-Linear Topology (Combinatorial Topology) (17) C.P. Rourke, B.J. Sanderson, Introduction to Piecewise-Linear Topology, Springer Study Edition (Springer, Berlin, 1972, 1982) (18) J.F.P. Hudson, Piecewise-Linear Topology (W.A. Bebjamin, New York, 1969) (19) E.C. Zeeman, Seminar on Combinatorial Topology. Mimeographed Notes (Inst. Hautes Études Sci., Paris, 1963) (20) M.M. Cohen, A Course in Simple-Homotopy Theory, GTM 10 (Springer, New York, 1973) Topological Linear Spaces (21) G. Köthe, Topological Vector Spaces I, English edition, GTM 159 (Springer, New York, 1969) Algebraic Topology (22) A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002) (23) W.S. Massey, A Basic Course in Algebraic Topology, GTM 127 (Springer, New York, 1991) (24) G.W. Whitehead, Homotopy Theory (MIT Press, Cambridge, 1966)

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Index

Symbols (K, K0 ) ≡ (L, L0 ), 42 (X, X1 , . . . , Xn ) ≈ (Y, Y1 , . . . , Yn ), 2 (X, X1 , . . . , Xn )  (Y, Y1 , . . . , Yn ), 7 (X, x0 ) ≈ (Y, y0 ), 2 (X, x0 )  (Y, y0 ), 7 C (0) , 28 Cε (L, K), 47 Cx , 22 D < C, 28 D  C, 28 K ∗ L, 29 K∼ = L, 43 K ≡ L, 42 K & L, 294 K &e L, 294 K   K, 33 KLε , 47 L ' K, 294 L e' K, 294 Nε (L, K), 47, 326 X ≈ Y, 2 X & 0, 290 X & Y , 290 X &e Y , 290 X  Y , 7, 17 Y ' X, 290 Y e' X, 290 A, 21 A ≺ B, 4 A ∧ B, 4 A[Y ], 4 Acl , 4 A|Y , 4 ⊕C, 377

σ < τ , 28 σ  τ , 28 σ [t], 221 σ τ , 28 γ ∈ x(γ ), 12 τ C, 377 1 L, 94 f 1 Li , 94 f

1 L, 94 1 Li , 94 f  0, 7 f  g, 6 f  g rel. A, 7 f U g, 8 f ε g, 8 v0 ∗ L, 29 A Aα , 459 Aα (τ ), 459 Absolute extensor (AE), 65, 482 Absolutely Fσ , 463 Fσ space, 459, 464 Fσ δ space, 459 Gδ , 19, 461 Gδ space, 459 Gδσ , 461 Gδσ space, 459 Borel space, 459 closed, 459 locally closed, 462, 463 open, 459

© Springer Nature Singapore Pte Ltd. 2020 K. Sakai, Topology of Infinite-Dimensional Manifolds, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-15-7575-4

609

610 Absolute neighborhood extensor (ANE), 65, 482 Absolute neighborhood retract (ANR), 65 Absolute retract (AR), 65 Absorbing set for C, 447 Absorptively C-universal, 439 Absorptively universal for C, 439 Abstract complex, 57 Additive class, 377 Adjunction space, 5 Admissible F -norm, 26 Admissible subdivision, 38 Affine function, 23 Affine hull see flat hull, 21 Affinely independent, 20 Affine set see flat, 21 Alexander horned sphere, 201 Alexander Trick, 267 Alexander Trick, Q-factored, 341 Ambient isotopy, 91 Ambiently invertibly isotopic, 91 Ambiently isotopic, 91 ANR Factor Theorem for manifold pairs, 422 Approximately soft map, 76 Arc, 6 Arcwise connected, 6 Arens–Eells Embedding Theorem, 65

B B(Q), the pseudo-boundary of Q, 84, 353, 356 Bd (x, ε), B(x, ε), the ε-neighborhood of x in X, 3 Bd (x, ε), B(x, ε), the closed ε-neighborhood of x in X, 3 BX , the unit closed ball, 11 Bn , the unit closed n-ball, 3 Baire Category Theorem, 19 Baire space, 18, 206 Banach space, 10 Barycenter, 36 Barycentric coordinate, 38 Barycentric subdivision, 36 Base point, 2 bd A, bdX A, the boundary of A in X, 2 βvK , 38 Bi-collar, 135 Bi-collared, 135 Bing Shrinking Criterion, 153, 157 Bi-topological closed embedding, 568 Bi-topological embedding, 568 Bi-topological homeomorphism, 568 Bi-topological manifold, 568 Bi-topological map, 568

Index Bi-topological open embedding, 568 Bi-topological space, 567 Borel field, 459 Borel set, 459 Borsuk conjecture, 57, 289, 294, 348 Boundary of a combinatorial manifold, 590 Boundary of a manifold, 4 Boundary of a PL manifold, 590 Boundary point of a manifold, 4 Boundary point of a PL manifold, 590 Bounded weak-star topology, 485 C C, the complex plane, 2 c, the space of convergent sequences, 12 C((X, x0 ), (Y, y0 )), 6 C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )), 6 C(A, K), 501 C(ℵ0 ), 377 c(), 11 C(τ ), 377 C(X), 69 c(X), the cellularity of X, 2 C(X, Y ), 6 c0 , the space of null-sequences, 12 c0 (), 11 C-absorbing set, 447 C-absorption base, 384 C-absorption property, 378 C-absorptive, 378 CB (X), 15 Cf |A (X, Y ), 206 cK (x), 31 C o (X), 69, 127 Cp (X), 15 CP (X, Y ), 206 CPf |A (X, Y ), 206 Cu (X), 14 CU (X, Y ), 208 C-universal, 203, 204, 387 Cscd , 377, 426 C(n) , 377 Cfd , 377, 426 Cσ , 377, 408 Canonical map, 59 Cap, 354 Cap set, 354 Carrier, 31 CE equivalent, 284 Cell, 27 Cell complex, 29 Cell-like map, 76 Cell-like space (compactum), 76

Index Cell-like subset, 76 Cellularity, 2 Cellular map, 41 CE map, 76 Characterization of Q, 284 Characterization of 2 (), 247 Characterization of ANRs, Hanner, 70 Lefschetz, 69 Characterization of simple homotopy equivalences, 344 Characterization Theorem — for (Q∞ , QN f )-manifolds, 569 — for (2 () × 2f )-manifolds, 437 — for (2f () × 2 )-manifolds, 437 — for (2f () × 2Q )-manifolds, 437 — for (R∞ , RN f )-manifolds, 569 — for Q-manifolds, 284 — for Q∞ -manifolds, 495 — for 2 -manifolds, 245, 248 — for 2 ()-manifolds, 239 — for 2f -manifolds, 438 — for 2f ()-manifolds, 436 — for 2Q -manifolds, 438 — for R∞ -manifolds, 495 — for combinatorial ∞-manifolds, 542 — for combinatorial n-manifolds, 585, 588 — for PL ∞-manifolds, 535 Circled, 23 Cl A, clX A, the closure of A in X, 2 Classification Theorem — for (Q, Qf )-manifold pair, 371 — for (Q, rint Q)-manifold pair, 371 — for (Q∞ , QN f )-manifolds, 570 — for (2 , 2f )-manifold pair, 371 — for (2 , 2Q )-manifold pair, 371 — for (R∞ , RN f )-manifolds, 570 — for Q-manifolds, 284 — for R+ -stable Q-manifolds, 197 — for E-manifolds, 145 Closed collar, 135 Closed Embedding Approximation Theorem — for Q-manifolds, 120 — for E-manifolds, 119 Closed hereditary class, 377 Closed manifold, closed n-manifold, 4 Closed over a subset, 169 Closed PL n-manifold, 590 Collapse, 290 Collapsible, 290 Collar, 129

611 Collared, 129, 134 Collaring Theorem — for Q-manifolds, 191 — for E-manifolds, 140, 182 Collectionwise normal, 17 Combinatorial ∞-manifold, 530 Combinatorial ∞-manifold, 542, 566 Combinatorial n-manifold, 529, 581 Combinatorial n-manifold without boundary, 591 Combinatorial manifold infinite-dimensional, 530 n-dimensional, 529 Combinatorially equivalent, 43, 529 Combinatorial manifold, n-dimensional, 581 Comp(X), 287 Compact absorption property, cap, 354 Compact-open topology, 8, 260 Compactum, compacta, 2 Completely metrizable, characterization, 19 Component of locally compact metrizable space, 207 Cone, 53, 69, 106 Cone, metrizable, 69, 106, 109 Congruent, 42 Conjugate space of a normed linear space, 484 Cont(X), 287 Contiguous, 32, 60 Continuum, continua, 2 Contractible, 66 Contraction, 66 Convergent (infinite sum), 12 Convex hull, 21 Convex linear cell, 27 Convex linear cell complex, 29 Convex set, 21 Coordinate-switching pseudo-isotopy, 110, 117, 118 Core, 22 Countable-dimensional (c.d.), 64 Countable discrete cells property, 217 Countable discrete n-cells property, 217 Countable locally finite approximation property, 238, 430 Countable locally finite cells property, 217 Countable locally finite n-cells property, 217 Countable Sum Theorem, 63 Cov(X), 4 Cover, 4 Covering dimension, (dim), 61 Covering projection (map), 295 Covering space, 295 Cozero set, 18

612 Curtis–Schori–West Hyperspace Theorem, 287

D ∂C, 22 ∂M, 4, 590 d(x, A), the distance of x from A, 3 Decomposition Theorem, 63 Deficient set, 108, 114, 129, 131, 179 Deformable, 68 Deformation, 67 Deformation retract, 67 Deformation retraction, 68 ∞ , 482, 530 n , 3 X , 73 Dens X, the density of X, 2 Density, 2 Derived neighborhood, 47, 593 Derived subdivision, 36 Diagonal, 73 Diamd A, diam A, the diameter of A, 3 Dim, 21, 30, 61 Dimension — of an abstract complex, 58 — of a cell (simplicial) complex, 30 characterization of —, 63 — of a convex set, 21 — of a flat, 21 — of a space, 61 Direct limit, 480 Discrete, 15 δ-discrete, 86 uniformly, 86 Discrete τ -polyhedral sequence property, 427 Discrete approximation property, 247 Discrete cell-tower property, 247, 427 Discrete compact polyhedra property, 229, 235 Discrete family, 216 Discrete f.d. τ -polyhedra property, 229 Discrete f.d. separable polyhedra property, 229 Discrete self-copies property, 237 Disjoint cells property, 220 Disjoint n-cells property, 220 Distd (A, B), dist(A, B), the distance of A and B, 3 Dual space of a normed linear space, 484 Dugundji Extension Theorem, 65

E (E, F )-manifold pair, 353 E-manifold, 81 E ∗ (bw ∗ ), 485

Index E ∗ (w ∗ ), 484 Edwards Factor Theorem, 277 Eilenberg–Otto Characterization on dimension, 63 Elementary collapse, 290 Elementary expansion, 290 Elementary simplicial collapse, 294 Elementary simplicial expansion, 294 Emb(X, Y ), 204 Embedding, E-deficient , 119 Z-, 181 strong Z-, 181 Embedding Approximation Theorem, 65 Embedding Theorem, 65 Epimorphism, 299 ε-close, 8 ε-homotopic, 8 ε-homotopy, 8 ε-neighborhood, 47, 326 Equi-connected (EC), 73, 559 Equi-connecting map, 73 Equi-continuous, 260 Essential family, 63 Euclidean half n-space, 3 Euclidean space, 3 Euclidean n-space, 3 Evaluation, 9 Evenly covered, 295 Expand, 290 Extension, 6

F F (C), 29 F (∂C), 29 F (∞ ), 530 F -hereditary property, 20 F -norm, 26 F -normed linear space, 26 FQ\{0} , 84 FQ×[0,1) , 84 Fσ set, 458 Fσ δ set, 458 F.d.cap, 354 F.d.cap set, 354 FX , 83, 392 Face, 22, 28 Fat mapping cylinder, 145, 277 Fathi–Visetti Deformation Theorem, 351 Fin(), 10 Fine C-universal, 387 Fine homotopy equivalence, 39, 71, 76, 145 Fine universal for C, 387

Index Finite-dimensional cell (simplicial) complex, 30 Finite-dimensional compact absorption property, f.d.cap, 354 Finite-dimensional (f.d.), 61 Finitely generated group, 302 Fitting open cover, 93 Fl A, 21 Flat, 21 Flat hull, 21 Flattened along a subcomplex, 504 Fréchet space, 27, 254, 259 Free arc, 287 Full complex, 30 Full realization, 69 Full simplicial complex, 30 Full subcomplex, 30, 46

G Gδ set, 458 Gδσ set, 458 G-hereditary property, 19 General Position Lemma, 44, 58 Generalized Keller’s Theorem, 285 Generalized Schoenflies Theorem, 158 Geometrically independent, see affinely independent, 20

H Handle Straightening Theorem, 326 Handle Straightening Theorem, Strong, 351 Hanner’s characterization of ANRs, 70 Hauptvermutung, 44, 501, 530 Hauptvermutung for combinatorial ∞manifolds, 530 Hausdorff metric, 287 Hausdorff’s Metric Extension Theorem, 67 Haver’s Theorem, 73 h¯ -refinement, 67 Hedgehog, 16 Heisey Theorem, 483, 487 Hereditarily paracompact, 18 Hereditary shape equivalence, 79 Hereditary weak homotopy equivalence, 77 Hilbert cube manifold, 81 Hilbert manifold, 81 Hilbert space, 13 Homeo(X), 266 Homeomorphism Approximation — for Q-manifolds, 352 — for E-manifolds, 145

613 Homeomorphism group, 266 Homeomorphism Group Problem, 266, 352 Homogeneity of Q, 84 Homogeneous, 83 Homotopic, 6 Homotopically soft map, 76 Homotopically trivial, 71, 167, 300 Homotopic relative to a set, 7 Homotopy, 7 Homotopy dense, 71, 181 Homotopy dominate, 55 Homotopy dominated, 55, 70 Homotopy domination, 55 Homotopy equivalence, 7 Homotopy equivalent, 7 Homotopy Extension Theorem, — for ANEs, 67 — for cell complexes, 32 Homotopy inverse, 7 Homotopy relative to a set, 7 Homotopy sense, 430 Homotopy type, 7

I I, the unit closed interval, 2 I (Q), the pseudo-interior of Q, 84, 183, 353, 356 I (Q)f , 355, 356 I-preserving, 88 I-stable class, 389 I∞ , 482 id, idX , the identity map, 2 Immersion, 319 Inessential family, 63 Infinite-dimensional cell (simplicial) complex, 30 Infinite-dimensional (i.d.), 61 Infinite-dimensional PL manifold, 530 Int M, 4, 590 Int A, intX A, the interior of A in X, 2 Interior of a manifold, 4 Interior of a PL manifold, 590 Interior point of a manifold, 4 Interior point of a PL manifold, 590 Invariance of Domain, 62 Invariant metric on a linear space, 25 Inverse limit, 461 Inverse sequence, 461 Invertible isotopy, 88 Invertibly isotopic, 88 Isotopic, 88 Isotopic, ambiently, 91

614 ambiently invertibly, 91 invertibly, 88 Isotopy, 88 ambient, 91 invertible, 88

J J (), 16 Join — of simplexes, 28, 586 — of simplicial complexes, 29, 548, 585 — of spaces, 548 Joinable, 28, 29

K Kadec–Anderson–Toru´nczyk Theorem, 259 Keller’s Theorem, 286 Klee’s Trick, 67, 140

L L, 207 L(τ ), 207 1 -product, 94 ∞ , the space of bounded sequences, 11, 12, 15 ∞ (), 11 p (), 13 p f (), 13 L.s.c., lower semi-continuous, 18 1 -completion, 557 2f -manifold, 554 1Q , 356 1f , 356 1f -product, 94 2 (), 13 2 ()-manifold, 81 2Q , 353 2Q -manifold, 365 2f , 355 2f (), 404 2f -manifold, 365 p Q , 356 p f , 356 Lσ , 436 Lσ (τ ), 436 λ-convex, 75 λ-extension, 75 λ-stable, 75

Index λ-stable refinement, 75 1 (τ ), 472 α (τ ), 472 Lavrentieff Gδ -Extension Theorem, 19 Lavrentieff Homeomorphism Extension Theorem, 19 Lefschetz refinement, 69 Lefschetz’s characterization of ANRs, 69 LF-space, 579 Lifting, 296 Lifting Criterion, 297 lim Xn , 480 − → Limitation topology, 204, 205 Linear cell, 27 Linearly accessible, 22 Linear manifold, see flat, 21 Linear metric, 25 Linear span, 10 Linear variety, see flat, 21 Link, 31 Lk(σ, K), 31 Local ∗-connection, 77 Local ∞-connection, 77 Local n-connection, 77 Local-compactification, 439 Locally arcwise connected, 6 Locally closed, 124, 462 Locally collared, 137 Locally compact, 462 Locally complete, 445 Locally contractible, 66 Locally convex, 27 Locally deficient, 116 Locally equi-connected (LEC), 73, 559 Locally finite, 15 Locally finite τ -polyhedral sequence property, 427 Locally finite τ -skeletal tower property, 245 Locally finite cell (simplicial) complex, 32 Locally finite cell-tower property, 247, 427 Locally finite compact polyhedra property, 229 Locally finite-dimensional cell (simplicial) complex, 32 Locally finite-dimensional nerve, 60 Locally finite displacing property, 238, 430 Locally finite family, 216 Locally finite f.d. τ -polyhedra property, 229 Locally finite f.d. separable polyhedra property, 229 Locally finite self-copies property, 237 Locally Finite Theorem, 63 Locally path-connected, 6 Locally Q-deficient, 118 Locally Q-stable, 117

Index Locally separable, 435, 464 Locally stable, 115 Loop, 297 Lower semi-continuous (l.s.c.), 18

M M(f ), 69 M0 , 83, 204 M0 -universal, 215 M0 -universality, characterization of —, 215 M1 , 83, 204 M1 (τ ), 83, 204 M1 (τ )-universal, 215 M1 (τ )-universality, characterization of —, 215 M1 -universality, characterization of —, 215 M1 (τ )-universality, characterization of —, 232 M0 -absorption base, 399, 404 M(n) 0 , 423 Mfd 0 -absorption base, 399, 404 (M0 )σ , 377 M1 (ℵ0 )-absorption base, 399, 405 M1 (τ )-absorption base, 399, 406 Mα , 459 Mα (τ ), 459 Mf , 53 Manifold modeled on E, 81 Manifold pair modeled on (E, F ), 353 Manifold, n-dimensional, 4 Manifold without boundary, 4 Mapping cylinder, 53, 69 Mapping cylinder reduced over, 270 Mapping Cylinder Theorem, 269 Mayer–Vietoris exact sequence, 300 Mayer–Vietoris Theorem, 299 Mazur Theorem, 14 Meshd A, mesh A, the mesh of A, 3 Metr(X), 2 Metric linear space, 26 Metric topology, 38 Metrizable cone, 69, 106, 109 Metrizable open cone, 69, 107 Metrization Theorem, Bing, 15 Nagata–Smirnov, 15 Urysohn, 16 Minkowski functional, 22 Mogilski Characterizations of 2f - and 2Q -manifolds, 438

615 Monomorphism, 299 Multiplicative class, 389 Multiplicative subset, 253

N N, the natural numbers, the positive integers, 2 Nd (A, ε), N(A, ε), the ε-neighborhood of A in X, 3 N(A, K), 501 Nd (A, ε), N(A, ε), the closed ε-neighborhood of A in X, 3 n-cell, 27 n-connected, 300 n-dimensional cell (simplicial) complex, 30 n-face, 28 n-manifold, 4 n-simplex, 28 n-torus, n-tori, 295, 319 Near-homeomorphism, 108, 145, 153 Negligibility Theorem, 248 — for (Q∞ , QN f )-manifolds, 571 — for (R∞ , RN f )-manifolds, 571 — for E-manifolds, 182 — for E-stable spaces, 124 Negligible, 124 Neighborhood retract, 65 Nerve, 59 Null-homotopic, 7

O OK , 31 OK (x), 31 ω, the non-negative integers, 2 !α (τ ), 472 Open cone, 127 Open cone, metrizable, 69, 107 Open Embedding Approximation Theorem — for E-manifolds, 144 Open Embedding Theorem — for (Q∞ , QN f )-manifolds, 570 — for (2 , 2f )-manifold pair, 371 — for (2 , 2Q )-manifold pair, 371 — for (R∞ , RN f )-manifolds, 570 — for Q-manifolds, 192 — for Q∞ -manifolds, 495 — for R∞ -manifolds, 495 — for R+ -stable Q-manifolds, 197 — for E-manifolds, 141 Open hereditary, 408 Open star, 31 Opposite face, 28

616 Ordered cell complex, 35 Ordered simplicial complex, 35 Order of an open cover, 61

P Paracompact, 17 Partial realization, 69 Partition, 63 Path, 6 Path-connected, 6 Perfectly normal, 18 Perfect map, 16, 206 Piecewise linear, 39 Piecewise linear R∞ -manifold, 580 Piecewise linear R∞ -structure, 580 Piecewise linear homeomorphism, 39 PL, 39 PL ∞-manifold, 530, 542, 566 PL n-manifold, 581 PL n-manifold without boundary, 590 PL Approximation Theorem, 44 PL bi-collar, 50, 311 PL bi-collared, 49, 310 PL characterization of ∞ , 536 PL collar, 50 PL collared, 49 PL embedding, 40 PL Embedding Approximation Theorem, 594 PL Embedding Approximation Theorem for Products, 45 PL homeomorphic, 40 PL homeomorphism, 39 PL manifold, n-dimensional, 581 PL map, 501 Pointed space, 2 Pointwise convergence topology, 9, 15 Polyhedrally approximately n-soft map, 218 Polyhedrally approximately soft map, 76, 218 Polyhedrally homotopically soft map, 76 Polyhedrally soft map, 76 Polyhedral neighborhood, 538 Polyhedron, polyhedra, 30 Product cell complex, 35 Product simplicial complex, 35 Product Theorem, 63, 64 Proper face, 28 Proper homotopy, 16 Proper homotopy dominate, 56 Proper homotopy dominated, 56 Proper homotopy domination, 56 Proper homotopy equivalence, 17 Proper homotopy equivalent, 17 Proper homotopy inverse, 17

Index Proper homotopy type, 17 Properly L(τ )-universal, 207 Properly contractible to infinity, 198 Properly homotopic, 16 Properly universal for L(τ ), 207 Property U V ∗ , 79 Property U V ∞ , 79 Proper map, 16, 206 Property U V n , 79 Pseudo-boundary of Q, B(Q), 84, 353, 356 Pseudo-interior of Q, I (Q), 84, 183, 353, 356 Punctured torus, 320

Q Q, the rationals, 2 Q, the Hilbert cube, 3 Q-deficient, 117 Q-factored Alexander Trick, 341 Q-manifold, 81 Q-stable, 117 Q∞ , 479 (Q∞ , QN f )-manifold, 568 Q∞ -manifold, 479 Qf , 353, 356 R R, the real line, 2 R , 10 Rf , 10 R∞ , 479 (R∞ , RN f )-manifold, 568 R∞ -manifold, 479, 542, 554 Rn , Euclidean n-space, 3 Rn+ , Euclidean half n-space, 3 R+ , the half (real) line, 2 R+ -stable Q-manifold, 191, 197, 560 Radial boundary, 22 Radial interior, 21, 355 Reduced Mapping Cylinder Theorem, 271 Reduced product, 108 Refine, 4 Refinement, 4 Refinements by open balls, 20 Refining simplicial map, 59 Regular neighborhood, 48, 593 Retract, 65 Retraction, 65 Riesz Lemma, 85 Rint C, 21, 365 Rint Q, 355

Index S s, the space of sequences, 3, 10, 15 SK , 31 SX , the unit sphere, 11 Sn−1 , unit (n − 1)-sphere, 3 s f , the space of finite sequences, 10, 355 s Q , 355 Same simple homotopy equivalent, 290 Same simple homotopy type, 290 Sd K, 36 SdL K, 37 Set of vertices, 30 Shape equivalence, 79 σ -completely metrizable, 435, 436 σ -discrete, 15 σ -locally compact, 436, 464 σ -locally finite, 15 σ -locally separable, 435, 436 Simple homotopy equivalence, 291 Simplex, 28 Simplicial approximation, 44 Simplicial Approximation Theorem — for maps, 44 — for PL embeddings, 501 Simplicial complement, 32, 501 Simplicial complex, 29 Simplicially collapse, 294 Simplicial cone, 29 Simplicial embedding, 41 Simplicially expand, 294 Simplicial homeomorphism, 41 Simplicially isomorphic, 41 Simplicial isomorphism, 41 Simplicial map, 40 Simplicial map between abstract complexes, 58 Simplicial mapping cylinder, 54 Simplicial neighborhood, 31, 501, 539 Simplicial subdivision, 33 Simply connected, 300 Skeleton, — of an abstract complex, 58 0-skeleton, 30 n-skeleton, 30 Soft map, 76 Splitting, 310 Splitting Theorem, 311, 318 St U, 5 St(C, K), 31 St(U, V), 5 St(Y, A), 4 Stn U, 5 Stability Theorem

617 — for (Q∞ , QN f )-manifolds, 571 — for (R∞ , RN f )-manifolds, 571 — for Q-manifolds, 117 — for E-manifolds, 116 Stable, 108, 114 Stable Hauptvermutung, 535 Standard ∞-simplex, 483, 530 Standard embedding of Tn × [−1, 1] into Rn+1 , 322 Standard n-simple, 3 Star of Y with respect to A, 4 Star-finite open cover, 60 Starring, 37 Star (subcomplex), 31 Stellar subdivision, 37 Straight, 325 Straight line homotopy, 32 Strong Z-set, 162, 171 Strong C-absorption property, 379 Strong deformation retract, 50 Strong Hauptvermutung for PL ∞-manifolds, 530 Strong Z-embedding, 181 Strong Z-set, 179 Strong Zσ -space, 412 Strong Zσ -set, 162, 181 Strongly C-absorptive, 379 Strongly C-universal, 204, 387 Strongly countable-dimensional (s.c.d.), 64, 426, 436 Strongly infinite-dimensional (s.i.d.), 64 Strongly properly L(τ )-universal, 214 Strongly properly universal for L(τ ), 214 Strongly universal for C, 204 Strong Universality Theorem — for Q-manifolds, 120, 191 — for E-manifolds, 119, 182 Sub-additive function, 95 Subcomplex, 29 — of an abstract complex, 57 Subdivide, 33 Subdivision, 33 Sublinear, 23 Subpolyhedron, subpolyhedra, 30 Subset Theorem, 63 Sup-metric, 8 Sup-norm, 15

T Tn , n-torus, 295, 319 τ (f ), 292 τ -DAP(C), 422

618 τ -discrete approximation property, 422 τ -discrete cells property, 217 τ -discrete n-cells property, 217 τ -locally finite cells property, 217 τ -locally finite n-cells property, 217 Theorem, Banach–Mazur, Klee, 27 Henderson–Sakai, 39 Kadec–Anderson, 254 Kruse–Liebnitz, 68 M. Brown, 137 Michael, 20 Stone, A.H., 15 Whitehead, J.H.C., 37 Tietze–Urysohn Extension Theorem, 15 Topological class, 377 Topological group, 253 Topological invariance of Whitehead torsion, 292, 344 Topological linear space, 23 Topological polyhedron, 30 Topological sum, 5 Toru´nczyk Factor Theorem, 95, 106, 107 Torus, tori, 295, 319 Totally bounded subset of a topological group, 253 Totally bounded subset of a topological linear space, 25 Totally bounded with respect to a metric, 26 Tracks of a homotopy, 7 Triangulable, 500, 542, 545 Triangulate, 30, 500 Triangulation, 30, 500 Triangulation Theorem — for (Q, Qf )-manifolds, 373 — for (Q, rint Q)-manifolds, 373 — for (Q∞ , QN f )-manifolds, 571 2 2 — for ( , f )-manifolds, 373 — for (2 , 2Q )-manifolds, 373 — for (R∞ , RN f )-manifolds, 571 — for 2 ()-manifolds, 153 — for 2f -manifolds, 153, 365 — for 2Q -manifolds, 373 — for R+ -stable Q-manifolds, 284 U U(f ), 204 U-homotopic, 8 U-homotopy, 8 U-homotopy equivalence, 145 U-homotopy inverse, 145 U-map, 208

Index Unified locally contractile (ULC), 74 Uniform ANR, 72 Uniform AR, 72 Uniform convergence topology, 8, 14 Uniformly continuous at a subset, 72 Uniformly discrete, 86 Uniformly separating cells property, 250 Uniform neighborhood, 72 Uniform neighborhood retract, 72 Uniform retract, 72, 95 Uniqueness of absorbing sets, 449 Uniqueness of absorption bases, 385 Unique path lifting property, 296 Unit closed n-ball, 3 Unit (n − 1)-sphere, 3 Universal for C, 203, 204, 387 Universal space, 467 Unknotting Theorem — for E-deficient closed sets, 141 — for Z-sets in a Q-manifold, 192 — for Z-sets in an E-manifolds, 182 Upper semi-continuous (u.s.c.), 18 U.s.c., upper semi-continuous, 18 U V ∗ , 79 U V ∞ , 79 U V n , 79 U V ∗ map, 79 U V ∞ map, 79 U V n map, 79

V Vertex, 28

W w(X), the weight of X, 2 Wall’s finiteness obstruction, 293, 294 Weak homotopy equivalence, 71 Weakly infinite-dimensional (w.i.d.), 64 Weak-star topology, 484 Weak topology, 5, 30 Wedge sum, 5 Weight, 2 Wh(X), 292 Whitehead group, 292 Whitehead–Milnor Theorem, 55 Whitehead topology, 30 Whitehead torsion, 292

Z Z, the integers, 2 Z-embedding, 181

Index Z∞ -set, 164 Zn -set, 163 Z-set, 162, 164, 167, 179, 499, 542 Zσ -space, 412 Z-submanifold, 181

619 ZfK , 55 Zf , simplicial mapping cylinder, 54 Zσ -set, 162, 167, 181 Zero set, 18