Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88), Volume 88 9781400881505

Table of contents :
Table of Contents
Essay I. Deformation of smooth and piecewise linear manifold structures
Essay II. Deformation of sliced families of manifold structures
Essay III. Some basic theorems about topological manifolds
Essay IV. Stable classification of smooth and piecewise linear manifold structures
Essay V. Classification of sliced families of manifold structures
Annex 1. Stable homeomorphisms and the annul us conjecture
Annex 2. On the triangulation of manifolds and the Hauptvermutung
Annex 3. Topological manifolds
Combined Bibliography for Essays I to V.
Index

Citation preview

Annals of Mathematics Studies Number 88

FOUNDATIONAL ESSAYS ON TOPOLOGICAL MANIFOLDS, SMOOTHINGS, AND TRIANGULATIONS BY

ROBION C. KIRBY AND

LAURENCE C. SIEBENMANN

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY

1977

Copyright © 1977 by Princeton University Press ALL RIGHTS RESERVED

Published in Japan exclusively by University of Tokyo Press; In other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication data will be found on the last printed page of this book

FOREWORD These five essays consolidate several parts of the subject of topo­ logical manifolds that were brought within reach by our two short articles [K ij] and [K Sj] published in 1969. Many of the theorems proved here were announced by us in [KS3] [KS4 ] [Si jq] . Certainly we have not labored in isolation; our bibliography bears clear testimony to this ! Preliminary versions of segments of these essays (particularly Elssay I ) have been in limited circulation since early 1970. Poly copied versions of all five were released in August 1972 (from Orsay), and reissued in November 1973. The final camera-ready manuscript was prepared by the second author, Wanda Jones, and Arlene Spurlock during 1974 and 1975 using the IBM selectric composer system (hitherto exploited but little by mathematicians, cf. [Whole, p. 4 3 5 ]) . G. Blattmann inked the drawings. Essential moral support and criticism were contributed by our colleagues, particularly by R. D. Edwards; and a multitude of errata and points needing revision were tracked down by: R. D. Edwards, A. Fathi, L. Guillou, H. Hahl, M. Handel, A. Marin, M. Scharlemann, R. Stern, J. Vaisala, Y.-M. V isetti.( It is a statistical certainty that many errata (or worse) persist. With our readers’ assistance we hope to keep a list available upon request.) Im portant direct or indirect support has come during this period from several institutions, including the NSF (USA), the CNRS (France), and our home institutions. For all these contributions, large and small, to the successful com­ pletion of this monograph, we are happy to express our heartfelt thanks. Robion C. Kirby University of California Berkeley, CA 94720 USA

Laurence C. Siebenmann Universite de Paris-Sud 91405 Orsay France

(v)

TABLE OF CONTENTS Essay I

Deformation o f smooth and piecewise linear m anifold structures, by R. Kirby and L. Siebenmann.

Essay II

Deformation o f sliced families o f manifold structures, by L. Siebenmann.

Essay III

Some basic theorems about topological manifolds, by L. Siebenmann and R. Kirby.

Essay IV

Stable classification o f smooth and piecewise linear m anifold struc­ tures, by L. Siebenmann.

Essay V

Classification o f sliced families o f m anifold structures, by L. Siebenmann.

Annex 1.

Stable homeomorphisms and the annulus conjecture, by R. Kirby, reprinted, with permission, from Ann. of Math. 89(1969), 575-582.

Annex 2.

On the triangulation o f manifolds and the Hauptvermutung, by R. Kirby and L. Siebenmann, reprinted, with permission, from Bull. Amer. Math. Soc. 75(1969), 742-749.

Annex 3.

Topological manifolds, by L. Siebenmann, reprinted, with permission, from Proc. Int. Congress. Math., Nice (1970), Vol. 2, 142-163, Gauthier-Villars, 1971.

Combined Bibliography for Essays I to V . Index Essay I includes (as Appendix B) a note by John Milnor on submerging the punctured torus. Annex 3 includes (tacked on at the end) a note on involutions by Michael Atiyah. Both are published here for the first time.

GUIDE Ample introduction to these five essays is provided by our three articles [K ij] [K Sj] [Si j q ] (of the combined bibliography) that have been reprinted as annexes to this volume. There the reader will find motivation and get a preliminary view of the whole subject. The interdependence of the essays is roughly as follows: Essay I Deformation of smooth and piecewise linear manifold structures.

Essay III Some basic theorems about topological manifolds

Essay II Deformation of sliced families of manifold structures

Essay IV Stable classification of smooth and piecewise linear manifold structures

EssayV Classification of sliced families of manifold structures

There are two itineraries we recommend: 1) To acquire an understanding of topological manifolds as such, using a minimum of machinery, read Essay I then Essay III . 2) To learn about classification of smoothings and triangulations read Essay V , referring back to the relevant parts of Essay II . Many readers will find itinerary 2) too steep. For them we suggest Essay I and then Essay IV ; the latter presents the basic classi­ fications in a leisurely and old-fashioned way. On the other hand, itinerary 2) will lead the reader to very sophisticated proofs of the results of Essay I , permitting him to pass, if he wishes, straight on to Essay III . Some readers will be most interested in understanding how the Hauptvermutung and the triangulation conjecture fail for manifolds. They should set out on a peripheral route [KSj ], [S ijg, § 2 ], [IV, App. B] [V, App. B ], referring also to [KiK] . On first reading, one can safely bypass the many alternatives digressions and generalizations offered. Some basic definitions conventions and notations are collected in § 2 of Essay I .

Essay I

D E F O R M A T IO N OF SMOOTH A N D PIECEW ISE L IN E A R M A N IF O L D ST R U C T U R E S

by

R. Kirby and L. Siebenmann

R. Kirby and L. Siebenmann

Essay I

Section headings in Essay I

§

1. Introduction

§

2. Some definitions, conventions, and notations

§

3. Handles that can always be straightened

§

4. Concordance implies isotopy

§

5. The product structure theorem

Appendix A.

Collaring theorems

Appendix B.

Submerging a punctured torus (by J. Milnor)

Appendix C.

Majorant approximation

3

§1. IN T R O D U C T IO N

This essay presents proofs based on handlebody theory in the sense of Smale [S n^] of three theorems concerning PL ( = piece wise-linear) or DIFF ( = differentiable C°°) manifold structures on topological manifolds. To state them in a simple form, let M denote a metrizable topological manifold without boundary and of (finite) dimension > 5. Concordance Implies Isotopy (see § 4 ). Let V be a CAT (= D IFF or PL) manifold structure on M X I , where L = [ 0 ,H C R 1 and let 2 X 0 be its restriction to M X 0 . Then there exists an isotopy (= path o f homeomorphisms) h t : M X I-*-M X I , O ^ t ^ I , such that h 0 is the identity map id\M X I and h { gives a CAT manifold isomorphism hi : (MXf)-yy^j (M X l)^ while h t fixes M X 0 pointwise fo r all t in [0,1] . Further, h ( can be as near to the identity as we please. The structure T is said to give a concordance from 2 to 2 ' where 2 ' X 1 = T I(M X 1) , and 2 , 2 ' are said to be concordant. The theorem implies that concordant structures 2 , 2 ' are isotopic , i.e. are related by an isotopy of id IM to a CAT isomorphism M jy . Concordance Extension (see § 4 ). Let F be a C AT manifold structure on U X I C M X I where U is an open subset o f M and let 2 be a CA T structure on M such that the restrictions T \U X 0 and 2 X 0 1 ( 7 X 0 coincide. Then T extends to a CAT structure T' on M X I such that r 'lA ( X 0 = 2 X 0 . Note the resemblance to the classical homotopy extension property. Product Structure Theorem (see §57. L et 0 be a C AT manifold structure on M X R s , s > 1 . There exists a concordant CAT structure 2 X R s on M X R s obtained from a CAT structure 2 on M by producting with R s . t The abiding lim itation to dim ensions ^ 5 cannot be rem oved entirely (see § 4, § 5, [Sig] ) ; perhaps it suffices here to assume M is non com pact or of dimension ^ 4 .

4

R. Kirby and L. Siebenmann

Essay I

These three results are parallel to existing theorems about compatible DIFF structures on PL manifolds due respectively to Munkres [ M ^ l jHirschfHi.^] and Munkres [Mu^] ; and Cairns and Hirsch [Ca ] [ t ^ ] [Hi^J [Hi^] . Our proofs, however do not much resemble existing proofs of these PL-DIFF analogues. Also we even­ tually manage to deduce them in dimensions > 5 , see §5.3 , §5.4 and [II, §4 ] . Historically, the above theorems were first sought after to permit a classification up to isotopy of manifold structures in the framework of Milnor’s microbundle theory. Essay IV explains this application in detail. Curiously, some of the most im portant applications of these theorems concern not manifold structures, but rather basic geometrical properties of topological manifolds. These include topological transversality theorems, topological handlebody theory, and topological facts about simple hom otopy type. They are the subject of Essay III . We remark that the proofs in this essay, in depending mainly on DIFF (or PL ) handlebody theory, thereby depend on transversality ideas and on some simple hom otopy theory. Our main technical tool is in fact the CAT ( = DIFF or PL ) S-Cobordism Theorem t (see f K ej ] [Mig] [Hu2/ [StgJ [R S g l ) . L et ( W; V, V ) be a compact connected C AT cobordismt V to V , which is relative in the sense that bW - in t (V C V ') = b V X I (where = denotes CA T isomorphism ; 3 indicates boundary ; int indicates form al manifold interior) . Suppose that the inclusions i : V Q. W , /': V Q. W are hom otopy equivalences , dim W > 6 , and i has zero torsion in the Whitehead group in Wh(7rt W) . Then ( W; V, V ) = V X ( I ; 0 ,1 ) "I When 7r ! W = 0 , so that Whiwj W) = 0 , the result is called the hcobordism theorem . The ‘ h ’ stands for hom otopy equivalence ; the ‘ s ’ stands for simple hom otopy equivalence , (i.e. one with zero torsion) .

CAT

^ This implies that V and V ' are CAT submanifolds o f 3W adm itting collarings in W (see A ppendix A) .

^ This isomorphism can clearly extend the identity isomorphism V ^ V X 0 ; it can also extend the given isomorphism 3W - int(V U V ') = 3V X I as a device exploiting a CAT collaring of 3V in V readily shows , cf. [Si^ , Prop. II , fig. 4] , or step 3 in proof of 4.1 below .

§1. Introduction

5

In addition , we use the non-compact version of the s-cobordism theorem [Siy ] (see 3.1.1) with V = M X R . It can be reduced by Stallings’ engulfing methods to the above compact version with V = M X S1 . To apply these s-cobordism theorems , we need the algebraic theorem of Grothendieck and Bass-Heller-Swan [BHS] [Bas] to the effect that K0Z[A] = 0 = Wh(A) for any finitely generated free abelian group A . Finally one must know that the annulus conjecture is true in dimensions > 6 in order to prove the Product Structure Theorem (not its companions though) . This in inevitably so , for the latter converts an annulus of dimension > 6 into a smooth h-cobordism . Unfortunately the proof of the annulus conjecture in dimensions > 5 given in [Kij ] relies on the principal theorems of nonsimply-connected surgery . As we shall note in § 5 , this unfortunate dependence of the Product Structure Theorem and its corollaries on sophisticated surgery can be artificially eliminated by restating it (and its corollaries) so as to replace topological manifolds everywhere by STABLE^ topological manifolds , i.e. manifolds equipped with atlases of charts related by homeomorphisms that are STABLE in the sense of Brown and Gluck [BrnG] . One can give a competitive alternative proof of the Product Structure Theorem , if one is willing to use the stability theorem for TOP /CATm ( m > 5 ) [K S j] [LR2 1 [IV, §9.4] [V, §5.2] and a (relative) hom otopy theoretic classification theorem for CAT structures derived by immersion theoretic m ethods.$ A significant advantage our present proof has is the greater simplicity of the prerequisites . The alternative proof necessarily uses the same handlebody theory to establish the stability theorem . In addition it uses immersion theory . And in the immersion theory are hidden two notable further pre­ requisites , namely Milnor’s microbundle theory and the isotopy extension theorem for the topological category . ^ The capital letters should serve to avoid confusion with other uses o f the word 'stable' . t Similar rem arks apply to the Concordance Implies Isotopy T heorem and the Concordance Extension Theorem .

6

Essay I

§2. SO M E D E F IN IT IO N S, CO N V EN TIO N S, A N D N O T A T IO N S

These are adopted for all five essays (although some inconsistencies persist). Experienced readers should manage to guess them all; less experienced readers will need to use this section as a glossary. A set consisting of a single element x is often denoted by x itself rather than by {x} , provided no confusion will result. The empty set is denoted by 0 (the G reek ‘phi’ will be of another shape:

DXI ; V +* VXI ; e -** a sufficiently small majorant . ■ General Case . (Relaxing the condition that 3M = 0 ) Let 3MXJ C M be a CAT collar neighborhood of 3M^ in M^ . Adjusting T by a (majorant) small self-homeomorphism of MXI which is the identity on MXO and near { C U (M -V )}X I (using the collaring uniqueness theorem Appendix A. 1 ) we can assume that T is a product along J near (3M)XI n DXI . Thus we can assume T has this prop­ erty from the outset. Two applications of Case 3 complete the proof. First apply it to T 1(3M X I) , and extend the resulting isotopy (ft say) naturally to one of id | (MXI) with support in the collaring (3M)XJXI of (3M)XI . Then a second application to fj^ T on MXI (with V~3M for V ) completes the proof. The details are much as for Case 3 . ■

31

§5. TH E PR O D U C T ST R U C T U R E T H E O R EM

THEOREM 5.1 (Product Structure Theorem) . L et M m be a TOP manifold and 2 be a C A T (= DIFF or PL) structure on M X R s , s > 1 . Let U be an open subset o f M so that there is a CAT structure p on U with 2 I £/ X R s = p X R s . Provide that m > 6 , or m = 5 and 3M C U . Then there exists a CA T structure a on M extending p and a conditioned concordance rel U X R s from 2 to o X R s . Complement 5.1.1. A structure a as described , with a X R s concordant rel U X R s to 2 , is unique up to concordance rel U. Remark 1. By Theorem 4.1 there is an e-isotopy from the identity to a CAT isomorphism between (M X Rs)£ and Ma X Rs . If C is a prescribed closed subset of U , the isotopy can fix C X Rs . Remark 2. Together 5.1 and 5.1.1 show that the rule o +> o X Rs is a one-to-one correspondence between concordance classes of CAT structures on M and on M X Rs , under the above dimensional restrictions . The relative version of this , i.e. the version rel U , is also im portant . Remark 3. Consider 5.1 without the proviso that m > 6 , or m = 5 and 9 M C U , a n d call that statem ent (P) . Using (a) the classification theorem [KSj ] [IV] , which follows from 5.1 , (b) the calculation that 7Tj(TOP/PL) and 7Tj(T0 P /0 ) are 0 for i ^ 2 and are Z2 = Z/2Z if i = 3 [KSj ] , and (c) the uniqueness up to isotopy of CAT structures on metrizable manifolds of dimension < 3 (cf. §3) , one easily finds that for and m + s> 6 (or m+s = 5 and dM C U ) the statem ent (P) holds if and only i f the cohomology group H 3(M, U; Z2) is zero . For example (P) fails if M3 = S3 , s > 2 , and U = 0 ! Statement (P) is undecided if m = 4 .

32

R. Kirby and L. Siebenmann

Essay I

Remark 4. Clearly 5.1 is modeled on the famous Cairns-Hirsch theorem stated at the end of this section (5.3) . It is the PL-DIFF analogue of our TOP-CAT theorem 5.1 . We obtain a new proof of it in high dimensions , based on 5.1 and 4.1 . We likewise obtain a proof of the PL-DIFF Concordance Extension Theorem ; however the PL-DIFF Concordance Implies Isotopy Theorem will elude us until §3.6 of Essay II . Remark 5. By a happy accident our proof of 5.1 (and 5.1.1) works equally well without the abiding assumption that M be metrizable (or even Hausdorff) , when we use Zorn’s Lemma in place of a countable induction over charts . Let us begin with PROOF OF 5.1.1 FROM 5.1. It is the one most usual when there is an existence theorem in relative form . Let o and a' be two CAT structures on M offered by 5.1 . The conditioned concordances o X Rs —2 —a' X Rs rel U X Rs provide a CAT structure F on R X M X Rs giving on I X M X Rs a concordance from o X Rs to o' X Rs rel U X Rs .

We arrange that F is a product along R and along Rs , on all of U ' X R s = ( R - [ 1/4, 3 /4 ]) X M X Rs U R X U X R S . Then F gives U ' X O a CAT structure 7 X 0 which 5.1 extends to all R X M X 0 giving a concordance o —o' rel U . ■ PROOF OF 5.1 (using 4.1 , 4.2 and the Stable Homeomorphism Theorem ) . In our proofs so far , handlebody theory has been the only difficult technique involved ; in particular , we have avoided immersion theory and surgery . However , now , the STABLE Homeomorphism Theorem [Kij ] , which involves the nonsimply connected surgery of Wall [Wa] , is needed in order to show that

§5. The Product Structure Theorem

33

(H) M admits a STA B LE structure 8 (= a maximal atlas o f charts to R™ related by S T A B L E f homeomorphisms) compatible with 2 in the sense that 2 is contained in the (maximal) STABLE atlas 8 X R s . The proof as we give it then yields a structure o automatically satisfying the condition (C) o agrees with the STABLE structure 8 o f hypothesis ( H ) . For the proof of 5.1 we admit (H) as a hypothesis . (It is redundant by [Ki] ] .) This causes all further mention of the STABLE Homeomorphism Theorem of [K ij ] to vanish . For convenience of reader any detail o f the p ro o f that could be eliminated by free use o f it will henceforth be enclosed in square brackets . [In short , what we now prove is the version of 5.1 for (compatible) CAT structures on STABLE topological manifolds — the STABLE-CAT version rather than the TOP-CAT version . | We may (and shall) assume s = 1 in proving 5.1 , because a finite induction using M X Rs = (M X Rs—1) X R retrieves the general case . The best part of the proof is The Case of 5.1 where M = Rm , m > 5 , or R™ , m > 6 , [and where 8 agrees with the linear structure ]) . [Since 8 X R agrees with 2 , our definition of STABLE makes it clear that J there is a concordance (not rel U X R) from the standard structure on M X R to 2 . Apply 4.1 to this concordance using for examples the substitutions C +* 4>, D +> M X [ 1,°°) , V +*• M X 0/2,°°) , in order to replace 2 by a CAT structure 2 1 that is standard near M X [1,°°), and equals 2 near M X (-°°,0 ] . Examine Figure 5-a below . • Since 2 l U X R = p X R , the set U X [0,1 ] inherits from 2 j a CAT manifold structure written (U X [0,1 ] ) ^ • Apply 4.1 , the Concordance Implies Isotopy Theorem , for the second time to find a ^ A hom eom orphism h : U -*■ V o f open subsets o f R™ is STABLE if and only if , for each point x E U , there exists an open neighborhood Ux in U and topological isotopy h t : U x -> R j 1 through open imbeddings from h 0 = h lU x to a linear imbedding h x . See [BrnG] [K ij] , where equivalent definitions are discussed , and also [III, A ppendix A ] .

34

R. Kirby and L. Siebenmann

Essay I

CAT isomorphism h : U X [0,1 ] ->■ (U X [0,1 so small that d(h(x),x) < e(x) where e :M X [0 , 1] -*■ [0 ,°°) is a continuous map with e' 1 (0,°°) = U X [0,1] t FIGURE 5-a

t If at this point one prefers to use the version w ithout e of the Concordance Implies Isotopy Theorem 4.1 , (but with a careful choice of V) , one succeeds easily enough at the cost o f weakening the final result o f 5.1 to be a concordance rel C X R to a product structure where C C U is a prescribed closed subset of M . The interested reader might pause to investigate two advantages that accrue, compensating this loss o f precision . First , the technically delicate parts of § 4 are avoided , while one obtains a version of 5.1 adequate for the applications in [III] to topological manifolds . Second , the argument w ithout e applies tel quel to prove the P L -D IF F Product Structure Theorem of [Hiy] from the (more basic?) PL - D IFF Concordance Implies Isotopy Theorem o f [Hi^] .

§5. The Product Structure Theorem

35

Extend h to a self-homeomorphism h of M X R so that one has h(x,r) = (x,r) if r < 0 or x ^ U , and h(x,r) = (p !h (x ,l),r) if x > 1 . With the help of the (local) CAT collaring uniqueness theorem A. 1 we arrange that h is a product with R near U X 0 and U X I . Then define 2 2 to make h : (M X R )^ ^ ( M X R)^; a CAT isomorphism. Clearly 2 2 IM X (1,°°) is of the form a X (1,°°) . This defines a . The properties of 2 2 indicated in Figure 5-a let one construct more or less standard conditioned concordances S 2 - » X R rel U X R and 2 2 —2 rel U X R using the following easy Windowblind Lemma 5.1.2. Let 2 ' and 2 ” be two C AT structures on M X R so that 2 '= 2 " on M X ( a , b ) , a < b , so that both 2 ' and 2 " are a product along R on an open subset U X R . Then there is a conditioned concordance 2 ' ^ 2 " rel U X R . Proof o f lemma . Using a conditioned CAT homotopy h^. ,0 < t < 1 , of id I R through imbeddings to an imbedding onto (a,b) , form an open imbedding H:IXMXR^IXMXR with Ht (x,r) = H(t,x,r) = (t,x,h^(r)) . The structure H- 1( I X 2 ' ) on I XMX R making H a CAT imbedding into I X(M XR)^< is a conditioned concordance 2 ' —H71(2 ') • It is rel U X R since 2 ' is a product along R on U X R . Similarly 2 " —H7!(2 " ) rel U X R . But H7H2') = H71(2 " ) , so we find 2 ' ^ 2 " rel U X R as desired . ■ The first case M = Rm or R™ of 5.1 is now established . ■ The general case follows by an easy chart by chart induction . We present the details in a novel fashion since the induction that the reader first thought of , probably does not work immediately if M is nonmetrizable . Let S be the set of pairs (U \ 2 ') consisting o f an open subset U' of M containing U , and a CAT structure 2 ' on U ' X R such that 2 ' = 2 on U X R U U ' X (-°°,0) , and 2 ' is a product along R on U' X (1,°°) . (Compare behavior of 2 2 in Figure 5-a .) According to Lemma 5.1.2 , the theorem is proved when we have found a pair (M ,2 ') 6 , or m > 5 and BM C C . Proof of assertion . By 5.1 .(ignoring s) there is a DIFF structure o' on M and a DIFF conditioned concordance T' rel C X Rs from 2 to o' X Rs . However , o’ may not be compatible with s , nor T' with I X s X Rs . Using Whitehead’s result find s ’ a DIFF triangulation of o’ equal to s near C , and find G' a DIFF triangulation of T' , giving a conditioned PL concordance from s X Rs to s' X Rs rel C X Rs . We shall construct a PL isomorphism F : (I XM X RS)G' -*■ I X Mj X Rs which fixes a neighborhood of 0 XM XRS U IX C X Rs and is a product with I and with Rs near 1 X M X Rs . Then F(r') = T will clearly be the required concordance compatible with I X s X Rs , and F( 1 X o’ X Rs) = 1 X o X Rs will define o . We proceed to define F as a composition H hXid (I X M X Rs)g » (I XM)? X R ------ ►I X M j X R . By Theorem 5.1 , Remark 1 , applied to I X M , there is a PL structure g that is a conditioned concordance s to s' rel C , and there is a PL homeomorphism H equal to the identity near {0,1} X M X RS U I X C X Rs . Finally by Theorem 4.1 , there is a PL homeomorphism h : (I X M)^ I X Ms so that h = id near 0 X M and h is a product with I near I X M . This establishes 5.3 in case m > 6 , or m = 5 and 9M C C . ■ 5.4. OBSERVATION . In the Concordance Extension Theorem 4.2.1 (rel a closed subset) suppose M is a PL m anifold. For ‘‘C AT structure” read “compatible DIFF structure” . There results a valid concordance extension theorem fo r compatible DIFF structures . Its proof is an easy consequence of 4.2 for DIFF reinforced by 4.1 , for PL and we leave it as an exercise . Again some low dimensions are not covered by this p ro o f.

§6. The Cairns-Hirsch Theorem

39

As a corollary one deduces a local version of the Cairns-Hirsch theorem analogous to 5.2 . These results provide , for dimension > 5 , the basis for Hirsch’s first obstruction theory [Hi^ ] for introducing and classifying up to concordance compatible DIFF structures on a given PL manifold . More elaborate theories of Hirsch and Mazur [Hi^.] [HiM] [Mor j ] [Hi-y ] [Hig] and of Lashof and Rothenberg [ LRj ] also use these results . It is of course a telling fault that we do not deal with compatible DIFF structures on PL manifolds of dimension < 4 . But the theory is rather joyless there ; compatible structures exist and are unique up to concordance [Cej] .

40

Essay I

Appendix A . C O L L A R IN G T H E O R EM S

We define a local collaring of a closed subset M of a metric space W to be an open neighborhood U of M X 0 in M X [0, °°) and an open topological imbedding f : U -* W satisfying f(p X 0) = p for all p in M . In case W and M are CAT objects (CAT = TOP, PL or D IF F ), and f is in addition a CAT embedding , then f is called a CAT local collaring . THEOREM A .l. Local collaring uniqueness . Consider the following data : M a CAT manifold (C AT = T O P , PL or DIFF) ; W a C AT manifold , with metric d , possibly with corners i f CAT = D I F F , and containing M as a closed su b se t; f g : U W two C AT local collarings o f M in W ; C a closed subset o f M such that f = g near C X 0 in C X f 0, °°) ; D another closed subset o f M ; e : W -+ [0, °°) a continuous function , positive on D . With these data there exists an isotopy h { , t < 1 , o f id IW fixing M so that f = h xg near D X 0 and d (h {(x),x) < e(x) fo r all x £ W . The isotopy htg is constant near C X 0 in C X [0, °°) n U . We will indicate a proof of this presently . Familiar (relative) collaring existence theorems follow from this , (see [Ar] [Brn] [Cny] for TOP ; [H u 2 ] for PL ; [Muj ] for DIFF.) It is not difficult to deduce a global uniqueness result . DEFINITION . If a CAT local collaring f : M X [0, °°) -*■W restricts to a closed imbedding of M X [0, 1] , we call the restriction f IM X [0, 1] a CAT clean collaring . THEOREM A.2 . Collaring uniqueness . Consider C AT clean collarings f , g : M X [ 0 , l ] - * W o f M C W . Suppose f = g near a closed subset C X [0, I ] . Let D C M be closed and let V C W be an open neighborhood o f f(D X [0, 1 ]) U g ( D X [ 0 , 11).

Appendix A.

Collaring theorems

41

Then there exists a C AT isotopy h { , f < 1 , o f id\W , fixing all points near f(C X 10, 1 ]) and all points outside V , such that f = h ig near D X 10, 1J . F u rth er, i f f = g near D X 0 , then h ( = id near M . The reader should be able to obtain the local collaring uniqueness theorem from the following key lemma , suggested by [Cny] . (There is an extra step unique to DIFF concerning angle of departure from the boundary ; cf. [Muj , § 6 ] .) LEMMA A.3 . Let U be an open neighborhood o f M X f-°°, 0] in M X R and let f : U -* M X R be an open C AT imbedding equal the identity on M X (~°°, 0] ; let D C M be a closed set and let e : M X R [0, °°) be a continuous function positive on D X 0 . Then there exists a C AT e-isotopy h { , 1 , o f id\M X R fixing M X (~o°, 0} so that h x = f near D X 0 . I f f is the identity on a neighborhood o f C X 0] in C X R for some closed set C C M , then the isotopy ht , t < / , can be so chosen that the same is true o f i t . Proof of Lemma (in outline). Concerning smallness, it will suffice to show that the isotopy h t can be made to have support in e' 1(0 ,°°) and can simultaneously be made as small as we please for the fine C° topology, equivalently for the majorant topology, cf. §4 and App. C .. Using normality, find a closed neighborhood Y of DXO in MXR contained in the open set e' 1(0 ,°°) ,disjoint from the closed set M X R - f(U ) , and so small that f I {YH(CXR) } = identity . Define X = f _1(Y) . Construct ‘sliding’ CAT isotopies ot : MXR-> MXR , 0 < t < 1 , of id (a) (b) (c) (d)

| MXR such that: at = identity on M X R -(X C iY ) ; at(xXR) = xXR and a t (xX (-oo,0]) D xX(°°,0] for all t and x a,(M X (—oo,0)) D DX (-oo,0] ; at is as small as we please for the majorant topology.

42

R. Kirby and L. Siebenmann

Essay I

Consider the rule ft -

1 ’ 0< t< l

.

By (a) , ft is an isotopy of f through open imbeddings U -►MXR , with support in XHY and disjoint from CXO . By (b) , ft fixes M X(-°°,0] . By (c) , fj is the identity near DX(-°°,0] . By (d) ft is as small as we please for the majorant topology. By (a) again, f^U = fU and fj. = f outside X . Since Y = f(X) is closed, and f(U) is open we can define ht , 0 < t < l , by the rules ht(y) = f ft_1(y) for y € f(U) lH(y) = y

for y £ f(X) .

This is the isotopy required. It is as small as we please for the majorant topology. ■

A.4 GENERALIZATIONS There are two ways in which it is often useful to generalize the above results . Neither generalization requires much more than notational change in the proofs . (a) TOP can be enlarged to the category of continuous maps of metric spaces ; PL can be enlarged to the category of piecewise-linear maps of locally compact polyhedra . (b) (Respectful versions) . One supposes W equipped with a family { Wa } of closed subsets . Then one insists in hypotheses and conclusions that each isotopy mentioned respect each set Wa or Ma X R as the case may be . For our purposes {W^} is usually the set o f fibers of a CAT submersion p : W -* A to a CAT object A .

43

Appendix B : S U B M E R G IN G A PU N C TU RED T O R U S

This contains verbatim a letter from J. Milnor of October, 1969 , which gives an elementary construction of a submersion of the punctured torus Tn—point into euclidean space Rn . It is used in §3 . A different elementary construction was found by D. Barden [Bar] [Ru] earlier in 1969 , and another by S. Ferry , [Fe] 1973t Milnor produces a smooth C°° (= DIFF) submersion . A secant approximation to it in the sense of J. H. C. Whitehead [Muj , §9] provides a piecewise-linear (= PL) submersion . “ Let M be a smooth compact manifold . HYPOTHESIS . M has a codimension 1 embedding in euclidean space so t h a t , for some smooth disk D C M and some hyperplane P in euclidean space , the orthogonal projection from M —D to P is a submersion . THEOREM . I f M satisfies this hypothesis , so does M X S 1 . It follows inductively that every torus satisfies the hypothesis . PR O O F. Suppose that M = M ^ - ^ embeds in R^ so that M —D projects submersively to the hyperplane Xj = 0 . We will assume that the subset M C R^ lies in the half-space x^ > 0 . Hence , rotating R^ about R ^—1 in R^+ ^, we obtain an embedding (x, 0) +> (x! x^_j , x^ cos 6 , x^ sin 0) of M X S' in R^+ l . This embedding needs only a mild deformation in order to satisfy the required property . Let ej ,...,6^+2 be the standard basis for R ^+ l . Let tq be the rotation e^ +»• e^ cos 0 + e ^ j sin 0 , e^ +* e^ for i < k , e^+ j

- e^ sin 0 + e ^ j cos 0 .

Let n(x) = n 1(x)e1 +"■+ n^fxje^ be the unit normal vector to M in r K t And still another by A. Gramain [Gra] 1973

44

R. Kirby and L. Siebenmann

Essay I

For x €= M-D we can assume that nj is bounded away from zero . Say n, > 2a > 0 . Suppose that M lies in the open slab 0 < xk < j3 of . Choose e > 0 so that the correspondence (x, t) ++ x + tn(x) embeds M X ( -e , e) diffeomorphically in this slab . Choose a smooth map t : S1 -* (~e,e) so that

2k

The required embedding M X S1 -*■

0

is now given by

( x ,0) -w r0(x + t(0 )n(x)) .

Computation shows that the normal vector to this embedding is p/llpll where p(x, 0 ) = (xk + tn k )r0 (n) Let v = ej - a e k + j . Then p • v = A + B where A = (xk + tnk )(nj - a sin0 nk ) and B = a cos0

> 0 .

Appendix B. Submerging the punctured torus

45

Thus if x G M - D we have A > (x^ + tnjc)(2a - a) > hence p • v > 0 . have

0

On the other hand , for any x G M, if 0 = 0 , we A>-0,

B > a(2p/a) .

Hence p • v > 0 for 0 = 0 , and therefore p • v > 0for allsufficiently small 0 ; say for 101 < r? . It now follows that the complement (MX S ‘ ) - ( D X [r?, 2ir- rj ]) projects submersively to the hyperplane v-^ . This completes the proof .

46

Essay I

Appendix C. M A JO RA N T A P P R O X IM A T IO N

This appendix collects some basic observations about majorant approximation of functions, and it proves a theorem that provides a local criterion for majorant approximating a ‘good’ function by a ‘better’ function. Let X and Y be topological spaces. If V is an open cover of Y and f , g are two continuous functions ( = maps ), we say that f and g are V -near if for each point x in X , both f(x) and g(x) belong to some one set of the cover Ir . We topologize the set C(X,Y) of continuous maps X - * Y in terms of the sets N( f , l r ) = ( g £ C ( X , Y ) | f and g are If -near} by defining a set N C C(X ,Y ) to be open if for each f E N there exists V so that N(f,V) C N . This is called the ( target) majorant topology. Fact 1 . I f Y is a fully normal topological space, each set N( f , \ s ) is a neighborhood o f f ; then fo r fixed f the sets AY/,If) form a basis o f neighborhoods o f f . P roof: Full normality means precisely that, for any open cover If , there exists an open cover lb = (W j} such that the starred cover lb* refines Ir . t (By definition lb* = {W*} , where W* = Uj (Wj I W j n W j * * } . ) Then observe that h E N ( g , l b ) =*■ N(h, lb) C N( g , b ) . It follows that, for any N C C(X ,Y ) , the subset NA of all g in N , such that there exists a N(g,\r) C N , is an open set. This in turn implies that each N(f ,lr) is a neighborhood of f . ■ Fact 2 . For f & C ( X , Y ) and g E C( Y, Z) the rule o f composition (f,g) +>- g f e C(X. Z) is a continuous mapping C ( X , Y ) X C ( Y , Z ) -*■C( X, Z) provided Z is fully normal. $ Full normality (as defined above) is easy to establish from metrizability. It also follows from the existence of fine partitions of unity (by regarding them as maps to nerves as in the proof of C .l below ). Two equivalences are worth noting . 1) full normality & paracompactness o existence of fine partitions of unity . 2) For spaces that are regular (all points have small closed neighborhoods), full normality is equivalent to paracompactness (= existence of fine locally finite open covers). See [StOj] or any treatise, shunning superfluous separation axioms.

Appendix C. Target majorant approximation

47

Proof: Given N(gf,lr) , find an open cover lb such that In* < If . Then f 'E N(f,g‘‘lb) and g 'E N (g,lb) imply gf ' E N(gf, lb) and f'g 'E N(gf',ln) , whence g' f ' E N(gf, in*) . ■ Fact 3 . The group o f homeomorphisms H(X) C C ( X , X ) is a topological group, fo r the inherited majorant topology, provided again that X is assumed to be fully normal. Proof: Given fact 2 we have only to verify the continuity of the rule f +* f _1 . But f -1 ,g-1 are 1/-near f - 1g, id are Ir-near g ,f are f(V)-near . ■ In case Y is a metric space there is an alternative definition of our topology on C(X ,Y ) as follows. Call positive continuous functions 5 :Y - * ( 0 ,° ° ] majorants , and for f in C(X ,Y ) define Ng(f) =

{ g E C(X ,Y ) | d(f(x),g(x)) < 5(f(x)) for all x in X } .

Fact 4 . As 5 varies, the sets N$ ( f ) give a basis o f neighborhoods o f f in C( X . Y ) . Proof: (i) Given 8 , one has N( f , V) C Ng (f) for any cover V of Y such that, for each V E V , diameter (V) < inf (5 (y) | y E V } . To obtain such a Ir , we simply cover each open set 5 _1(l/n ,°°) with balls of radius < l / 2n . (ii) Given V , one has N§(f) C N(f,lr) for any majorant 5 such that, for all y E Y , the ball B g ^ fy ) of radius 5(y) about y lies in some V E If . T o obtain such a 5 first find a locally finite refinement lb = {Wj} of If (recalling that metric =*• paracompact) ; then define : Y -*• [0,°°) to be the distance to Y-Wj ; and finally let 5(y) = max §,(y) . Since lb is locally finite the maximum 5(y ) exists i and is continuous. ■ The following lemma shows that, in case f : X -> Y is proper (i.e. K compact => f _1K compact) , one could as well replace majorants 5 on Y by majorants 5' on X in the definition of Ng(f) , substi­ tuting < 5 '( x ) for < 5 ( f ( x ) ) . Lemma . L et f : X ~*Y be a proper continuous map to a metric space Y , and let 8 f : X -+ (0, 00 ) be continuous. Then there exists a contin­ uous map 8 : Y - > ( 0 , ° ° ) such that b ( f ( x ) ) < b ,(x) for all x E I . Proof . Since Y admits fine partitions of unity to help us build 5 , it is enough to check that each y E Y has a neighborhood Vy such that inf (5 '(x ) | x E f ” 1 (Vy) }> 0 . Since f _1(y) is compact, it has an open neighborhood Uy such that inf (5 '(x ) | x E Uy } > 0 . But,

48

R. Kirby and L. Siebenmann

Essay I

as f is proper, f maps closed sets to closed sets, so we can take Vy = Y -f(X -U y ) . ■ Putting these introductory observations aside, we proceed to formulate the promised approximation theorem. Let X and Y be topological spaces. We shall be working with the collection § of continuous maps f : U -►V where U is open in X and V is open in Y . Letting S(V) denote the set of all such f having as target the set V , We observe that the rule V +* g(V) , together with the restriction maps S(V) -*■ S(V ') for V' C V , constitute a sheaf o f sets t on Y*, that is also denoted by § . Let £ ^ C S C § be two subsheaves of § . (Thus, for example, &(V) is a subset of S(V) for all open V C Y . ) Also let 7r : Y -* Y* be a continuous map to a space Y* . It serves to give a second, coarser topology on C(X,Y) ; the simplest case Y = Y* with n = identity is perhaps the most usual, and we choose our language to conform to it. We pose the following Approximation Problem . Given f : X - > Y in &(Y) and given an open cover Ir o f Y*, under what conditions can we fin d a map g : X -> Y in &+(Y ) such that f and g are \s-near in the sense that fo r all x in X , the two points f ( x ) and g(x) lie in some one set ir '1 V with F G l t ? Example . Given a homeomorphism f : X -> Y of smooth manifolds and open cover If of Y*, when can we find a diffeomorphism g : X -*■ Y that is V -near to f ? This question fits into the above frame­ work when one defines £ [respectively &+ ] to be the subsheaf of § consisting of all maps that are homeomorphisms, [respectively diffeo­ morphisms]. The answer we can give in general terms, is merely that it usually suffices to verify the strongly relative local criterion below. Let be t A sheaf of sets 6n (or over) Y is a contravariant functor § from the inclusions of open subsets of Y to the category of sets, such that the following ‘unique pasting’ axiom is verified. For any open V C Y and any open cover {V j} of V the elements of §(V ) correspond bijectively via restriction to the indexed collections {xj | x^ E § ( V j) } such that for any pair of indices i,j the elements , Xj restrict to the same element of SCVjOVj) .

Appendix C. Target majorant approximation

49

a fixed open cover of Y* . (When Y* is Hausdorff ( = T2 ) and locally compact a popular choice for is the open sets in Y* that have compact closure.) The ‘strongly relative’ local criterion is stated as follows in terms of the data X , 7t:Y Y* , £ , £ + , : (9)

L et C and D be closed subsets o f Y* with D contained in some U E . L et V be an open neighborhood o f D - C. Consider any f : X ~ * Y in &(Y) that is in &+ near C t • Then there exists g : X -*■ Y in £ equal to f near C U ( Y * - V) such that g is in &+ near C U D .

APPROXIMATION THEOREM C.l . Given the data X , ir: Y -> Y* , £ , , as presented above, suppose that (9) is verified. Provide that Y* is fu lly normal and paracompact. Then, given f : X - * - Y in £ , together with any open cover If o f Y* , there exists f + : X Y in &+ , such that f and f + are If -near. Furthermore suppose that f is in £ + near a closed set C C Y* . L et D be any closed set in Y* , and V any open neighborhood o f D ~C . Then there exists g : X -*■ Y in £ , If -near to f , such that g is in &+ near C U D and equals f near C U ( Y* - V) . Example . In § 4 , this theorem can provide part of the proof of the Concordance Implies Isotopy Theorem. With the notation used there, Y* will be the manifold M , X and Y will be IXMXI and n will be the projection to M = Y* . For V open in Y , an element f : U -*■ V of £ (V) will be a homeomorphism respecting projection to the first interval factor, and equal to the identity both on U H (OXMXI) and near U H (IXMXO) ; this f lies in &+ (V) if it gives a CAT iso­ morphism U n ( l X M X I ) r to V n ( l X M 2 XI) . Thus a f : X- » - Y in &+ is a topological isotopy of id | (MXI) rel MXO to a CAT isomorphism (MXI)p -*■ (M ^XI) . Rem arks. We have made the approximation theorem just general enough to cover the handful of applications in these essays. One can generalize it by allowing £ , &+ to be arbitrary sheaves of sets, given together t This means that for some open neighborhood f ”^7T~^N^ ^7T~^N^ , obtained by restricting f , lies in ‘abuse’ o f language will persist.

of C , the map . Similar

50

R. Kirby and L. Siebenmann

Essay I

with sheaf morphisms § that are not necessarily subsheaf inclusions. This generalization is somewhat more awkward to discuss t , but it permits one simplification: the (generalised) case where Y = Y* and 7r = identity immediately implies the most general case. PROOF OF THE APPROXIMATION THEOREM . We need only prove the first assertion of C.l ; the stronger asser­ tion involving given subsets C , D and V of Y* can then be deduced by applying the first assertion to suitable subsheaves &' and &+ of & and &+ , namely: &' = the maps in & equal to f near C U (Y*~V) (wherever defined).

= the maps in S ' that lie innear D (wherever defined). This works because the local criterion (9) is verified for S ' and S+ in place of S and S + above. To prove the first assertion we begin by using paracompactness of Y* to find an open cover to = (Wj) of Y* that refines the two covers and V . Since Y is fully normal as well as paracompact, there exists a parti­ tion of unity {ojj} on Y* with (0,1 ] C Wj for each i . This can be interpreted as a continuous map « : Y*

| Uj |

to the nerve of the cover to ; namely the one such that, for each y in Y* and each index i , the point a(y ) has cq(y) for its barycentric coordinate with respect to the vertex wj corresponding to Wj . The appropriate topology for | to | is the ‘weak’ topology t , for which a set is closed if and only if it meets each closed simplex of 1to | in a closed set. The continuity of the functions cej implies that o T ^ a) is closed for each closed simplex o and that a. is continuous over each such a . Local finiteness of the partition { a j } implies local finiteness ‘value-wise’ of the collection ( a -1 (a)} in Y* —i.e. , given x in Y* t For exam ple, f : X “* Y in £ comes to mean an elem ent f of &(Y) whose image in 8(Y ) is a continuous map X “*Y . $ The three standard topologies on | Ud| (cartesian, m etric, weak) coincide precisely if to is star-finite . The covering to can always be chosen to be star-finite precisely if Y* is strongly paracom pact. Every locally separable m etric space, and also every locally com pact H ausdorff space is strongly paracom pact, see [N a 2 , p .172, p .182 ] .

Appendix C. Target majorant approximation.

51

there is an open set U containing x so that { a - 1( o ) n U } ( o varying) is a finite collection. Thus, a set in Y* is closed precisely if it meets each set a -1 (a) in a closed set. It follows that a -1 (A) is closed if A is closed; so the continuity of a is now explained. Writing I lb I = K for convenience, suppose for an induction over the skelleta , r > —1 , that we have built fr : X- »- Y in £ that is in £ + near KV) . The induction begins with f_j = f . For any closed simplex a of | lb I we have a standard open "spindle neighborhood’ N(a) of in ta = ( o~da) , namely the open star of in ta in the first barycentric subdivision of |lb 11 .

Observe that, if N (a) meets N ( t ) , then either a < r or r < a

.

For each closed (r+l)-simplex a of llbl= K , apply the criterion (?) with the substitutions C 4* a r 1K(1’) , D +> a -1 (a) , V a - I N(a) , to obtain a map ga : X - * Y in £ equal to f r near Y* - c r 1(N(a)) and lying in £ + near c r ^ K ^ U a ) . Since the open sets a - 1 N(a) are pairwise disjoint, we can $ piece together a unique map fj+ j : X - * Y such that f r+1 coincides with f r near Y* — Nr+l where Nk = U { N(a) I dima = k , a < K } and fr+l coincides with gG over a r ^ f a ) . Clearly fr+1 is in over a~1(K^r+1^) . This completes the induction to construct a sequence f0 , fj , f2 , ... of maps X ^ Y . f If O has vertices wQ , ..., w§ then N(a) is the set of all points z E |j^| such that if W| is not a vertex of a then w-(z) < min {wQ(x) , ..., w s(x) } . This N(cr) is clearly open in the weak topology; it is often not open for the other two. t The collection of all Ot l N(o) , O a simplex of | Ip | , is easily seen to be a locally finite open cover of Y* , because lb is locally finite. But beware that, in general, the open covers (N (a )} and (N r ) of | lb I are not locally finite.

52

R. Kirby and L. Siebenmann

Essay I

The collection { a-1 Nf | r = 0 , 1, ...} is locally finite in Y* because to is locally finite . Thus any point y G Y* has a neighborhood Vy meeting only finitely many sets a - 1Nr . It follows that the sequence fr i Vy , r = 1 , 2 , , is eventually constant. This shows that g = lim fr is a well defined map X -*■ Y in & . It clearly lies in &+ . Finally, we show that g is V-near to f . For any point x G X , suppose the eventually constant sequence f ( x ) , fQ( x ) , f j ( x ) ,... moves just before fr(n)(x) , n = 0 , 1 ,... , s . Then fr(n_!)(x) and fr(n)(x) lie in some a - lN (an ) . Thus N(an ) meets N(an+1) , n = 0 , . . . , s - l , and clearly this implies that aQ < a l < ... < as . I f Wj is a vertex of oQ , then the barycentric coordinate for wj is positive on N(a0) U... U N(as) . Thus a-_1(0 ,l] contains both f(x) and g(x) , i.e. f and g are O-near, where G = {Oj- 1(0 ,l]} . Since Q < ID < V , the proof is complete. ■

Source majorant approximation (supplementary remarks) . There is a companion to the majorant approximation problem dis­ cussed above; it arises when one wishes to measure progress in the source space X rather than the target Y . To simplify we assume henceforth that X and Y are metric and that X is locally compact. One gives to C(X,Y) the source majorant topology , letting a basis of neighborhoods of f : X -*■ Y be the sets N (f;e) = { gG C(X,Y) I d(f(x),g(y)) < e.(x) for all x in X } where e : X -*■ (0,°°) ranges over all positive continuous functions (majorants). This is also known as the ‘fine Whitney topology’ or the ‘fine C° topology’, cf. [ Mu j , p.29] . It is in general finer than the (target) majorant topology first considered, as the example X = R , Y = [0,1 ] amply illustrates. Of course, when X is compact, both majorant topologies coincide with the compact-open topology. Fixing attention on the source space X , let U be any open subset of X , and denote by S(U) the set of continuous functions U -►Y . With obvious restriction maps, the sets S(U) form a sheaf of sets on X as U varies; it is denoted § . We consider subsheaves

Appendix C. Source majorant approximation

53

S + C 8 C § of ‘good’ functions & and ‘b etter’ functions &+ . Pause here to note that & could for example be the open immersions to Y , but it could not be the open embeddings. (Just inspect the basic axiom for sheaves on X !) Approximation Problem: Is &+(X) dense in &(X) fo r the source majorant topology? To be sure, one can pose this problem in a strongly relative form, but, as noted in the proof of C .l, such generality is illusory. We indicate how to solve this problem in two steps. Step I). &+(X) is indeed dense in &(X) provided the following strongly relative local criterion is verified: (6) Given f : X - * Y in & that is in &+ near a closed set C C X , and given a compactum D C X and an open neighborhood V o f D , there always exist approximations g to f that are in &+ near C U D and are equal to f near C U (X - V) . Note that this property (d) remains just as strong if we suppose that V has compact closure, or if we suppose that V is contained in some set of a fixed open covering of X . It is still unsatisfactory that in (d) we should have to hypothesize approximations; nevertheless it is encouraging to note that it does not m atter which topology we use on £(X ) , since V may as well have compact closure. The proof of I) is left as an exercise, cf. [Si j 2 >Proof of 6.3 ] . Hint: It can exploit a well known ‘concentric annuli’ trick to express X as Xq U Xj where Xq and Xj are closed sets each a discrete sum of compacta. (For X = R2 these compacta could be concentric annuli.) Step II). Verification of (d) . Frequently, when one is faced with a specific approximation problem, the criterion (d) is directly verifiable . This will be the case for approximation of maps by ones transversal to a TOP microbundle

54

R. Kirby and L. Siebenmann

Essay I

[III, § 1] ; of for approximation of continuous functions by TOP Morse functions [III, §2] . However, it is reassuring to observe that C.l reduces (d) to a criterion not involving approximation. Assertion: (d) is implied by the weaker version (d)* where g is no longer required to be an approximation to f but instead one in­ sists that g(V) lie in a prescribed neighborhood o f f(V) . Proof: Regard (d) as a target majorant approximation problem as follows. With the notation of (d) let £ ! be the sheaf on Y consist­ ing of continuous maps h : U' -*■ V' with U' open X and V' open in Y , so that U' V C> Y is in £ and h equals f near U 'n ( C U ( X - V ) } . Also let C £ ! be the subsheaf of maps h such that U '^ V ' C ^ Y is in £+ near U 'n ( C U D ) . Then in view of C.l , the criterion (d) for £ + C £ is implied by (9) for £ ^ C £ ! . This (9) is in turn impjied by (d)* . ■

Essay 11

D E F O R M A T IO N OF S L IC E D F A M IL IE S OF M A N IF O L D ST R U C T U R E S

by

L. Siebenmann

L. Siebenmann

56

Section headings in Essay II

§

0. Introduction

§

1. Bundle Theorems

§

2. Sliced concordance implies sliced isotopy

Essay II

57

§0. INTRODUCTION A sliced family of CAT ( = DIFF or PL ) structures on a mani­ fold M with parameters in A (a simplex or any CAT manifold) is a CAT manifold structure T on the product AXM that is sliced over f A in the sense that the projection (A X M )p-^A is a CAT submersion. Then, for each point u in A , T gives to uXM a CAT structure Tu . Beware that a family T is decidely more than the collection { Tu I u G A } of its ‘members’, whenever dimM > I . % A key result of §1 is that the projection p j:(A X M )p -* A is a CAT bundle projection provided dimM ^ 4 ¥= dim9M . When A is contractible, this means that there is a CAT isomorphism h : (AXM)p AX(M7 ) that is sliced over A (in the sense that p jh = pj ) . This result requires no dimension restriction if M is com­ pact, since every proper CAT submersion is easily seen to be a CAT bundle projection. In the non compact case we shall use an engulfing procedure in order to revert to the compact case; a simple covering trick is involved. As first sketched in [KS4 ] , this argument enjoyed con­ siderable excess generality that was then intended to let one pursue the study of limits of homeomorphisms, begun in [Si j j ] (see [S ijj,en d ] for an example). We retain that generality here, but we have tampered with the original proof to make it self-contained, and in addition we have found a short-cut for the case at hand. R. Lashof and D. Burghelea have in the meantime devised an alternative argument [B uL ,I]. The above bundle theorem of § 1 is extensively used in classifying sliced families in Essay V .

t Beware of confusion with the notion of a “slice” in the theory of group actions. $ There is for example a DIFF structure T on T ] XS 6 so that Tu is standard for all u e A = T 1 = R/Z , but ( T 1 XS6 )r ^ T 1 XS 6 . To build T , let it be standard near the complement of a disc (—lA?A) X R 6 in T 1 XS 6 , with S 6 = R 6 U 00 ? on which disc we make it the image of the standard structure under a TOP Alexander isotopy H with compact support, to id I R 6 from h I R 6 , where h is an exotic DIFF automorphism [Mi^Jof S 6 rel 00 ; more specifically H can map (t,x) +* (t , 8 t h ( x / 8 t) for 0 < t < 1/4 and (t,x)+*(t,x) for - I / 4 < t < < 0

.

58

L. Siebenmarm

Essay II

In §2 we refine the bundle theorem in case A is a simplex or cube, showing that there is a majorant-small TOP isotopy hq , 0 < t < 1 , of id I(AXM) sliced over A , so that hj T is of the form AX2 /with 2 a CAT structure on M . If T was of the form AX2 near AX2 , where A is a retract of A , this isotopy can be the identity near AXM . There are further refinements, giving, in all, a complete analogue for “sliced” concordances of the Concordance Implies Isotopy Theorem of [I, § 4 ]. This result of §2 is useful in converting information presented in terms of such sliced concordances T into geometrically useful isotopies. For example it lets one approximate topological isotopies of the identity by CAT isotopies of the identity. The 1972 version of this essay contained two further sections t that studied the space of CAT concordances rel 9 of the structure of a CAT manifold M , by generalizing directly the proof of the Concordance Implies Isotopy Theorem of Essay I (cf. the announce­ ment [KS4 ]). In particular, provided d im M > 5 , this space (semisimplicially or semi-cubically defined using sliced families) is contrac­ tible for CAT = PL and 1-connected for CAT = DIFF . The reader will find that such results can equally well be obtained by combining §2 with the classifications of Essay V . Certainly the approach via Essay V is more sophisticated; but on the other hand it is probably more enlightening.

t To which, no doubt, stray references persist. Perhaps these sections will be published in Manuscripta Mathematica .

59

§1. B U N D LE T H E O R E M S

A CAT map p :E ->B (for CAT = DIFF , PL , TOP , T2 + ) is called a CAT submersion if for each point x in E there is a CAT object U , an open neighborhood BQ of p(x) in B , and a CAT open imbedding f:B 0 X U -*E onto a neighborhood of x such that pf is projection B0 XU -HB0 C B . Such an imbedding f is called a submersion chart. When f is normalized so that U is an open subset of p_1p(x) and f(p(x),u) = u for u in U , then we sometimes call f a product chart about U for p . Note that every fiber of a CAT sub­ mersion is a CAT object. Also, for any open set EQC E , the restric­ tion p |E Q is a CAT submersion, and its image p(EQ) is open in B . Using the CAT isotopy extension principle ( CAT =£ T2 ) one can show that any compactum in a fiber p~*(y) of a CAT submersion p is contained in the image of some submersion c h a r t . H e n c e , if the submersion p has compact fibers and is closed, i.e., maps closed sets to closed sets, it follows that p is a locally trivial CAT bundle projection. The aim of this section is to give conditions assuring that a CAT submersion with noncompact fibers is a locally trivial CAT bundle projection. Our main result is a technical one, formulated so as to avoid restric­ tions of dimension or category.

t T2 is the category of all Hausdorff topological spaces and continuous maps . PL could be enlarged in this section to contain all locally compact polyhedra and piecewise linear maps. £ For CAT = T2 this applies nevertheless, if for example p” *(y) is locally triangulable; see [ S i ^ l f ° r the relevant isotopy extension principle. ^ See [ S i ^ 6.14, 6.15] for this reduction to the CAT isotopy extension theorem. But note that, for CAT = DIFF or PL , very direct proofs exist. For CAT = PL , [III, Lemma 1.7] provides one. For CAT = DIFF , any sm ooth neigh­ borhood retraction r:E o “> p“ k y ) provides one since r is a DIFF submersion near p” *(y) .

L. Siebenmann

60

Essay II

TECHNICAL BUNDLE THEOREM 1 .1 . Consider a CAT object E ( CAT = D IF F , PL , TOP or T2 ) , equipped with (a) a CAT submersion p : E ^ A onto a C AT object A . (For u G A write Fu = p -1(u) fo r the fiber o f p over u . ) (b) a merely continuous map i t : E -+ R . (For a,b in R write F ja ,b ) = F ^C tt- 1(a,b) ; adopt similar notation with any subset o f R in place o f the open interval (a,b) ) , We suppose verified an ‘engulfing’ condition somewhat weaker than the statement that each Fu(a,b) is a CAT product with R : (*) For any u G A and any pair o f integers a < b , there exists a CAT isotopy ht ,0 < t < = l , o f Fu with compact support in Fu (a-1 ,b + l) such that hj F (-°°,a ) D F (-°°,b ] For CAT = T2 we m ust assume that each fiber Fu is locally triangulable t , and that, fo r each pair o f integers a < b , the set Fu(a,b] is connected and compact. $ Also we m ust assume that A is paracompact and $ o f covering dimension d < °° . Under these hypotheses p : E -*■ A is a locally trivial CAT bundle. Remarks on 1.1 . 1) If CAT = DIFF , PL , or TOP the special case where A is a simplex (or cube) implies the general case. 2) The TOP version is clearly subsumed in the T2 version. 3) The T2 version could probably be proved differently in case E and A are complete metric spaces, by using a selection theorem of E. Michael [ Mic ] . 4) Whether the hypothesis dimA < °° is necessary is unknown. For the applications of the T2 version to the closure of the homeo­ morphism group of a manifold envisaged in [Si j j ] , it is a most undesirable restriction. f Or more generally verifies the general isotopy (local) extension principle of [S i12 ; § 0 , § 6] for parameters in A . J This compactness, in fact, follows easily from (*) . i i.e., each covering of A admits a refinement whose nerve is a simplicial complex of dimension ^ d , and d is the least such number. One could equiva­ lently use finite coverings o n ly, see [Nag, p .2 2 ] [Dow] .

§1.

61

Bundle theorems

5) The requirements on some new category CAT , in order that 1.1 hold, could be axiomatized without difficulty. 6) The fibers of p may indeed not be a product with R . For examples in dimension > 5 detected by class group obstructions, see [Sig] . A suitable 7r : Fu -> R can be gotten by identifying Fu with the natural infinite cyclic covering of a band formed by gluing together the ends of Fu , see [Sig,§2] 7) In stating this theorem in [KS4 ] , we forgot to mention the isotopy ht to h j in (*) ; our proof definitely needs it. PROOF OF THEOREM 1.1 . UCertain parts of the proof not needed to establish the im portant DIFF and PL Bundle Theorem 1.8 below are marked off by double brackets. J We shall show that p factorizes as p = p '° q :E ^ - B ^ > A

,

where q is a CAT infinite cyclic covering, classified by a map f : B -»• S1 , while p' is a closed CAT submersion with compact fibers, and hence a CAT bundle projection . It follows that p is also a CAT bundle projection. Proof: As the condition is local in A , we can assume p ' is trivial, i.e., B = F'XA so that p' becomes projection to A . To verify local CAT triviality of p at u E A , consider Fu = q- 1(F'Xu) and the infinite cyclic covering qQ : FUXA — -^ -)xl-d> F'XA = B

•

This covering qQ is clearly classified by the map fQ : F'XA S1 sending ( x ,v ) +> f(x,u) , and fQ agrees with f on F'Xu . Since close maps F' S1 are canonically hom otopic, these classifying maps f and fQ are hom otopic when we cut A down to a small neighborhood of u . T hen, by the bundle hom otopy theorem [Hus] , q and qQ give isomorphic coverings of F'XA , and so E = FUXA making q correspond to qQ , and hence p'q = p correspond to the projection P % = P2 : FUXA A * Tfds proves the wanted CAT local triviality . The proposed factorization of p will be obtained by patiently constructing an infinite cyclic covering translation on E .

62

L. Siebenmann

Essay II

PART A). The following global sliced version # (y) o f the engulfing property (*) holds true, with 7 = dimA+1 = d+1 . # (7 ).'

For any pair o f integers a K b , there exists a C AT isotopy h f . O K t K l , o f id\E sliced over A , such that h jir '1(~°°,a) D ir '1 (-°°,b ] , and the support o f the isotopy h ( is contained in it' 1[a~y,b+ y] .

To prove this we must consider also a weaker condition # r( y >c) for C a closed subset of A and r€E(0,°°] . This differs only through weakening the inclusion condition to be hj 7r_1(-°°,a) D 7r_1(-°°,b ] H p-1 C

,

and suppressing even this condition except when [a~ y,b+ 7 ] C [-r+ 2 ,r—2]

.

Clearly # 00( y , A ) = # (7 ) ; also # T(y,C) => # s(6 ,D) if r > s , y < 8 , and C D D . Addition Lemma 1 .2 . The conditions # r ( y,C ) and # / 8 ,D) together imply the condition # r ( y+ 8, C U D) . Proof o f 1.2 : Given a pair of integers a < b , the isotopy asserted by # r(y+8 ,C U D) can be defined as h ^ h fo h f

, 0 [0,1 ]

,

with support contained in Au , we then define a map h t on Im aged) ht :v?(x,v) +* -[0 ,l] with ay(Cy) = 1 , each having as support a set Ay contained in U ^ y ) , so that Aj = UjAy is again a disjoint discrete sum. Then the isotopies corresponding to the ay , j varying, compose disjointly to establish

#r(i,q ) • Finally, as promised, d applications of the addition lemma 1.2 deduce # r (d + l,A ) and hence # (d + l) . ■ I It remains only to prove facts 1 and 2 in full generality. To prove Fact 1 in full generality, one needs: Lemma 1.3 . Let 0 , sliced over Ak and equal the identity near Ak XC , form a semi-simplicial group ( css group). Any css group enjoys the Kan extension condition [May, p. p. 67] , which is just the statement italicized for A = simplex . When A = cube , a similar, less formal argument works; it is left as an exercise. The following alternative version of 2.1 is sometimes more convenient. ALTERNATE SLICED CONCORDANCE THEOREM 2.2 . (same data) L et A C A be any continuous deformation retract o f A . f Suppose T = AX2 near AXM , and suppose dimM ^ 4 ^ FdimFdM - C ) . Then there exists a sliced TOP e-isotopy h( , 0 < t < 1 , o f id \(AXM) , rel AXM U AXC , to a C AT isomorphism h{: AXM^^(AXM)r . Remark 2 .3 . Along with 2.1 and 2.2 , (or from them) one can obtain more complicated versions of each, parallel to the most general Concordance Implies Isotopy Theorem of [I, §4] , involving a closed set D in M and an open set V 3 D . One insists that ht be rel A X (M-V) , while merely requiring hj to be a CAT isomorphism near A X (C U D ) . The arguments proving the Concordance Implies Isotopy Theorem in §4 of Essay I , amply show that to establish 2.1 and 2.2 (and also 2.3 ) , it will suffice to prove the following HANDLE LEMMA 2.4 . Make the hypotheses o f 2.1 ([ or 2.2 ]], but forget about the map e . Suppose further that C = 9M ,and let D C M be a compact set. t

Readers familiar with shape theory will find that it is enough to suppose

A is a compactum in A having the shape of a point.

75

§2. Sliced concordance implies isotopy

Then there exists a compact support sliced isotopy h { , 0 ^ t < : l , o f i d\ ( AXM) , rel AX9M and fixing AXM , such that hj : AXM -+ (AXM jp is a C AT imbedding on and over a neighbor­ hood o f A XD . ([ I f we have the hypotheses o f 2.2 , the isotopy h t can be the identity on a neighborhood o f AXM ( independent o f t ). 1] Remark: The case (M, D) = Bk X(Rn ,Bn) , k+n = m , of a true handle is really no easier to prove once 1.8 is available. PROOF OF THE HANDLE LEMMA . To save notation we identify M = M ^ making M

CAT .

The Bundle Theorem 1.8 provides a sliced CAT isomorphism ip : AXM —* (AXM)r

.

If we have the hypotheses of 1.1 , every sliced CAT autom orph­ ism of AXM extends to a sliced CAT automorphism of AXM . Thus we can arrange that (1)

3 straightening theorem (stated in Appendix B) to help reduce to the corresponding PL result 1.6 , which we then easily prove ‘by hand’ . Such a TOP transversality result would seem at first sight to be of limited interest —although it is pleasant enough to state and not unreasonably difficult to prove —simply because normal bundles for TOP submanifolds often fail to exist or fail to be (isotopy) unique (see [RS4 ] , [S te m ]) . On the contrary , the viewpoint that evolved in discussion with A. Marin maintains that it should (or at least can) play the central role . This is well-illustrated already in the PL category where we observe (1.8) that the Rourke-Sanderson imbedded block bundle transversality theorem follows from the PL imbedded microbundle transversality theorem —which , in turn , we have deduced from the PL analogue of the Sard-Brown theorem . In the TOP category A. Marin has somewhat similarly obtained a general transversality theorem for submanifolds through replacing block bundle

84

L. Siebenmann and R. Kirby

Essay III

transversality by a stable microbundle transversality • (Recall that stably normal microbundles do exist and are isotopy unique , cf. [IV , Appendix A] . ) We conclude by introducing Marin’s work , which is still too difficult to present here in fu ll. In Appendix C we whall present an ad hoc transversality lemma for immersions designed to adapt directly smooth surgery methods to TOP manifolds . Where it applies many readers will prefer it to the more ambitious approach of this section . We draw the reader’s attention to two general position theories , perhaps as im portant as the transversality results we present here . One for polyhedra in codimension > 3 is mentioned in Appendix B (conclusion) ; another for arbitrary (!) closed subsets is mentioned in Appendix C (see C.3 and [Ed^J ) . M IC R O B U N D L E T R A N S V E R S A L IT Y FOR M APS

Recall that a TOP n-microbundle £n over a space X can be defined as a total space E(£) D X together with a retraction p : E(£) -*■ X t h a t , near X , is a submersion whose fibers p~ *(x) , x £ X , are (open) n-manifolds . A DIFF n-microbundle over a manifold is defined similarly working within the category DIFF of smooth maps of smooth manifolds . Similarly a PL microbundle over a polyhedron . Consider a pair (Y,X) of topological spaces where X is closed in Y and equipped with a normal microbundle £n , i.e. E(£n ) is an open neighborhood of X in Y . Consider also a continuous map f : Mm -*■Y from a TOP mmanifold to Y . Suppose f~*(X ) is a TOP submanifold L C M m and i>n is a normal n-microbundle to L in M such that f | E(i>) is a TOP microbundle map to E(£) (i.e. f gives an open TOP imbedding of each fiber of v into some fiber of £ ). Then we say that f is (TOP) transverse to £ (at v ) . We say f is transverse to £ on U open in M , if f | U is transverse to £ . Transverse near C means transverse on an open neighborhood of C . If Mm and £n are DIFF we define DIFF transversality of f at a DIFF normal microbundle v for L = f ^(X) . This means that f | E(i>) is by assumption a DIFF microbundle map to E (£ ). We do not however assume f : M -*■Y is DIFF ; it is convenient to allow it to be merely continuous outside E(i>). Similarly PL transversality . Let C and D be closed subsets of a (metrizable) TOP m-

§1.

Microbundle transversality of maps

85

manifold Mm , and let U and V be open neighborhoods of C and D respectively . Let £n be a normal n-microbundle to a closed subset X of a space Y . FIRST TRANSVERSALITY THEOREM 1.1. Suppose f :M m ->-Y is a continuous map TOP transverse to £ t on U at v0 . Suppose m =£ 4 =£ m —n , and either 3M C C or m —1 ¥= 4 =£ m —1—n .$ Then there exists a hom otopy f^ : M -*■Y , 0 < t < 1 , o f f0 = f fixing a neighborhood o f C U (M—V) so that f t is transverse to £ on an open neighborhood o f C U D at a microbundle v equal v0 near C . Furthermore , i f Y is a metric space with metric d , and e : M -* (0, °°) is continuous , then we can require that d(f|.(x) , f(x)) < e(x) fo r all x £ M and all t (0,1 ] . There is (in 1974) no reason to believe that the above dimension restrictions are necessary . ^ This question is in tight connection with the question whether Rohlin’s theorem (on index of almost parallelizable closed 4-manifolds) holds for TOP , see [Sig, §5] [Mat] [Sch] . We will say no more about the e-smallness condition . It can be carried through the proof , beginning with an application of Sard’s theorem to get a similar DIFF result involving e . On the other hand one can always deduce the e-condition from the version w ithout e , by exploiting the strongly relative nature of the theorem. Cf. [ I ,§4] and [ I , Appendix C ] . The idea of our proof is to use the Product Structure Theorem of [I] to reduce the proof to chart-by-chart applications of the following easy Theorem 1.2 (DIFF Transversality). Consider f : Mm ->■ Rn a continuous map o f a DIFF manifold M ; C,D closed subsets o f M ; U,V open neighborhoods o f C , D respectively. I f f is DIFF transverse on U to 0 (i.e. to the trivial micro­ bundle 0 Q .R n -^0 ) at v0 , then there exists a hom otopy ft : M ^ Rn , 0 < t < 1 , o f f = f 0 fixing a neighborhood o f C U (M—V) 50 that fj is DIFF transverse near C U D to 0 at a DIFF microbundle v equal to v0 near C . t Beware of saying transverse to X (ommiting £ ), see discussion above 1 .4 . ^ The case m—n < 0 is trivial to prove , even if m = 4 , (D. E p stein ).

86

L. Siebenmann and R. Kirby

Essay III

Proof o f 1.2. This theorem would follow easily from the Sard-Brown theorem [Mig] , if we had (1) assumed f everywhere D IF F , and (2) required for transversality near S C M only differentiability near f ] (0) IT S and surjectivity of the differential d ft of f x near f _1(0 ) n s . To see the difficulty related to (1) note that f is not necessarily DIFF near C . So we cannot simply make f DIFF everywhere on V w ithout changing it near C . The difficulty is not serious . Let C’ C U be a closed neighborhood of C in M . Alter f on E(t'o) U (V—C') = V' by a homotopy ft , 0 < t < l , to make it first DIFF on V' [Muj , §4] , then transverse to 0 on V' , cf. [Mig] , in the sense of (2 ). Neither change need alter f near C (because the transversality condition (2) is stable for the fine C ^ topology), nor outside V' ( since we can perturb less than e where e is a continuous function M -»• [0,°°) with e~~*(0, °°) = V ') . Now one sees that ^ is transverse to 0 in the sense of (2) on C' U V D C U D . Thus to attain DIFF transversality as we defined i t , we need only to equip the DIFF manifold L = f~ *(0) fl (C' U V) with a DIFF normal microbundle v equal to v0 near C . This amounts to simply finding a DIFF neighborhood retraction j* L C E -* L , equal to the projection r 0 of v0 near C , since any such r is a DIFF submersion near L by the implicit function theorem . ■ Another tool we need is the Pinching Lemma 1.3. L et vn be a microbundle over a paracompact space L . Consider D closed in L and a neighborhood Z o f D in EO) . There exists a hom otopy p t , 0 < t < l , o f id | E(>) respecting fibers o f v and fixing all points outside Z and all points in L so that p ~ * L is a neighborhood o f D . Proof o f 1.3. First suppose EO) = Rn , Z = Bn and C = L = 0 E Rn . Then the construction of p t is triv ial. Second suppose EO) is an open sub-microbundle of L X Rn with L = L X 0 . It is this case we shall use for 1.1 . Let f : L -* [0, oo) be a continuous function positive on D so that { (x,y) E L X Rn ; |y| < f (x) } C Z . Let rj : L -> [0, 1 ] be 1 near

§1.

Microbundle transversality of maps

87

I) and zero near f ^(0 ). Writing a^ for the hom otopy obtained for the first case we can define p^ on all L X Rn by Pt (x>y) = ( x ,? (x )a 7?(-x )t( f ( x r 1y )) if f(x) > 0 and by pj.(x,y) = (x,y) otherwise . (The continuity is obvious because of rj ) . In the general case p t can be an infinite but locally finite composition of homotopies obtained by the second case . We leave this unused generality to the reader . ■ PROOF OF 1.1 Step 1. The case M open in Rm , Y = E(£) = Rn and X = 0 . Proof: Step 1 follows immediately from the DIFF transversality theorem 1.2 as soon as we find a DIFF structure 2 ' on M such that for some open neighborhood N of f ^(0) n C , the microbundle i>0 n N : E(i>0 ) H N - > LQ n N is DIFF and f : M s ^ R n is DIFF transverse to 0 near C at ^ H N . Here L0 is the base space of "oTo find S ' we apply the Local Product Structure Theorem [I, §5.2] , to E(»>0 ) with the structure 2 inherited from M C Rm . For this we regard E(y0 ) canonically as an open sub-microbundle of Lq X Rn by the rule E(vQ) (p ( x ) , f(x)) (L0 X Rn , E(i»q ) , 2 , C n LQ , V' ) where V' is an open neighborhood of C in E(i>0 ) , whose closure in EO'q) is closed in M . Thus [I , §5.2] provides 2 ' on E(i>0) which extends ( by 2 ) outside V' C M to 2 ' on M C Rm as required . If m —n = 3 and LQ has closed compact components , [I , §5.2] does not immediately apply . We can quickly recast the argument however by first deleting from M the closed set F consist­ ing of all compact components of LQ meeting C . Then enlarge C to C' by adding a small closed neighborhood G of F and cut back U to an open neighborhood U' of C' such that all compact components of f —* ( 0 ) n U ' lie in F . Then we can repeat the whole argument with M-F , C '-F , and U '-F in place of M , C , and U, getting the wanted hom otopy w ithout encountering invalid cases of [I, §5.2] . ■

88

L. Siebenmann and R. Kirby

Essay III

Remark for m -n < :3 : Even the nonstandard cases of the Product Structure Theorem [I , §5] used in the above proof , viz. for structures on L ^ X R n , k + n > 5 , k < 3 , with 9L = 0 and L lack­ ing compact components if k = 3 , can be proved w ithout recourse to a microbundle classification of structures , by solving a sequence of handle problems in dimension n + k corresponding to the critical points of a proper Morse function on L lacking critical points of index 3 . See [K Sj] and [Sin ] . Step 2 . The case M open in Rm , Y = E (£ ), and £n a standard trivial bundle (over any space X ). XO n pi Proof: £ is X -*■ X X Rn -*■ X . So we can write f : M -»■ E(£) = X X Rn componentwise f = ( f j , f2) . By step 1 there is a hom otopy rel C U (M—V) from f2 : M -*■ Rn to a map f2 which is transverse to 0 on a neighborhood W of C U D at a normal microbundle v to L = f2_1(0) n w so that v = vQ on a neighborhood U, C U of C . We can assume W C U , U V . The resulting hom otopy from f = (f2 , f2) to f ' = (ft ,f2) does not solve our problem since f x is perhaps not constant on the fibers of v near L . To remedy this , find a neighborhood Z of L fl (C U D) in E(i>) which is closed in M , and , noting that EO) C L X Rn by x +» (p (x) , f2( x ) ) , apply the Pinching Lemma 1.3 to obtain a fiber preserving pinching hom otopy , 0 < t < 1 , of id | E(u) which extends by the identity outside Z to all M . Then (fjpj. , f2) , 0 < t < 1 , is a hom otopy of f* to f" = (f iP x , f2 ) . If N C Z is a neighborhood of L fl (C U D) so that p t (N) C L , the map f "= (fi Pi, f^) is transverse to £ at c H N since fjP ! is constant on fibers of v fl N . The hom otopy f* to f" is constant near C since f ' is already constant on fibers of v Cl U j (where f ' = f and v = i>Q) . Hence the homotopy f' to f" has support in Z -U j C W -Uj C V . (The support is the closure of the set of points moved) . Recall that W C l^ U V by choice . The composed hom otopy f to f ' to f" establishes Step 2 . ■ Now we let Y grow larger than E(£) . Step 3 . The case Mm open in Rm , and £n a trivialized bundle , i.e. E(£n ) contains X X Rn as an open sub-microbundle. Proof: Apply Step 2 with the substitutions

1. MicroburrdJe transversality of maps

89

Proof: Apply Step 2 with the substitutions M C D

f _ 1 (X X Rn ) , +* {C U f _ 1 (X X(R n - B n) ) } n f - 1 (X X Rn ) , D n f _ 1( X X R n ) ,

etc. to obtain a hom otopy which extends as the constant hom otopy outside f _ ^(X X Rn ) to the required hom otopy to transversality . The normal microbundle v obtained may at first be smaller than vQ near C if X X Rn # E(£) , but this is trivial to remedy by adding on to this v the restriction of vQ to a small neighborhood of C . ■ Next we allow £ to be arbitrary and M to be an open manifold . Step 4 . The case 3M = . Proof: Let X be covered by open sets X a , a in some index set , where £n is trivialized over each Xa , i.e. a microbundle map Xa X Rn C* E(£) is given extending the inclusion Xa X 0 = Xa C ,E (£ ). There is a locally finite collection of co-ordinate charts in M , R . » j = 1, 2, • - • such that D C Uj B™ , each R1-1 C V , and each set J p{f(Rr|1) H E(£)} C X lies in some set Xft , which will be denoted Xi-

Suppose now for a construction by induction on i > 0 that we have constructed a continuous map fj : M -> Y transverse to £ on an open subset Uj C M , at vx , where Uj D Cj = C U {B™ U • • • U B1? } . At i = 0 , we begin with UQ = U and fg = f . Apply Step 3 with the substitutions M- ^ Rj + j ; C Cj n R|+ j ; D- ^B™] ; V +> 2 B -T].1 Y +> (Y - E(£)) U E(£ |Xi+ j ) ; £ ^ £ | X i + 1 ; . Th e r e results a hom otopy ft' , i < t < i + l , constant outside Rf+i , an open neighborhood Ui + 1 of Ci + 1 , and a microbundle + 1 , such that f/+1 is transverse to £ on Ui + 1 at *>i + 1 , while Ui + 1 and + 1 coincide outside R™j . This completes the induction to construct f^ , 0 < t < oo , Uj and vx for 0 < i E Z . To complete the proof of Step 4 , we have only to let ft , 0 < t < 1 , be the unique conditioned hom otopy so that ft = ^ ( t ) j 0 < t < 1 , where a(t) = t/( 1—t) , and near any compact set let v equal vj for j large enough . ■

90

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Essay ill

Step 5 . The general case. P r o o f: We are now allowing M a non-empty boundary . One simply applies Step 4 twice , once to 9M , trivially if 9M C C , (hence the condition ni — l # 4 :?fcm — 1 —n , or 9M C C) ; then once again to intM . To prepare the application to intM one uses a collaring 9M X [0, 1) of 9M X 0 = 9M in M so chosen that v0 coincides near 9LQ H C with (i>0 | 3Lq) X [0, 1) under this collaring . In providing such a collaring one should apply the bundle homotopy theorem to the restriction of vQ to a collar of 9LQ in LQ , and then apply relative collaring theorems . ■

M IC R O B U N D L E T R A N S V E R S A L IT Y OF S U B M A N IF O L D S

Two CAT submanifolds Uu and Vv (all manifolds without boundary) in a CAT manifold Ww are locally CAT transverse if the triad ( W ; V , V ' ) is locally CAT isomorphic to a triad (RW;L U, L'v) given by transversally intersecting affine linear subspaces with dim(Lu n L'v) = u + v - w J.F.P. Hudson [HU3 ] (cf. [RS5 ] ) esta­ blished the disturbing fact t h a t , for CAT = PL or TOP , there exists no relative transversality theorem involving this notion : If U and V are locally transverse near a closed set C C W it is in general impossible to move U and/or V relative to C (or even alter them rel C) to make U and V everywhere geometrically transverse . In his examples U and V are certain topologically unknotted euclidean spaces of codimension > 3 and closed in Rw = W , with (u + v) — w = 4k + 1 , k > 1 , while C is a neighborhood of °° . In the PL category there is nevertheless a perfectly satisfactory relative transversality theorem due to Rourke and Sanderson [RS q ,II] , i For manifolds with boundary we use a model o f the form (Rw ; Lu , L'v) X [0,°°).

§1.

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91

which we will reprove as 1.8 . It involves a more strigent (non-local) notion of transversality called block-transversality. In the topological category , the theorem which we are led to prove first is analogous to that of Rourke and Sanderson ; but (TOP) microbundles replace (PL) block bundles . DEFINITION 1.4. Consider a CAT map f : Mm -* Y and a CAT normal nmicrobundle | n to X in Y as given for our definition of transversality o f f to £ above 1.1 . We suppose that (Y,X) is a CAT manifold pair and that f is a proper inclusion f : M Q . Y onto a (clean) CAT submanifold M = fM of Y . We shall call M CAT imbedded-transverse to £ in Y if f : M Q. Y is CAT transverse to £n (see 1.0) ,and M n X is aCAT submanifold of X (and hence also of Y ) . Note that M is then CAT locally transverse to X in Y . We say M is CAT imbedded-transverse to £ in Y near C C Y if , for some open neighborhood YQ of C in Y , one finds YQ (T M imbedded-transverse to £ Cl YQ in YQ . TOP IMBEDDED MICROBUNDLE TRANSVERSALITY THEOREM 1.5. In this situation , let M m be imbedded-transverse to %n near a closed subset C C Y , and let D C Y be a closed s e t . Suppose M , X and Y w ithout boundary. Suppose also m F 4 F m —n (as for Theorem 1.1) , and either dim Y —dimM F J or d im Y F 4 F d i m X . t Then there exists a hom otopy f f , 0 ^ t ^ l , o f f : - M - > Y rel C so that : (a) f \ ( M) is imbedded-transverse to £ near C U D , (b) the hom otopy f t , ( X ^ t ^ 1 , is realized by an ambient isotopy F t o f id I Y , i.e. f ( = F {f , and (c) the ambient isotopy F ( is as small as we please with support in a prescribed neighborhood o f D —C in Y . Results for manifolds with boundary can be deduced with the help of collaring theorems . (See last step of proof of 1.1 . ) ^The case m - n = 0 , dim Y =£ 4 , partially excluded in 1.5 , can be proved easily with the help o f the Stable Homeomorphism Theorem o f [K i| ] , see [S ijo , § 7 .2] . The case m — n < 0 is covered by Hommas’s method [Horn] , or by general position (see Appendix C) .

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Just as for 1.1 , the dimension conditions may prove unnecessary . The parallel DIFF and PL theorems hold true (w ithout the dimension conditions) ; the PL version is proved below as 1.6 , and the DIFF one similarly . The pattern of proof for the given (TOP) version is as for 1. 1, but it is the PL version we fall back on ; the DIFF version would appear to leave us with a metastability condition (m-n) < 2q-3 on the co­ dimension q = dimY - dimM , instead of q > 3 . The condition q > 3 is used to straighten a PL piece of M in a PL piece of Y ; this is nontrivial (cf. Appendix B) and contributes to make the proof more ‘expensive’ than for 1.1 . PROOF OF THEOREM 1.5 . We perform a cumulative sequence of normalizations until we reach a case that is clearly implied by the PL version . Each successive normalization ( a ) , . is added w ithout loss ( o f generality) — in the sense that the general case remains a consequence . (ce) Without lo ss, D is co m p a ct. (P ) Without loss , we can leave aside the smallness conditions (c) on the isotopy F t i f instead we make it have compact support in Y . (7 ) Without loss , M and X are PL (indeed have a single ch a rt). ( 5 ) Without loss , Y is open in a PL product X + X R n and X = Y fl (X + X 0) while %n is the inherited trivial normal microbundle to X . The normalizations (a) —(5) are clearly possible because o f the strongly relative nature of the theorem . (We could in fact have Y = X X Rn without loss , but that would soon become a nuisance .) ( e ) Without loss , M is a PL product along R n near C n X . where M is already transverse to £. Condition (e) is realized through altering the PL structure on M using the (local) Product Structure Theorem [I , §5.2] , which requires m =£ 4 =£ m -n (and for m -n = 3 the device of neglecting compact transverse dim 3 intersection components , as for 1.1 , Step 1 ) . At this point one has an open subset LQ of M n X containing (M n X) n C with PL manifold structure so that , near LQ , M coincides as PL manifold with LQ X Rn .

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( f ) Without loss , L q Q . X is PL and locally fla t (although n o t closed) . (77 ) Without loss , M C> Y is PL and locally f l a t . With the normalizations (a) —(17) the theorem obviously follows from its PL version ( 1.6 below) . It remains to verify (f) , (77) , and for this we distinguish two cases . Codimension M > 3 . In case dim Y - dim M > 3 , we realise (J) and (17) without loss , using the codimension > 3 straightening theorem first proved by R.T. Miller (see Appendix B for the precise statem en t). A first application alters the PL structure on X to realize (f) . Then M Q Y is PL near C n X . A second application alters M C > Y by a compact support ambient isotopy of Y rel C so that M C> Y is PL locally flat near D f l X . By then cutting back Y (and hence X etc.) to a sufficiently small open neighborhood of D O X , we realize (tj) w ithout loss , as w e ll. This proves 1.5 for dimY —dimM > 3 . ■ Codimension M < 2 . In case q = (dimY - dimM) < 2 and dimY ¥==4 =£ dimX , we shall realise ( f ) and (r?) by building a normal bundle to M and exploiting the Product Structure Theorem , as follows . The PL structure on M promised by (7 ) should be discarded now . In codimension q < 2 , there is a relative existence theorem for normal R^-bundles ; for q = 2 it requires ambient dimension # 4 as the proof in [KS5 ] is based on handlebody theory and torus

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geometry ; for q = 1 it follows trivially from collaring theorems ; and for q < 0 it is vacuous . Applying this to LQ in X , then to M in Y rel C H X we obtain a normal R^-bundle to M in Y such that is a product along Rn near C H X . As our theorem normalized by (a) , . . . .(e ) is still sufficiently relative we can assume w ithout loss that M «= Rm ; then can be trivialized making EO) « M X R9 . This lets us apply the (local) Product Structure Theorem twice . First we use it to concord the PL structure on X and on X+ so that L q CL^ X is PL locally flat, for some (new) PL structure on Lg . Next we use it to find a concordance rel C H X of the structure just obtained on Y C X+ X Rn so that M C ^ Y becomes PL locally flat for some (new) PL structure on M while M coincides PL with L0 X Rn near C n X . To this second concordance we apply the Concordance Implies Isotopy Theorem [I, §4.1 ] to produce a compact support ambient isotopy of Y rel C making M CL*. Y locally flat PL near D ft X . (We don’t alter the structure on Y C X+ X R^ at this p o in t) . Finally we cut back Y to a sufficiently small open neighborhood of D Cl X in X+ X R^ to realise (f) and (tj) w ithout loss of generality . This closes the proof of 1.5 for dimY - dimM < 2 . ■ For completeness we present the PL IMBEDDED MICROBUNDLE TRANSVERSALITY THEOREM 1.6. The transversality theorem 1.5 holds true in the PL category w ithout any restrictions concerning dimension . The proof is based on the following simplicial lemma , which will play the role the Sard-Brown theorem had in 1.1 . Lemma 1.7. (well known) Every simplicial map f : K-» A^ o f a finite complex onto the standard d-simplex is a PL ( trivial) bundle map over int . There are standard bundle charts which respect every subcomplex o f K . Proof o f 1.7. Topologically this is clear since K is canonically a sub­ complex of the join K0 * K j * . . . * Kj , where vq, . . ..v^ are the vertices of

Kj = f-^(v j) , . For the PL version we can

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clearly assume that K is all o f this join . Then there is a canonical homeomorphism A 1 ) . If Aq is any closed linear d-simplex in A we form the standard simplicial subdivision of the regular convex linear cell complex Aq X Kg X . . , X K j and replace i// on this by the simplex-wise linear map equal \jj on the vertices . Then g> : A0 X KqX . . . X Kd - * H ( A ) is a typical standard PL bundle chart respecting each subcomplex of K. ■ PROOF OF 1.6. Without loss (of generality) we normalize imposing the conditions (a) , (|3) and (5) used in the TOP p ro o f. Further normalization is required . (i) Without loss , we can om it the condition that ft , Ft be rel C . (Recall that FjM is required to be transverse to £n near CUD). To see this choose a small closed neighborhood C' of C in M near which M is transverse to £n and apply the new version after replacing C by C' and D by D—C' , and then delete C from Y (and from everything e lse ). We can now choose a compact polyhedral neighborhood K of D in Y and endow it with a finite PL triangulation such that M n K is a subcomplex and the projection p2l K to Rn is simplicial for some triangulation of Rn . Then we can replace Y by the interior K of K in Y establishing via the simplicial lemma 1.7 that : (ii)' Without loss , over the interiors o f the open n-simplices o f a triangulation o f Rn , the PL projection Y R n is a PL bundle with charts that respect M . We would like such a chart over a neighborhood of the origin 0 in Rn . So we choose an open relatively compact neighborhood X x of D n X in X , then a small neighborhood R x of 0 in Rn so that D H p71R 1 C Xj XRj C Y C X +XRn . There clearly exists a PL isotopy Gt , 0 < t < 1 , of id IY such that Gt has compact support in X, X R! , G^. maps each fiber (x X Rn) fi Y of £n into itself and (most im portant) for all x in a

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neighborhood of D H X in X one has Gj (x,v) = (x,g_1 (v)) (*) where g:Rn ->R n is a PL automorphism of Rn with g(0) in the interior of an n-simplex of the mentioned triangulation of Rn . See Figure 1-b . Note that, provided Rj is small, since Gj respects each fiber of £ , the image G j M will be transverse to £n near C , as is M itself. Thus, in view of (*), we can have in place of (ii)' the condition: (ii) Without loss, fo r some small open neighborhood X QXR0 o f (D n X) X 0 in Y , there exists a PL submersion chartf 3DQ = g'9D0 = g'DQ n y>(D - 6 Q) it agrees with the structure of N H U . This is accomplished by an obstruction theory for smoothing using 7ri(TOP/O) = 0 , i = 0,1 , see [KSj] [IV] [V] [S ij0 ] . More simply one can solve a zero- and a one-handle problem for two handles with core in y>(3DQ) , by the method of [KSj ] applied to DIFF .t Now (N H U ) and B ^ together give a DIFF structure S ' on a small neighborhood V of y?(D - DQ) fl g'(DQ) in N extending that of U near (D-D0) . Finally alter ^ U g ' extending tp to a DIFF embedding p' : D-> V jy . Then Whitney’s DIFF process applies to eliminate the pair (a, b) of intersection points . ■ Remark : The argument outlined in [S ijq , §7.3] , although tougher , is more elementary in that it does not use the surgery concealed in [K Sj] . *

*

*

The 1-parameter theory of generic functions due to J. Cerf has (in 1975) not yet been reworked in the TOP context. Nevertheless, the TOP version of its main theorem ( [Cey ] , [HaW]) has been established via a reduction to the smooth case, see [BaLR, pp. 148-9] (E. Pederson’s appendix): Theorem: L et Vn , n > 5 , be a connected compact TOP manifold such that 7Tj V acts trivially on 7T2 V . Then the components o f the space o f automorphisms o f Vn XI fixing (9Vn)XI U Vn X0 pointwise, are in bijective correspondence with the elements o f an algebraically defined abelian group Wf^TTj V) © Whj (7t2 V \Tt\ V) that is 0 when­ ever 7Tj V = 0 . In case V admits a DIFF structure, the reduction is a simple application of the Concordance Implies Isotopy Theorem (Essay I) and its 1-parameter version (see Essays II, V). In general one seeks to replace V by a subhandlebody of index < 3 , which does admit a DIFF structure. (When n = 5 , Pederson hypothesises that Vn is a handlebody.) t For 0 - and 1-handles , a PL solution easily gives a DIFF solution ; so [KSj ] could be applied directly .

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§4. THE SIM P L E HOM O TO PY TYPE OF A T O PO LO G IC A L M A N IF O L D ^

We shall define a prefered simple type for a topological manifold M by properly imbedding M in a high dimensional euclidean space Rn with closed normal disc bundle D ; the (local) Product Structure Theorem of [ I , §5.2] applied near 3D in Rn gives D a triangulation ; then by definition the preferred simple type of M is the •one making i : M C* D a simple equivalence . If this procedure is carried out again yielding i* : M CL* D* we observe using the Concordance Implies Isotopy Theorem of [I , §4] that (after stabilisation) there is a PL isomorphism f : D D* so that fi is proper homotopic to i* . This implies that our preferred simple type is well-defined . The proofs are very easy ; most space is devoted to recollecting basic facts about simple types , triangulations , normal bundles and imbeddings . First the basic facts about simple hom otopy types (from [C hi^] and [ Siy ]) . A simplicial map f : X -> Y of finite (unordered) simplicial complexes is an elementary expansion if f is injective and Y - f(X)

^ Since this essay was written T. Chapman has succeeded in defining the simple type more generally for any locally compact metric space X such that X X [ 0 ,1 ] ° ° is a manifold with model the Hilbert cube [ 0 , 1 ] ° ° , see [ C h j H C f ^ ] I Si j 4 ] . (For example X can be a TOP manifold or locally a finite CW com plex; it seems that X can be any locally compact ANR , see [Mi^HWesJlEd^D.This flows from Chapman’s triangulation theorem and Hauptvermutung for Hilbert cube manifolds M : one has M = K X [0, 1 ]°° for some locally finite simplicial complex K ; and if Kj X [0, 1 ]°° ->■ K 2 X [0, 1 ]°° is a homeomorphism (or even proper and cell-like [ C h o D , then the proper hom otopy equivalence K 1 K2 determined is a simple equivalence of complexes . Several other interesting finite dimensional arguments have since appeared , each showing at least that a homeomorphism Kj -> K 2 of locally finite simplicial complexes is a simple equivalence [ Ed 7 ] [Edq] [CI1 3 ] . The argument [Edo] [ Si j 4 ] of R. D. Edwards is based on this section .

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consists of exactly two open simplices , one being a face of the other . In the category C of injective simplicial maps (inclusions) of locally finite unordered simplicial complexes we must specify the sub­ category £ of expansions . It is the least family in C containing isomorphisms and elementary expansions that is closed under 1) finite composition , 2) disjoint union , 3) pushout . Closed under pushout means that if X = Y U Z in C and Y Pi Z CL* Y is in £ then Z C* X is also in £ , see [Si^] . A map f : X - > Y of C where X is compact clearly belongs to £ if and only if it is a finite composition of isomorphisms and elementary expansions . A proper hom otopy equivalence f : X -> Y of locally finite simplicial complexes is simple if it is proper homotopic to a finite composition X = X x -> X2 ... Xn = Y where each map is an expansion or a proper hom otopy inverse to an expansion . Clearly each expansion is a proper homotopy equivalence - i.e. a hom otopy equivalence in the category of proper continuous maps . It is an elementary fact that a PL homeomorphism is a simple equivalence . It is enlightening to recall (from Whitehead) that when X and Y are properly simplicially imbedded in Rn and n > 2 max{dimX , dimY} + 1 , then X and Y have PL isomorphic regular neighborhoods if and only if X and Y are simple hom otopy equivalent . A simple type for an arbitrary (metrizable) space X is represented by a proper hom otopy equivalence f : X -* Y to a locally finite simplicial complex Y (if such f exist) . Another such equivalence f : X -* Y' represents the same simple type if there exists a simple proper homotopy equivalence s : Y Y' so that sf is proper homotopic to f' . A space equipped with a given simple type will be called a simple space . Every locally finite simplicial complex X has a canonical simple type represented by the identity map X X ; so X is a simple space . A proper homotopy equivalence f : X -> Y of simple spaces is called simple if when g : Y -> Z represents the given simple type of Y , then gf : X -> Z represents that of X . When X and Y are locally finite simplicial complexes this is equivalent to the earlier definition of a simple proper hom otopy equivalence . THEOREM 4.1 . Every separable metrizable topological manifold M has a preferred

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simple type —so that M is a simple space . It can be defined by the following CAT(= DIFF or PL) rule : Im bed any topological closed disc bundle over M as a (clean) C AT codimension zero submanifold o f an euclidean space and select a CAT^ triangulation T o f i t ; then the inclusion map M ^ - T represents the preferred simple type . Other equivalent definitions of simple type will be discussed in § 5 . COROLLARY 4.1.1. A ny homeomorphism h : M - + M ’ o f separable metrizable topological manifolds is a proper simple hom otopy equivalence. Proof o f 4.1.1. from 4.1. This is seen in the nature of the rule defining the preferred simple type . ■ We can establish the preferred simple type of 4.1 using only DIFF or only PL arguments ; for the readers convenience , the specifically DIFF arguments are marked off by square brackets [ 1 an d /o r double stars ** . [ I f we use DIFF as our working t o o l , Whitehead’s C°° triangulation theory is of course used , in particular to prove : * *LEMMA 4.2. I f T and T' are two (Whitehead) DIFF triangulations o f a DIFF manifold possibly with corners , then the identity map T T' is a simple hom otopy equivalence ; in fact it is isotopic to a PL homeomorphism . The proof is in [W hj] [Muj ,§ 1 0 ] .

■

**4.3. Straightening corners . Let B2(0) = {(r cosip , r simp) E R2 I 0 < r < 1 , 0 < ip < 0 } . If has corners and N = B2 ( it/ 2) X C C is DIFF tubular neighborhood of a manifold C of corners we can straighten (or unbend) corners along C altering X to Z* on B2 (7r/2) X C defining Z* so that p X (idlC) : (B2(ir/2) X C )2 * -► B2 (tt) X C is a diffeomorphism where p : (r cos

T* is simple . (In fact it is not hard to see a topological isotopy of it to a simplicial isomorphism) . ■ ] An essential tool will be the theorem of M. Hirsch and B. Mazur stating that a bundle with fiber euclidean space contains a disc bundle after stabilization , which is up to isotopy unique after (more) stabilization . Its compactifying role is crucial ; recall that the role of torus in [I , §3] was similarly compactifying . So it is appropriate to recall the precise statement required and indicate the proof . Consider a locally trivial bundle E with fiber Rn and zero section , over an ENR X ; let X be identified with the zero-section so that X C E and the projection p : E -> X is a retraction that completely specifies the bundle . We say E has fiber (Rn , 0) since the group of E as a Steenrod bundle is the group (with compact-open topology) of all self homeomorphisms of (Rn , 0) , i.e. of Rn fixing 0 . We will use similar terminology for disc bundles . PROPOSITION 4.4 (see [Hi6 ] , also [KuL! 2 1) •t A) (Mazur) The stabilized (R n+^ ,0 ) bundle E X R over X contains a (Bn+^ ,0 ) bundle D C E X R over X . This D is understood to come from a reduction o f E to the group o f hom eo­ morphisms h o f R n+^ so that hBn+^ = B n+^ and h(0) = 0 . B) (Hirsch) I f E itself contains two (Bn, 0) bundles D x , D 2 , then D x X B l , D 2 X B 1 are isomorphic (Bn+^ ,0 ) bundles. In fact the (R n+^y 0) bundle [1,2] X (E X R ) over [ l y2] X X contains a (Bn+^ ,0 ) bundle D' extending k X D ^ X B 1 over k X X , k = l , 2 - called a concordance from D x X B 1 to D 2 X B 1 in E X R . Proof of A) . Let 2 n+ * be thesphere R11"1"^ U oo . There is clearly a unique (2 n + 1, 0 , oo) bundle S_ so that the complement of the oo-Section is E X R . The closure in S of EX [-1, °o) is a (Bn+ 0) bundle D . t Concerning reduction of (the group of) a bundle like E , see [Hus] .

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This D contains a disc bundle lying in E X R . Indeed , by Alexander’s theorem [A lj] , any (Bn + ^ ,0 ) bundle reduces to the group of homeomorphisms h of Bn + ^ so that h(x) = lxlh(x/lxl) for x f 0 . So reduced , D contains a (Bn + ^, 0) bundle in E X R o f any radius in (0, 1). ■ Proof of B) . Let S be the ( 2 n , 0, °°) bundle associated to E , where 2 n = Rn U oo . By Alexander’s theorem , there is a reduction p xof S to the group of those homeomorphisms h of 2 n fixing 0, °°with h(x) = Ixl h(x/lxl) for x f 0, °° , a reduction such that Dj is the unit disc bundle . _ We assert that there is an isotopy h ^ , 0 < t < 1 , of id I S = h0 respecting fibers and fixing the zero-section , such that hj (Dj X B1) = D , where S and D were defined for A) . Then , using the _ reduction of D mentioned for A) we have a (Bn + ^,0) bundle ViD of radius xh and can easily deduce from h t an isotopy h^. of _ id I S = h^ respecting fibers and zero section so that h'j (D! X B1) = ViD and h^(Dj X B1) C E X R , for all t . This hj. gives a concordance from Dj X B1 to VzD in E X R . Similarly we get one from ViD to D2 X B1 , so from Dj X B1 to D2 X B1 as required .

Figure 4-a

( Illustration fo r the paragraph overleaf )

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To construct h^ note that Pi

Essay III

gives an obvious quotient map

q x : R+ X 8D! — *■ E

(polar coordinates)

where 5Dj is the boundary sphere bundle of Dj , and by natural extensions quotient maps R+ X R X 5 D 1 - > E X R and q : {(R+ X R)U oo} x S D j - ^ S , so that with the notation of Figure 4-a D is the quotient of Q X 5D! and Dx X B1 of Qj X 8D! . We leave the reader to produce an isotopy gj. , 0 < t < 1 of the identity of the Gauss sphere R2 U o° so that g1(Q i) = Q , gt( Q i) C (R + X R ) U oo , gt(0) = 0 , and g^. X (id I 5D) passes to the quotient by q defining the asserted isotopy hj. . (Just isotop Q x onto Q'j , then Qj onto Q , taking care to respect (y-axis) U °° throughout ! ) ■ PROOF OF 4.1 . Given M , let M' = {M U 3M X [0, 1)}/ {3 M = 3M X 0} be M with an open collar tacked on the boundary . Find an imbedding i : M' -»• Rn , n large , with a closed locally flat disc bundle D' C Rn , and let D C Rn be the restriction of D' to M . We arrange that D is a clean submanifold of Rn . [We recall simple ways of finding i and D' . Using an atlas and a partition of unity imbed M' in some Ra . Then using a proper map M' -*■ R we get a proper imbedding M' -*■ Ra + ' = Ra X R . We identify M' now with its image . By a tricky elementary argum entt[M ij] [Hi^] . M' has a normal microbundle in some Rb , b > a + 1 . This is an open neighborhood of M' with a retraction to M' that is a submersion near M' , all fibers being open manifolds near M' . This microbundle (indeed any) contains a euclidean space bundle E' by the Kister-Mazur theorem [Kis] [K uLj] . Then E 'X R C Rb+1 = Rn contains a disc-bundle D' over M' whose boundary sphere bundle 3D' is a copy of E' with each fiber one-point compactified , see 4.4 . As D' is a clean submanifold of E' X R , we can assure that D and D' are clean submanifolds of Rn by choosing E' so small that its closure in the normal microbundle is its closure in Rn” ^ = R*3] . ^ The argument of Hirsch is explained in [IV , Appendix A] ; Milnor’s argument is different .

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Note that the manifold boundary 3D is bicollared in Rn and apply the Product Structure Theorem [I, §5.2] to obtain a CAT structure 2 on Rn concordant to the standard one so that 3D ^ and are CAT submanifolds . Then fix a CAT triangulation of as a simplicial complex T . If CAT = PL this means that the identity map 8 : T -> is PL ; existence of T is an elementary fact [ H ^ ] • [ I f CAT = DIFF , this means that 8 : T -►D ^ is smooth C°° non­ singular on each closed simplex ; existence of T is Whitehead’s theorem [M uj] .J By definition the canonical simple type fo r M is given by the inclusion M T. By the CAT Concordance Implies Isotopy Theorem [ I , §4.1] , there is a small isotopy of id I Rn carrying D^; onto a CAT submanifold of Rn . Thus the definition in the statement of 4.1 is seen to be equivalent . Note that we have already proved Local finiteness theorem 4.1.3 . Every metrizable topological manifold is proper hom otopy equivalent to a locally finite simplicial complex . ■ Preliminary to showing that the chosen simple type is canonical (i.e. independent of choices) here are two remarks : (1) The final choice o f T is irrelevant. (Recall that a PL iso­ morphism is simple [and for CAT = DIFF use 4.2} .) (2) Our choice o f simple type is unaffected by the following stabilization which we describe first for CAT = P L . Replace Rn by Rn X R ; D by D X B1 ; 2 by 2 X R ; T by T X B1 (with standard subdivision , B1 having vertices - 1 , 0 , 1 ) . This holds because T X O -*■ T X B 1 is an expansion . And the stabilized construction is again of the sort described . [ I f CAT =* DIFF this stabilization is to be corrected as follows : (a) Alter 2 X R on Rn+^ by a concordance , unbending the corners of (2 ID) X B1 using a tube about 3D X 3B1 respecting D - thus getting 2 * for instance, (b) Replace T X B 1 by any DIFF triangulation T* of (D X B1)^ * . Note that T X B 1 is not a DIFF triangulation of 2 * . However the identity map T X B1 -» T* is a simple equivalence by 4.3 so the original reasoning now applies .]]

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Now suppose the construction of i : M' -* Rn , D' , D , 2, T has been carried out in two ways indicated by subscripts 1 and 2 . We must show that M

Tx

, M —^

T2

define the same simple type . After stabilization as described in (2) above , we can arrange that Rn i = Rn2 = Rn , that ij , i2 : M' -»■ Rn are proper , that there is a clean proper imbedding j : M' X [1 ,2 ]

Rn X [1 ,2 ]

equal i x X 1 on M' X 1and equal i2 X 2 on M' X 2 , and also that there is a normal disc bundle B' to j(M' X [ 1, 2 ]) in Rn X [1 ,2 ] extending Di X 1 and D2 X 2 . We will give more details for construction of j and B' presently . Clearly , we can arrange t h a t , as a bundle , B' is a product along [1 ,2 ] near Rn X 1 and Rn X 2 .

Figure 4-b. There exists a CAT structure 6 on Rn X [ l,2 ] extending 2 j X 1 and 2 2 X2 , also a product along [1,2] near both Rn X 1 and Rn X 2 . Then the restriction B of B' to j(M X [ 1 ,2 ] ) is CAT in (Rn X [ 1, 2])0 near there . So the (local) Product Structure Theorem [I , §5.2] lets us alter 6 in Rn X ( l , 2 ) so that B is (everywhere) a CAT submanifold B^ , which cannot in general be a CAT bundle . Using the bundle hom otopy theorem , identify B to D t X [1 ,2 ] sending j(M X [ 1 ,2 ] ) to i j (M) X [1 ,2 ] in the standard way ,

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Dj X 1 to itself, and D2 X 2 to D x X 2 . Then the Concordance Implies Isotopy Theorem applied to (D, X [ 1, 2 ] )^ provides us with a CAT isomorphism h making the triangle

M

h

proper hom otopy commutative , because , as a map D 1 , h is TOP isotopic to the identity . In view of preliminary remark (1) , we have proved that the ciioices do not affect the simple type assigned to M - i.e. the type is canonical . ■ We conclude with Notes on construction o f j and B ' (cf. Figure 4 -b ). j is a concordance from i x to i2 , constructed after stabilization via preliminary remark (2) , much as we suggested ij and i2 be constructed . The concordance B' from D'j to D2 is a bit harder to fin d . Hirsch’s elementary relative normal bundle existence theorem [Hi^] , [IV] coupled with the Kister-Mazur theorem [K uLj] , does provide , after stablization , a concordance F say from E t to E2 , viz. a normal euclidean space bundle to j(M' X [ 1, 2 ]) in Rn X [1 ,2 ] restricting to E ^ X k over i^CM') , k = 1,2 , here E^ being the bundle in which was found . Just as one found D' in E one can find a disc bundle B" in F after one stabilization , that is a clean submanifold of Rn X [ 1, 2] . This B" can , as u su a l, be a product along [ 1, 2 ] near 1 and 2 . If B'M GjM 'X 1) is concordant to D'! X 1 as a disc bundle in E, X 1 and similarly in E2 X 2 , then clearly B" can be corrected near 1 and 2 to give B' as required . Now 4.4 asserts this after one more stabilization ; so then we get B' . ■

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§ 5 . SIM PLE TY PES D EC O M PO SIT IO N S A N D D U A L IT Y

This concluding section gives the essential links between our definition of the canonical simple type of a TOP manifold and TOP handlebody theory (or Morse function theory) and TOP surgery . More specifically (a) We prove that various definitions of simple type for a topological manifold - using (arbitrary !) triangulations , handle decompositions , or Morse functions - all give the same result when­ ever they apply (b) We show that , with the canonical simple type , any compact topological manifold (or triad) is a simple Poincard duality space i.e. roughly speaking the Poincard duality cap product comes from a simple chain hom otopy equivalence . We continue to give two alternative arguments using DIFF and PL methods respectively . The oldest way to define the simple type of a manifold M involved choosing i f possible a CAT (= DIFF or PL) structure 2 on M , then a CAT triangulation M ^ ^ T by a simplicial complex T . That this gives the canonical simple type for M is a special case of PROPOSITION 5.1. If D is any closed disc bundle over the TOP manifold M and D has a CAT (= DIFF or PL) structure and a CAT triangulation ; then M C_> D gives M the canonical simple type of § 4 . Proof : CAT imbed D properly in euclidean space with a closed cleanly imbedded normal CAT disc bundle D* (see [Hi^ g] for PL) and CAT triangulate D* so that the projection p : D* -* D is a piece wiselinear map . I for CAT = DIFF assure this by building the Whitehead C°° triangulation [M uj] [W hj] of D* using only charts on D* presenting p as a linear projection .J Now M C* D* gives by ^ Where triangulations are concerned T. Chapman’s or R. D. Edwards’ more recent results are stronger (see §4) . We recommend to the reader Chapman’s brief proof [CI13 ] that a proper cell-like mapping of simplicial complexes is a simple equivalence .

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127

definition the canonical simple type . But the inclusion D C* D* is simple , as an induction up through the skeleta of D readily shows (alternatively the PL retraction p :D * -> -D is collapsible [ C h n j] ) . Hence M C_> D also gives the canonical simple type . ■ It is im portant in handlebody theory to have a rule that assigns a simple type to a manifold X for each suitably well behaved filtration of X . We present a simple geometric one that will apply to both handle­ body decompositions and arbitrary triangulations . Let us operate in the category S' of locally compact metrizable spaces and proper continuous m a p s. (A map is proper if the preimage of each compactum is co m p act). Thus hom otopy means proper homotopy; the word equivalence means proper hom otopy equivalence . An inclusion map X C-> Y is a cofibration if any map f() : X X [0, 1] U Y X 0 -> Z extends to a map Y X [0, 1] -> Z . We are here working in S' , to be sure . But note that if an extension f : Y X [0, 1] -*■ Z of a proper map fQ exists so that f is continuous but not proper , then one can deduce from it a proper extension using the fact (easily proved) that f is necessarily proper on a sufficiently small closed neighborhood o f X X [0, 1] U Y X 0 in Y X [0,1 ] . Recall that the set S(X) of simple types on a locally finite simplicial complex is an abelian group and that the rule X +* § (X) is functorial on proper hom otopy classes of maps . An equivalence f X -►Y of simplicial complexes represents an element [f] £ g (X ) , also denoted [Y, X] if f is an inclusion . Both addition and functoriality come from astutely forming amalgamated sums [Siy] [Chn2 ] : if f : X C+ Y and g : X C* Z with W = Y U Z and X = Y n Z , then g* : g (X) -*> g (Z) sends [f] = [Y, X] to [W, Z] ; if g : X CL*. Z is also an equivalence one defines [f] +[g] = [Y, X] + [Z, X] to be [W, X] = [Y U Z, X] . We extend this trivially to a functor g : 3" 0 {Abelian groups} where J is the category of proper hom otopy classes of maps between spaces X such that there exists some (unspecified) proper equivalence fi : X -»■ X' to a locally finite simplicial complex . We simply define g (X) = g ( X ') . This makes sense because , if f" : X -* X" is another such equivalence , there is a uniquely determined isomorphism g* : g (X') = g (X ") where gf' =“ f" . Here indicates proper hom otopy .

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Beware that the group §(X ) is not naturally identified to the set of simple types on X unless X is a simple space . For X in 3 or 3”q , the simple types form a merely affine set (possibly empty) , and clearly it does not vary functorially with X in S' . F o r X G S q , our group § (X) can be defined more intrinsically as a group whose elements are ordered pairs of simple types on X . DEFINITION 5.2 . A spaced simple block decomposition of a space X in S' , or an s-decomposition for short , consists of two things . First is a finite filtration of X x 0 c x j c X! c x | c x 2 c x j c . . . c x n c x j

= x

by closed subsets Xj C Xj1- (equal 0 if i < 0) enjoying the properties : (a) Xj CL* X f is an equivalence . (b) writing Bj = Cl(Xj - x£_j) (Cl indicating closure in X ), and 9_Bj = B- H X£_j , the inclusion 9 JB- C* Xj is a cofibration . Second is a prescription of simple type for each “block” Bj and each 9_BiLEMMA 5.3 . Suppose Z G J is a union Z = Zj U Z 2 o f closed subsets so that Z Q = Z x Cl Z 2 CL* Z is a cofibration . I f Z Q , Z j , Z2 are simple spaces , then there is an equivalence o f triads f : ( Z ; Z x , Z 2 ) -*■ (Z ' ; Z \ , Z'2) to a simplicial triad in S' , giving simple equivalences f : Zfc -► Z ^ , k = 0 ,1,2, where Z'Q = Z \ n Z '2 . The resulting simple type for Z is independent o f choices. Proof o f 5.3 . Form a hom otopy commutative diagram

Z'i « (5.3.1)

Ji

Z'

J2

♦ z;

f, Z,

Z\ o ■ We see that Z\ can be a simplicial mapping cylinder of a map to Z \ 0 namely jj in (5.3.1) made simplicial - so Z x 0 C* Z\ is an expansion . And f I Z x can be fi 0 • ■ That the simple type given by f : Z -»• Z'is independent of choices follows from the sum theorem of [ Si-y ] . This states t h a t , if g : (Z' ; Z\ , Z2) -> (Z" ; Z " , Z2) is an equivalence of tripples of complexes , then in S (Z') one has J

(5.4)

[g] = ii*[gi ] + i2*[g2 ] - io * tg 0 ^

where g^ representing [g^] E § (Z^) is the restriction of g to an equivalence Z^ , and ij, : Zj, -► Z' induces ij^^ : § (Z^) -► -> S (Z') . Thus g is a simple equivalence if gj , g2 and g0 are . This completes the proof of 5.3 . ■ REMARK 5.4 . N otice that the sum form ula carries over automatically to triads o f simple spaces where all the subspaces are closed and their inclusions cofibrations. ■ We now deduce THEOREM 5.5 . In a s-decomposition as above , each preferred simple ty p e .

and each x t

has a

Proof o f 5.5 . By hypothesis X0 = B0 has a preferred simple type . Suppose inductively that has one , where n > k > 1 . Give to ^ It transcribes without change into the proper category .

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X j_j the simple type making X^_ j -»■ x £ _ j Now apply 5.3 to the tripple

a simple equivalence .

130

(Xk ; Xk-1 ’ Bk )

•

9- Bk = Xk - l n B k>

to define the simple type of Xj. . This completes the induction to make all X j , Xj" simple spaces . ■ REMARK 5.5.1. Suppose in the above theorem that X is a simplicial complex which is simplicially s-decomposed in the sense that (a) each Xj and Xj1’ is a subcomplex and Xj C* X* is a simple equivalence . (b) Bj and 3Bj are subcomplexes and are assigned their natural type . Then it is clear that this theorem assigns to X its natural type (as com plex). ■ DEFINITIONS 5.6. Consider f : X -►X a map in J of spaces with s-decompositions with the same number n + 1 of blocks , such that for each i , 0 < i < n , the map f gives by restriction a hom otopy equivalence of triads (Xj ; x£_j , Bj) -*■ (Xj ; X jlj , Bj) . Then we call f a blocked equivalence of s-decompositions . The s-decomposition of X is called complete if inclusion induces the zero map S (Bj) -► § (X) for each i . THEOREM 5.7 (with data of 5 .6 ). The class o f f inS (X ) is b0 + ( b 1 - b \ ) + ( b2 - b ' 2) + . . . + ( b n - b' n) , where bj is the image in &(X) o f the class in § (B I;) o f the map f : B-I -> B:I and b'{ is similarly the image in § (X) o f the class o f f : b _ B j -*■ 3_ B j . Proof o f 5 .7 . Use induction on m , applying the sum theorem 5.4 extended to simple spaces . ■ COROLLARY 5.7.1. I f an s-decomposition o f a space X is complete , the simple type it gives X (via 5.5) is independent o f the simple types chosen fo r the blocks Bj and for d_Bj C B j . ■ PROPOSITION 5.8. L et X be a space in 3 with an s-decomposition. There always exists a simplicially s-decomposed simplicial complex X (see 5.7) and a blocked equivalence f : X-+ X . Proof o f 5.8. If this is true for decompositions of length n , it follows for length n + 1 by applying Lemma 5.3 and Remark 5.3.2 . ■

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131

DEFINITION 5.9. An s-decomposition of a TOP manifold Mm : M0 C M j C Mx C M | C . . . C Mn C M j = M is called a TOP sdecomposition i f : a) Each Mj and Mj" is a clean TOP m-submanifold of M . b) Each ‘space’ Sj = Cl(Mj" - Mj) is a clean submanifold of M j", and ( S j, 9_Sj) « 3_Sj X ([0, 1], 0) where 3_Sj = Sj n Mj . Here indicates homeomorphism . (Note that Sj may well be empty .) c) E a c h ‘block’ Bj = C l(M j-M jl]) is a clean submanifold of M j. And both Bj and 3_Bj = Bj H M^.j have the canonical simple type of § 4 , (which is necessarily true if (B j, 3_Bj) is a disjoint union of handles) . Example 5.9.1. From a TOP Morse function f : (Mm ;V ,V ') -*■ ([0 , 1 ] ; 0 , 1) and a gradient-like field on a compact triad (see §3) , a familiar procedure [Mig , §3] produces a TOP s-decomposition of M with Mq a collar of V , and with one block Bj for each critical value being a disjoint union of handles , copies of of f , each (B j, 3_Bj) (B ^ , 3B ) X Bm -k for various k . Example 5.9.2. Using §2 one obtains an s-decomposition for any TOP manifold Mm , m > 6 . An s-decomposition of M is complete as soon as each component of each block lies in a contractible co-ordinate chart (see 2. 1. 1) . THEOREM 5.10. Every TOP s-decomposition o f a TOP manifold M yields (via 5.5) the canonical simple type fo r M defined in § 4 . Proof o f 5.10. Assume indicutively that this holds for such decompositions that have < n blocks . (It holds trivially for < 1 block). Let M have n blocks ; then by inductive hypothesis M^_j gets canonical type . Choose an embedding of Mm in some euclidean space of high dimension with normal cleanly embedded disc-bundle D . We find a CAT (= DIFF or PL) structure 2 on DI Mn arranging , by the (local) Product Structure Theorem [I , § 5.2 ] , that (D lM ^_j) is a clean CAT submanifold of (DI Mn )£ . Then we find a CAT triangulation of the CAT triad (D lM n ; D lM j.j , D lB n . This clearly determines by inclusion the

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canonical simple type for all parts of the triad (Mn ; M*_j , Bn) , even for the intersection 9_Bn . Thus the canonical type for Mn agrees with the type given by 5.5 . The same is true for M* = M because the inclusion Mn -*■M* is a simple equivalence for the canonical simple types . Indeed Mn C* M* is proper homotopic to a homeomorphism . This completes the proof of 5 .1 0 . ■ COMPLEMENT 5.10.1. Suppose a TOP manifold M has an sdecomposition , which is not TO P, but gives a TOP s-decomposition o f M X Br by producting with a disc Br , r ~^ 0 . Then this sdecomposition o f M again gives the canonical simple type o f M . X0 Proof o f 5.10.1. The inclusion M — ►M X B is a simple equivalence both if we use the canonical types of § 4 and again if we use the sdecomposition types of 5.5 (see 5.7) . Thus the result follows from 5.10 applied to M X Br . ■ f THEOREM 5.11. A n arbitrary triangulation o f a TOP manifold M as a simplicial complex gives to M the canonical simple type o f §4 . Proof of 5.11. Let M ® be the i-skeleton of M for a fixed triangulation of M , and define an s-decomposition of M by setting Mj = Mj” equal the union of all closed simplices of M" which meet M(i) . Here M” = (M '/ denotes a second derived subdivision of M . Since M0 C M j C M 2 C . . . is a filtration of M by subcomplexes of M" , the simple type of the filtration is by 5.5.1 the simple type defined by id : M -*■ M" or again by id : M -> M . But this s-decomposition also gives the canonical simple type , by complement 5.10.1 , since we shall prove the ASSERTION . M0 X B3 C M, X B3 C. . .C Mm X B3 TOP s-decomposition o f M X B3 .

is an unspaced

The proof is broken into two steps . ^ Omit on first reading , as much stronger results are now available , eg . [CI1 3 ] . However this is a part of R. D. Edward’s proof that a homeomorphisms of polyhedra is a simple equivalence [S i1 4 ] [Ed2 1 •

133

§5. Simple types and triangulations

Note that Bj has one component for each i-simplex o of M . It is the closed star neighborhood St(a , M") of the center a of o' (the vertex of M' in the interior of a ) . Step 1. The com ponent (II g , d_II g) o f ( I f , d _ I f ) . corresponding to an i-simplex o o f M , is homeomorphic to ( B f b B 1) X cL . Here B l = (i-disc) , and cL is a copy o f the closed cone on the link L k ( o , M ) o f o in M. Also Bj has collared frontier in M - Mj_j ■ Step 2. Bj X B 3 is a manifold and d_Bj X B 3 is a clean sub­ manifold o f its boundary. Proof of step 2. We verify this for any one component of (B j, 9_Bj) , which we express as (B1 ,9 b 1) X cL using step 1 . We know that -jO O (intB1) X cL , with c indicating open cone , is an open subset of M . Thus cL X R1 ,L X Ri+1 , L X Ri+1 X [0, 1) , cL X Ri+1 , 9B1 X cL X R2 , 9B1 XcL X B3 are successively seen to be manifolds , from which step 2 follows . ■ For completeness we give Proof of step 1. For any simplicial complex M this step succeeds and the result is in fact PL . In the first derived M' of M we have the join Na = (9a)' * a * L where L is the subcomplex with typical simplex (di , o2 . • ,0^) spanned by the (bary-) centers of simplices with o < o x < o2 < . . .< • Here < indicates “ a subcomplex o f ’ . Recall that L is isomorphic to the 1st derived of Lk(o, M) by pseudo-radial projection sending the vertex oj of L to r j , where Oj = o * r j . As Na is a neighbor­ hood of Ha in M it will suffice to examine Na n (Mj , ). This is naturally (PL) homeomorphic to the pair (A U B , A) in Na described as follows . Use barycentric co-ordinates of the join NQ to express a typical point p of Na as p = u #x + v < j + w z

E (9a)' * a * L = Na

with x E (9a)' , z E L and u, v, w in [0, 1 ] with u + v + w = 1 . Then define A = {p I u > 1/2}

;

B = {p I w < 1/4} .

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Figure 5-a See Figure 5-a (le ft). In fact the two pairs Na H (M- , M j_j) and (A U B, A) are identical when (following Zeeman [Ze , Chap. 3 ]) we make special (non-standard) choices for the centers f of simplices t of N a < M' to define the second derived - namely , so that if f = u ’x + v ff + w*z , then (a) u = 0 , 1/2 or 1 , and (b) w = 0 or 1/4 when v f 0 . (In general , the PL homeomorphism comes from shift of centers , see Figure 5-a comparing right and le f t) . With the special choices of centers Hff = A n B = {p I u < 1/2, w < 1/4} , 9_Ha = (p I u = 1/2 , w < 1/4} , 9+H = (p I u < 1/2 , w = 1/4} , and No n ( M - M M ) = (p I u < 1/2}. Step 1 follows ; a suitable homeomorphism Ha -* c(9a) X cL « B1 X cL is given by u*x + v*a + w*z +> (2u*x, 4w*z) , and a suitable (PL) collar of 9+Ha in is (p I u < 1/2 , 1/4 < w < 1/ 2 } . ■ Theorem 5.11 is now proved .

■

Next is the result about s-decompositions necessary for the TOP s-cobordism theorem . THEOREM 5.12. Let M 0 C Mq C M 1 C M [ C. . .C M + m+1 = Mm be a TOP s-decomposition o f a compact connected TOP m-manifold M such that ( B^, 9_B.) is a disjoint union o f (i-l)-handles fo r i > 1 .

§5. Simple types and handle decompositions

135

I f j ■Mo M is a hom otopy equivalence , its Whitehead torsion' t(j) E Wh( irl M) = S (M) is the torsion o f the finitely based acyclic complex over Zf rt i M] derived from the induced filtration M 0 C M j C. . . C Mm o f the universal covering M o f M . Proof o f 5.12. By 5.8 , there is an s-decomposed finite complex Z and a blocked equivalence f : M -* Z , giving , by 5.7 , a simple equivalence Mj -* Zj for each i . Then the induced map f* : C*(M, M0)

C * (Z ,Z 0)

of complexes associated to the filtrations of the universal covers is an isomorphism commuting with the action of covering translations . As f is blocked , it maps a prefered basis on the left to a prefered basis on the rig h t. [Recall t h a t , for each handle component (H, 3_H) of (Bj+ j , 3_Bj+i) »one forms a basis element for C*(M, M0) as follows : One chooses at will a lifting (H, 3_H) in M , and at will an integral homology generator on (H ,3_H ) ~ (B1, 3B1) , giving an element of Hj(Mj+ j,M j" ) = Hj(Mj+ j,M j) =

Cj(M,M0)-

This is the basis

element for (H, 3_H) . Similarly in C(Z, Z0) . ] Now the torsion r(j) of j : M0 C* M equals the torsion of Z0 CL* Z , which Milnor’s algebraic subdivision theorem [Mi^ , §5.2] proves to be that of C*(Z, Z ) . As the latter is based isomorphic to C*(M, M0) , the proof is complete . ■ We close this final section by explaining why Poincard duality expresses itself at the chain level by an algebraically simple chain homo­ topy equivalence . Complete proofs would involve the commodious machinery of simple hom otopy theory and thus require an extravagant amount of space . We we will give proofs in outline , as in [Wa] , leaving the reader to fill in the details ; some points are clarified by Maumary [Mau] . Consider a TOP compact manifold triad (M ; M_ , M+) , 3M = M_ U M+ , M_ n M+ = 3M+ . There always exists a finite simplicial ^ The parallel statement and proof apply to Reidemeister representation torsions [Mig , § 8 ] .

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136

complex triad (X ; X_ , X+) and by 5.3 a simple hom otopy equivalence of triads f : (M ; M_ , M+) -» (X ; X_ , X+) . It is understood here that M , M_ , M+ and M_ n M+ = 9M+ have the canonical type of §4 . The map f gives a simple equivalence of all four of these . THEOREM 5.13. In this situation ( X ; X_ , X +) is a (simple) Poincari duality triad in the sense o f Wall [Wa , p .23] . As this is independent of the choice of (X ; X_ , X+) , see [Wa , p.24] , it is convenient to say that (M , M_ , M+) itself is a simple Poincari duality tria d . Proof of 5.13. Let D be a normal disc bundle to M embedded as a clean CAT (= DIFF or PL) submanifold of some Rn , using the (local) Product Structure Theorem [KS$ ,5 .2 ] we arrange that the three manifolds D+ = D IM+ , D_ = D I M_ and the sphere bundle D are clean CAT submanifolds of 9D (possibly with corners if CAT = DIFF) . Then we choose a CAT triangulation of D giving CAT triangu­ lations of the submanifolds D , D+ , D_ . This makes D a simplicial complex . The proof centers on a homotopy commutative diagram of chain hom otopy equivalences (with shift of dimension) , over the group ring A = Z [ tt,M] C*(M, M_) (☆) [M ]n

C *(M ,M .)

r* ------->•

C*(D, D_)

[D]Ci — — * C*(D, D U D+)

i* [M ]n \^ ^

«--------------------------------r ...

nu

C*(D, D+)

As (X; X_, X+) can be (D; D_, D+) our task is to show that i*[M] Cl is always a simple homotopy equivalence . With the one im portant difference that we use singular theory , we adopt the conventions of Wall [Wa , Chap. 2] . For example , for any pair (X, Y) with X mapping naturally to M (as does D) , C*(X, Y) is the complex of singular chains of X modulo that of Y . Here X is the covering of X induced from the universal covering M -*■ M . Similarly for Y. This complex C*(X, Y) is a free right A complex

§5. Simple types and Poincare duality

137

via the free right action of 7rjM on X as covering translations . C*(X, Y) is Hom^(C*(X, Y), A ) , converted from a left to a right A module using the involution of A given by g +> w 1(g)g” 1 G A , g E 7rjM . Recall that w 2 : ir1M -> Z2 is the 1st Stiefel-Whitney class of M . C*(X, Y) is thought of as a graded group with C1 in dimension -i and with differential 5 : C1+1 6 , but not always for dim M < 6 [ Sio ] .

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If we can prove that the Thom equivalence HU is simple , it will follow that [M ]n is simple . To check the simplicity of (TU is easy if M admits a handle decomposition on a (collar of) M+ . Since the result is clear for just one handle , it can then be established conveniently by induction on the number of handles . When M does not admit such a decomposition , we can reduce to the case where it does ( § 2, § 3 ) , by producting everything with a disc to reach dimension > 6 . Alternatively reduce to this case by pulling back D over a copy M' of D ; with this approach (M,M+) gets replaced by a CAT pair (M', M+) , and § §2 and 3 are not needed . Thus Poincard duality is simple .

■

139

Appendix A ON A V O ID IN G S U R G E R Y BY USE OF ST A B L E ST R U C T U R E S

A .I. It is regretable that the very basic results of this essay depend via the Product Structure Theorem [KS5 , §5] on the sophisticated nonsimply-connected surgery used in [Kij ]. One aim of this appendix is to show that this unfortunate dependence disappears if we reformulate all our results in the realm of STABLE topological manifolds of Brown and Gluck . Thus the basics of the theory of STABLE manifolds are relatively elementary . What at present requires the sophisticated surgery is the fact [Kij ] that in high dimensions STABLE manifolds are precisely TOP manifolds . A nother aim is to show that this surgery is entirely unnecessary to well-define the simple type of a TOP manifold as in § 4 . A.2. Recall that a homeomorphism h : U -►U' of open subsets of Rm is called STABLE if for each point x G U , there is an open neighborhood Ux of x in U so that h lU x is isotopic through open embeddings to a restriction of a linear isomorphism Rm -►Rm . When such an isotopy exists , it can be made to fix x . The topological isotopy extension theorem [EK] shows that the above definition is equivalent to the more complex definition of Brown and Gluck [BrnG] . Then the STABLE homeomorphisms of open subsets of Rm form a pseudo-group [KN] , so we derive the notion of a STABLE mmanifold without boundary . If hj. : M -*■ M ', 0 < t < 1 , is an isotopy through open embeddings where M and M' are STABLE manifolds , and h0 is a STABLE isomorphism onto its image in M' , then h x is likewise STABLE . The proof is straightforward . Hence if h : M -*■M' is a homeomorphism of connected STABLE manifolds (without boundary) that is STABLE on some open subset , then h itself is STABLE . This is the cardinal fact about STABLE structures .

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STABLE m-manifolds with boundary can be defined similarly ( as in [ I , §5] ) by using R+1 in place of Rm . There is an equivalent definition as follows . A STABLE structure on a manifold M with boundary 9M consists of a pair (2 ,a) consisting of a STABLE structure 2 on in-tM and a STABLE structure a on 9M such that , for one (or any) collar 9M X [0,1 ] of 9M = M X 0 in M , the STABLE structures 2 and a X (0,1) coincide on 9M X (0,1) C intM . The pair ( 2 , a) together with the collar determine an unique STABLE structure 2 ' in the first described sense , with model R+1 • And 2 ' clearly gives back 2 and a by restriction . The choice of collar does not affect 2 ' , since collars are (locally) unique up to isotopy [ I , Appendix A ] . We now give boldly the recipe for translating the results of this essay into the realm of STABLE manifolds : In the definitions and hypotheses , as well as in the conclusions o f each theorem , each manifold and homeomorphism which is mentioned or could naturally be m entioned is to be supposed S T A B L E . Every condition o f mutual compatibility which can be naturally expressed fo r the STABLE structures is to be insisted upon . In practice this recipe should lead to a perfectly unique maximally STABLE version of each notion and re s u lt. The reader is however cautioned not to overlook manifolds given implicitly which should be supposed STABLE . Let us illustrate by describing a STABLE Morse function f : Mm R where M is a STABLE manifold w ithout boundary . Let C be the set of critical points . For each point y G R , f -1 (y) - C is a “ naturally expressed” manifold , so it is supposed to have a given STABLE structure . (This is extra equipment necessarily given with f !). These structures are naturally subject to a compatibility condition : for each x G M-C , there must exist U open in Rm - ^ X R = Rm and a STABLE embedding h : U -►M-C so that x G hU , and fh = p2 : Rm -*■ R , and so that , for each y G R , the restriction of h to U n t R 1" - 1 X y) -*■ f -1 (y) - C is STABLE . Thus flM -C is a STABLE submersion in the best sense . About each critical point p for f there is a STABLE chart h : Rm M with h(0) = p so that fh(x) = f(p) + Q(x) for x near 0 , where Q (xx xm ) = ± x\ ± x | ± . . . ± . Further , hear 0 , h is a STABLE embedding of each smooth surface CT1(constant) into a level surface o f f I (M-C) .

Appendix A.

Eliminating surgery by using S T A B L E structures

141

The STABLE results are in each case^ to be proved like the corresponding TOP result in § § 1-5 . Whenever the Product Structure Theorem was applied for the TOP result the STABLE version of it [ I , §5] (using no surgery!) can be substituted when we rely on the compatible system of STABLE structures that is carried (and developed) throughout the STABLE proof . A. 3. The TOP h-cobordism theorem without surgery?? It seems plausible to us that there is a still undiscovered p ro o f, without difficult surgery , of at least that part of the Stable Homeomorphism Theorem (for dimensions > 5 ) [Kij ] called the Annulus Theorem , which in dimension n is equivalent to surjectivity of 7r0(TOPn_ j) -»-»■ 7r0(TOPn ) . To support this prejudice , note that the surjectivity irj(TOPn_ |) -*-► 7r-(TOPn ) , 0 < i < n > 6 , requires only the DIFF or PL s-cobordism theorem and some geometry . (See [SijQ , §5] for exam ple). Anyone fortunate enough to discover such an elementary proof (for n > 6 ) could for example show (cf. [BrnG]) that any simply connected h-cobordism is STABLE and so acquire an elementary proof of the TOP h-cobordism theorem . Simple type without surgery . We turn now to the m atter of well defining the simple hom otopy type of a TOP manifold (in §4) without reliance on surgery The surgery intervened in the two applications of the Product Structure Theorem of [I] : one application for constructing the simple type , one for proving its well-definition . We can clearly replace these two applications by applications of the STABLE Product Structure Theorem (independent of surgery) as soon as we can prove PROPOSITION A.4. L et M be a TOP manifold o f which an open subset M 0 D int M has a given STA B LE structure. I f dM C M gives a tt j surjection ( o f connected spaces) , one can fin d (by elementary means) a unique STABLE structure on M X [-1 ,1 } extending the given STABLE structure on M 0 X l - l , 1) . ^ There is one exception . The discussion o f elimination o f double points by Whitney’s method should follow (S ij q , §7.3] not our treatment in §3 , which relied on surgery. ^ T. A. Chapman’s recent more general results [Chj ] [C l^ l [S i{4 ] certainly do not require surgery either .

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Indeed, in revising the proof of 4.1 one applies this once to D and to B . An allowable stabilization of each by X [-1 ,1 ] is introduced . The above proposition follows from the next proposition applied to 9(M X [ - 1 ,1 ] ) . To see the implication recall the cardinal fact that STABLE structures on a connected open manifold coinciding near some point coincide throughout . PROPOSITION A.5. Let M be a connected open TOP manifold and let M 0 C M be a connected open subset with a given STABLE structure . I f 7Ti M 0 -* 7txM is a surjection , one can fin d (by elementary means) a unique STABLE structure on M that extends that o f M 0 . Some preliminary remarks (from [B rG ]) are required for the p roof. Given any topological manifold M without boundary there is a canonical covering space p : M -►M where M is equipped with a STABLE structure . M is in a sense the space of STABLE structures on M ; to construct it , form the disjoint sum of all UQ where U is an open subset of M and a is a STABLE structure on U , then identify x E Ua to another such point x' E \3' * whenever , as points of M , x equals x' , and o equals a' near x in M . To see that the natural map p :M M is a covering , use the cardinal fact again . (By the STABLE Homeomorphism Theorem of [Ki ] , we know , for dimM f 4 , that M = M and p is the identity . But we avoid using this theorem and treat p : M -*■ M as a possibly nontrivial covering .) Proof of A.5. A STABLE structure on an open subset U of M is the same as a continuous section of p : M -*■M defined over U . Thus in the proposition we are given a section s defined over M0 . Given x E M , let X : [0,1 ] -*■ M be a path/•N joining the base point x0 = A(0) E M0 to */ X( 1) = x . There is a lifting X of X to M unique once we lift x0 to s(x0) . Let s(x) = X(l) define s : M -> M . To see that s(x) is independant of the choice of X , note that a different choice X' gives a loop "X"1© X '" in M which deforms to a loop in M0 ; but as the loop in M0 lifts to a loop in M , so does X” 1© X' . This s is the required continuous section . ■

143

Appendix B. S T R A IG H T E N IN G P O L Y H E D R A IN C O D IM E N S IO N > 3.

This appendix briefly discusses (but does not prove) a strong codimension > 3 straightening theorem due to R. T. Miller , J. L. Bryant and others . It was first attained via simplicial techniques largely unrelated to these essays ! We made use of the version for manifolds in our treatm ent of imbedded TOP microbundle transversality for manifolds in § 1 of this essay . Consider a polyhedron P topologically imbedded as a closed subset of a TOP manifold M . We say that P is locally tame near C C M if C is covered by open charts U o f M with PL manifold structure , such that P H U C_> U is piecewise linear from the PL structure of P to that of U . THEOREM B .l.

[Bry2 ] [BS, Theorem 3] [M ilj]

Let M m be a PL manifold with metric d and P C M a topologically imbedded polyhedron closed and locally tame in M . Provide that m - dimP > 3 , and dM -

5 hence­ forth . The case where P is a PL manifold was solved n e x t. The Stable Homeomorphism Theorem of Kirby [K ij] 1968, reinforced by E. Connell’s PL approximation theorem for STABLE homeo­ morphisms [C nlj] 1963 (or by [ I , § 4 .1 ] ) , provided small ambient isotopies making P CL* M PL on any given open ball . Using the version with majorant approximation of the PL local unknotting

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theorem for codimension > 3 proved in R. T. Miller’s thesis 1968 [Mil j ] , one easily puts together these local solutions to get a global one , cf. Rushing [Ru] . The proof when P is a non-manifold was reduced by J. Bryant (and J. Cobb) [Bry2 ] 1970 to the manifold case . There are several variants and alternatives to this proof , see [Ru] [KS2 ] [RS^] [Edw j ] for information ; torus methods have succeeded , but only for the case of manifolds . From our point of view it would be desirable for some future exposition to (a) suppress the surgery entering via [ Ki j ] ^ , and (b) suppress the delicate PL topology used in [Milj ] and [ B ^ ] . For manifolds , the result should follow gracefully from the PL and TOP unknotting of spheres in codimension > 3 , and topological geometry . We have taken the trouble to mention the general version of B. 1 (treating polyhedra) since it is particularly useful - for example in adapting to TOP manifolds the classical PL general position principles [HU2 , Chap. 5] for mappings of polyhedra to PL manifolds . We refer to [SGH] for a typical example of this applied in TOP engulfing. Of course for B. 1 to apply , the polyhedra must have codimension > 3 . The intuitive idea behind these adaptions is that , so far as locally tame sub-polyhedra of codimension > 3 are concerned , the charts of a TOP manifold are by B. 1 as good as PL com patible. This naive idea (an old one) is capable of making many other PL results topological.

^ This is accomplished by very exacting PL geometry in [Mil^l

145

Appendix C. A T R A N S V E R S A L IT Y L E M M A U SEFU L FOR SU R G E R Y .

In smooth surgery of dimension > 5 manifolds , as presented for instance in [Wa] , the surgery obstructions are met and their invariance ascertained through making certain smooth immersions suitably transverse to themselves and to each another . We shall present lemmas which permit one to carry over the same procedure to topological manifolds . The point is that the manifolds immersed in the TOP manifolds undergoing surgery are standard smooth manifolds such as X Rn , X Rn , etc . immersed with codimension zero^ ; and the expected intersection dimensions of the ‘cores’ S^X 0 , B^X 0 , etc . are < 2 (even < 1 it seems) . This will permit us to adjust the immersions producing a smooth open subset of the target manifold so that the immersions become smooth open immersions into this open set . Then the procedures of smooth surgery apply . The tools we need for this smoothing are —(a) a relatively new topological general position principle from [Ed5 ] [Su] [Ur] . —(b) handle smoothing of index < 2 (as in [KS j ] , cf. [Sijq , § 5 ]) corresponding to the vanishing of 7Tj(TOP/DIFF) , i < 2 . Now (a) could be replaced for surgery applications by the MillerBryant straightening theorem B.l above ; however we prefer to introduce the reader to this new and extremely useful topological principle . As explained to us by R. D. Edwards , it requires nothing stronger than simple engulfing in the sense of J. Stallings . The reader will recall that (b) requires the full force of DIFF surgery ; but that is a reasonable price to pay for TOP surgery . Finally some hints for those readers who would prefer to set up TOP surgery without using this appendix . First recall that the surgery obstructions are encountered with immersions for which the expected ‘core’ intersections are of dimension 0 . This case is easily dealt with ^ i.e. k + n is the dimension of the target and the immersion is open ; it is thus equally a submersion .

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using only the Stable Homeomorphism Theorem of [Kij ] (with [I, § 4 .1 ]) ; indeed we have already done as much in §3.4 and [ Si j q , § 7 ]. Thus invariance of the surgery obstructions is the only real problem . But there is a stronger (and perhaps more difficult...?) result known , the invariance of surgery obstructions for Poincard spaces . See [J] [Qn] [Hod] [LLM] for the multiple approaches to this . In the simply connected case , there is a brief homological argument [BrH] for invariance of surgery obstructions . The promised sequence of lemmas has to do with the following situation : Vw is a TOP manifold without boundary ; M C Vw is a closed subset^ and g : Vw -> Ww is an open immersion (= submersion) into a TOP manifold W such that glM is proper Recall that one says M is k-LC imbedded* ( k > 1) in V if V -M is k-LC in V , i.e. if for each point x £ M and each neighborhood N of x in V , there exists a smaller neighborhood N ' C N of x so that each continuous map -*■ (N' - M) is null-homotopic in N - M . Next recall LC^ means j - LC , for j = 0 ,1 ,.. .,k . These are local properties. One says that a closed set M C V has Stan’ko hom otopy dimension of imbedding m in V and one writes t demM = m in case -1 < dimM < w and M is LCw -m -^ imbedded in V but not b e tte r . If dimM = - oo^ or w we set demM = dimM . Here dim is the usual covering dimension of M [Na] . Observe that dem = dim for a subpolyhedron of euclidean space ; dem < k closed subsets share some key properties of dim < k sub polyhedra . Note that demM < w - 1 (not - 1) .

Appendix C. Demension and topological general position

147

Given the facts set out above , one easily verifies that any closed subset of a dem < k set in V is of dem < k ; also as V is complete metrizable , a closed countable union of closed dem < k sets is of dem < k . GENERAL POSITION LEMMA C .l . Consider an open immersion g : V w Ww o f TOP manifolds , w > 5, without boundary . L et M be a closed set o f dem m in V such that g \M is p roper. L et C C M be a closed subset such that g IC is in general position in the sense that the closed subset o f V S(g IC) = (x E C I ] y G C , y ^ x , g(y) = g(x)} is o f dem < 2m - w . Finally let D C M be co m p a ct. Then there exists a regulart hom otopy g t : Vw -*■ Ww , < 1, o f g 0 = g fixing g IC and with compact support in V , to an immersion gi : Vw -*■ M'u which is in general position (as above) on C O D . Further g ( can be as near to g as we please . Proof o f C.l . In view of the relative nature of this result we can assume that gl D is injective . This uses the basic fact (proved readily by contradiction) that if glD is injective then g' is injective near D for every immersion g' : Vw -►Ww sufficiently near to g . In case g I D is injective , we choose an open neighborhood V0 of D - C in V , having compact closure in V , so that D - C is closed in VQ , and g lV Q is injective . Then we use [Ed^] [Su] [Ur] (the noncompact general position theorem involving dem for embedded closed sets is further explained in C.3 below) to obtain a small isotopy h^ , 0 < t < 1 , of id I g(VQ) such that g(D - C) meets the dem < m set gC in general position , i.e. dem{x E D —C 1 3 y ^ C , hjg(x) -- g(y)} < demC + demD —w . If ht is identity g^ = h tg of g as

chosen (majorant) sufficiently near id I g(VQ) , it extends as the outside g(VQ) to an isotopy h^ of id IW . Then setting on V0 and gt = g elsewhere in V we have a regular homotopy required . ■

^ A regular hom otopy is here just a hom otopy through open immersions.

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We will want to a sharper result if M C V is a manifold with several components o f various dimensions . Then we need a finer notion of general position as follows . Suppose M = Mt U . . .U Mk , where each Mj is closed in Vw and of dem < m ^. C.2. Complement to C .l. The general position lemma holds true when the following meaning is assigned to general position fo r g IC (and similarly fo r g I(C U D)) : For each ordered pair o f integers i, j in [l , k] ( i - j allowed) we suppose , writing C- fo r C th a t: Sjj(gl C) = {x € Cj I 3 y e Cj , y f x , g(y) = g(x)} is o f dem 5 , are pushed to meet in general position . This is just what is needed for C.l and C.2 above . For any continuous e : VQ -> (0, °°) , (called a majorant) one can find a closed polyhedron P in VQ with dimP < demX = x so that X lies in the e -neighborhood of P and can be ambient isotopically moved arbitrarily near P by e-isotopies ; P will be called an e-spine o f X in VQ . (The notion is due to J. Bryant [Bryj ] and Stan’ko [§ tj]). This P can be the x-skeleton of a sufficiently fine PL triangulation of VQ ; because the dual skeleton P* (of dimension w - x - 1) can be PL pushed a little to miss X , by the device (R. D. Edwards’) of simplex-bysimplex engulfing^ of P* with VQ- X , using Stallings’ engulfing and connectivity facts we have mentioned above C .l. ^ One can add that w = 4 is permissible provided each mj is > 2 ; the technique explained in C.3 below applies unchanged . ^ This argument fails if demX < 1 , i.e. if dimP* > w - 2 ; but the low dem permits another argument [B r y jllE d j] . This low dem argument breaks down for w < 4 : for w = 3 the general position result is false as stated , cf. [McR] [Ur] .

149

Appendix C. A transversality lemma for surgery

Similarly Y has an e-spine , y = demY . We observe that , if f and g are e-pushes^ of X and Y respectively sufficiently near to P and Q , then fX n gY has P Pi Q as e-spine . Also dim(P H Q ) < x + y - w i f P and Q are in PL general position , as we may suppose . Composing such pushes for infinitely many such pairs of en-spines , where the en : V Q (0 , °°) are majorants tending to zero and carefully chosen by induction■£, one obtains e-pushes f ^ , so that f^X n gooX has en~spines of dimension < x + y - w for all n . Then f^X n gooY is easily seen to be as highly LC^ imbedded as its en-spines , which implies that dem ff^X n g^Y ) < x + y - w . ■ SMOOTHING LEMMA C.4. Consider an open immersion g : Vw-+ Ww o f a C AT (= DIFF or PL) manifold w ithout boundary Vw to a TOP manifold Ww . L et C C V be a closed subset such that g is ‘weakly sm o o th ’ near C in the sense that fo r some open neighborhood o f C in V , g imposes a CAT structure Cl on g V ^ making V(j -§■ (%Yq)q, a C AT immersion . L et D C F be a compact set such that S = { x G D I 3 y G C U D , y f x , g(y) = g(x)} is o f dim < 2 . Finally provide that vv > 5 . Then there exists a CAT concordance T : 2 ~ 2 ' rel C with compact su p p o rt, from the given CAT structure 2 o f V to a new CAT structure 2 ' such that g : Kyy W is‘weakly sm o o th ’ near CUD. Proof o f C. 4 . As this lemma is strongly relative , we can assume that g is injective on an open neighborhood V p of D in V . Let VQ = gVpj arid 2 0 be the structure on VQ making V q ->■ (V q)^ a diffeo­ morphism . Then it suffices to find a concordance r o : 2 0 ~ 2 ^ rel gC with compact support to a structure 2 ^ coinciding with Cl near gS . Indeed the wanted concordance T can then be defined to have compact support V p and make (V p X I)p ^ (VQ X I)p a diffeomorphism . ^ We mean homeomorphisms o f VQ that are e-isotop ic to

id IVQ .

t To arrange that , gM are hom eom orphism s , one can apply to the 1-point compactification o f VQ the fact that for any metric compactum (C, dc ),th e space o f homeomorphisms C -*■ C has complete metric d(h, h') = sup{dc(h(x), h'(x)) + dc(h-1 (x), h '- 1 (x)) I x GC }.

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The existence of r o follows from the next lemma using substitutions we shall indicate presently : LEMMA C.5 . Let A be a closed subset o f (covering) dimension < 2 in a TOP manifold W w ithout boundary and o f dimension > 5 . L et Ua , Vp be open neighborhoods o f A in W equipped with C A T (= DIFF or PL) structures a and (3. Suppose a = j3 near a closed set B C A . Then there exists a concordance T : a ~ a ' rel B to a C AT structure a' that equals (3 near A . Proof o f Lemma C.5 . According to the classification theorem [IV, 10.1 -1 0 .9 ] , the obstructions to finding T lie in the & ch cohomology groups H*(A, B ; ffj(TOP/CAT)). These vanish for i > 3 since dimA < 2 ; and they vanish for i < 2 since 7Tj(TOP/CAT) = 0 for i < 2 by [K Sj] ,cf. [Si1 0 ,§ 5 ] . ■ Remark. - In case demA < 2 , one can (with due care) prove this more directly using an e-spine P2 of A in Ua and solving CAT handle problems of index < 2 , as the following sketch suggests for demA = dimP = 1 . ■

Here now are the substitutions in C.5-for the construction of T in the proof of C.4 : W +* W ; U +* VQ ; V +* gVc ; ; j3 +>■ CL ; A **■ gS ; B +> g(S (T C) . Lemma C.5 yields a concordance Z Q ~ relg(S C iC ) , to a structure equal £2 near gS . By the Concordance Extension Theorem [I, §4.2] we deduce another concordance r o : ~ 2 ^ rel g(C H V q ) and with compact support , to 2 ^ equal J2 near gS , as required to prove C.4

Appendix C. A transversality lemma for surgery

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Applying the topological general position lemma C.l & C.2 then the smoothing lemma C.4 together with the (relative) Concordance Implies Isotopy Theorem [I, §4.1] , and finally a DIFF transversality theorem (essentially the Sard-Brown theorem [M iy]) one routinely deduces : TRANSVERSALITY LEMMA FOR SURGERY C.6 . Consider an open immersion g : Vw -+ Ww o f a DIFF manifold w ithout boundary Vw to a TOP manifold Ww . Let M C V be a DIFF submanifold (with components possibly o f various dimensions f t , so that g\M is p ro p er. Let D C . M be co m p a ct, and let C C M be a closed su b se t. Suppose th a t, fo r some open neighborhood V q o f C in V , the restriction g I V q is weakly smooth in the sense that it imposes a DIFF structure LI on g V g making V q ^ (gV(j)^i a DIFF immersion . Suppose also that g \C is in DIFF transversal position in the sense th a t, i f g(x) = g(y) fo r x f y both in C , then the images under the differential Dg o f the tangent planes T^M and T^M are linear transversal. Finally provide that w > 5 , and that fo r all x 6 C U D and y ( E D , one has dimT%M + d im l^ M - w < 2.% Then there exists an arbitrarily small regular hom otopy gt : yw Ww ' 0 < r < 7 , o f g = g0 relC with compact support in V , such th a t, on some neighborhood V(y\j£) o f C C D in V , g x is weakly smooth , and g \C C D is in transversal position , in the sense explained a b o ve. ■ For manifolds with boundary one deduces a parallel result exploiting suitable collarings of the boundary . (Beware a dimension 3s 5 condition for 3 V ) . This lemma helps one establish for TOP manifolds just as for DIFF manifolds not only the basic theorems of Wall’s book [Wa] but also many recent geometrically proven theorems on surgery such as the splitting theorems of Farrell & Hsiang [F H j] and of Cappell [Capj ] . ^ Against our usual conventions . $ Generalization : If g ID is an imbedding . gt I D remains an imbedding throughout any small regular hom otopy o f g . Thus , it suffices in this case to assume dimT x M + dimT y M - w < 2 merely for x £ C (not C U D ) , and y £ D .

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It can do no harm to describe how a TOP surgery is done (cf. [Ls] [Waj] ), at least in the absolute case. One is faced with a degree one map f : Ww -* Xw from a closed TOP w-manifold W to a closed manifold or Poincar 6 space X ; and as extra data one has a TOP microbundle £ and a stable microbundle map V?: r(W) £ covering f . The pair (f, is called a normal map . Given an element x G 7rk + 1 (f) , k < w , we shall see that a surgery on x is possible precisely if a certain regular homotopy class of immersions Sk X Rn Ww , n - w - k , contains an imbedding.(A. Haefliger introduced these immersions in the early 6 0 ’s.) The element x G 7rk+ j ( f ) is represented by the comutative square at left, which we cover by a commutative square of stable microbundle maps at right : f V ► X t (W) £ W ----------3g

J

9H = Sk XRn C

|

§

> Bk + 1 XRn = H

97 f r(9H )

{ 7

■> r(H )

Given g , the map 7 exists since the handle H is contractible; then 7 determines 8 7 uniquely. (By convention, for a manifold M with boundary, the tangent bundle t(M ) is r(M+) , where M+ is the open manifold obtained by adding an open collar (8M)X(-o°,0] along 8 M = 8 MXO ; and r ( 8 M) r(M) is defined using a collaring of 8 M - see [V, §4] . ) The stable map 8 7 is nevertheless , up to micro­ bundle homotopy, in a unique class of non-stable maps, because 7rk TOP = 7rk TOPw for k < w , by a stability theorem [KS] ] [SijQ, §5.3] [V, §5] . Thus, by TOP immersion theory [Ls] [Las 2 ] [V,App.A] , we can arrange that 8 g is an open immersion while 8 7 is its differential. By a parallel argument this process determines a unique regular hom otopy class of immersions 8 g (except for reflection in Rw_k ). Incase 8 g is an imbedding, surgery on x is done as follows: To WX[0,1] , attach the compact handle Bk XBn C H , n = w - k , by imbedding ( 8 Bk )X B n into into WX1 using 8 gXl , to get Z say. Referring to the right hand square, observe that, as soon as is made a product along the first stabilising factor, \p and 7 together will determine a stable microbundle map : r (in tZ ) -> £ , over the map F :Z X determined by f and g . ( Pause to check this! ) Certainly can then be modified to extend as 4> : r(Z + ) £ , leaving it unchanged near WXO . The surgered norm al map (f\) . A key question to decide, for k < w / 2 , is whether immersion 8 g is regularly homotopic to an imbedding. It always is by general position (C.l) if k < w/2 . For k = w/2 our transversality lemma (C. 6 ) leaves 8 g | Sk X0 with just isolated double points of smooth transversal intersection. A suitable way of counting these (relative to a fixed fundamental class) gives Wall’s self-intersection number ju(8 g) ; and a further application of C . 6 , with 1-dimensional self-intersections, reveals that ju(8 g) is regular homotopy invariant, i.e. ju(8 g) is p(x) . For w > 5 , Whitney’s process (see [IV, § 3.4] [SijQ, § 7.3] ) permits a regular homotopy of 8 g to an imbedding precisely if p(x) = 0 . These p(x) determine a nonsingular quadratic form on 7rk + j( f) when f is k-connected, which, modulo hyperbolic forms, is Wall’s obstruction (for w > 5 ) to surgering (f, 6 (or > 5 if 9M = ). In Essay I we have used results of handlebody theory to establish the Product Structure Theorem [I, §5.1] relating smooth C°° (= DIFF) or piecewise-linear (= PL) manifold structures on M to those on M X R , where R = (real numbers}. It says , roughly , that the classifications up to isotopy are the same , a bijection of isotopy classes being given by crossing with R Recall that two CAT structures ( CAT = DIFF or PL ) 2 and 2 ' on M are isotopic if there is a TOP isotopy (= path of homeomorphisms) ht : M - » M , 0 < t < l , from h0 - id IM , the identity map of M , to hj : “*■ a homeomorphism giving a CAT isomorphism from M with the structure 2 to M with structure 2 ' . Our goal in this essay is to use our Product Structure Theorem and some well-established ideas of earlier smoothing theories to prove a serviceable version of the CLASSIFICATION THEOREM (see § 1 0 ). L et M be a (metrizab le) topological manifold o f dimension > 6 (or > 5) i f the boundary 3A/ is em p ty). There is a one-to-one correspondence between isotopy classes o f PL [respectively DIFF] manifold structures on M and vertical hom otopy classes o f sections o f a (Hurewitz) fibration over M . I t is natural fo r restriction to open subsets o f M . This fibration is the pull-back , by the classifying map M -*■B jQ p for the tangent microbundle r(M) of M , of a fibration Bpj^ -*■ ®TOP tresP • B'q -»• Bt o p ] hom otopy equivalent to the usual ‘ forgetful ’ map Bpp -»■ Bj q p [ resp. B q -► B-pQp ] , of stable classifying spaces . The fiber , called TOP/PL [ resp. TOP/O ] , has hom otopy groups 7Tj(TOP/PL) equal zero if i f 3 and equal Z2 if ^ Strangely enough, this is false in some cases where M has dimension 3 or 4, see [I, § 5 ] , [Sig] .

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i = 3 , see [K Sj] , [ resp . 7Tj(T0 P /0 ) = © ^ T j if i > 4 , and ^(TOP/O) = TTjdOP/PL) if i < 6 ] , c f . [Si1Q] or § 1 0 . The version of this theorem for PL structures was announced in [KSj ] . The proof outlined there proceeded handle by handle and used results of handlebody theory , surgery and immersion theory . But as we suggested there , this was more extravagant than necessary . It has turned out that one can purify the arguments of [KS j ] by following either one of two radically different general methods . One method (pioneered by C e rf, Haefliger , Lashof , and Morlet [Ceg] [Haj ] [Las | ] [Mor2 3 4 ]) is to rely heavily in one way or another on ideas of immersion theory ; we shall exploit this method in Essay V getting more precise (unstable , sliced) classification theorems , at the cost of introducing numerous semi-simplicial spaces and fibrations . As we have stated, the course we follow here relies on the Product Structure Theorem , hence chiefly on handlebody theory . We were encouraged to adopt it by some extremely direct and natural applications in [III] of this Product Structure Theorem to TOP manifolds , which , we hope , assure its role as a landmark in the theory . An analogue of the Product Structure Theorem for compatible DIFF structures on PL manifolds , called the Cairns-Hirsch theorem has become well known since 1961 [Ca] [Hi2l [Hi^] . It allowed a classification of compatible DIFF structures quite analogous to the theorem of this article The question of existence was first decided by Milnor [Mi3 4 5 ] , who introduced microbundles for this purpose . The corresponding uniqueness question proved technically more delicate ; it was worked out by Hirsch and Mazur and by Lashof and Rothenberg . (See Morlet [Mor j ] for an outline of the theory as of 1963 . See also [MazP] [HiMj ] [LR j ] [Wh] and [HiM2 l .) After this work , it became quite clear to the well-informed that a similar classification for DIFF structures on TOP manifold would be feasible if and when our Product Structure Theorem could be proved . This made us hesitate to write down the details . But once started we made the exposition both leisurely and elementary . We are fortunate that no technical bugbear as nasty as the functorial triangulation of vector bundles [HiM j ] [LR j ] ever intervenes . We now briefly sketch the proof to come . ^ This classification was constructed against the background o f T h o m ’s conjectures |Th-,] and an obstruction theory of J.M unkres [Mu^] .

§0.

Introduction

157

Milnor’s method for introducing a CAT (= DIFF or PL) structure on the TOP manifold M involves the observation that when M is embedded in euclidean space Rn as retract of a neighborhood N by r : N -» M , then the pull-back r*r(M) over N of the tangent microbundle r(M) of M contains a copy of M X Rn as an open neighborhood of embedded M . There results , given the Product Structure Theorem , a mapping from concordance classes of CAT microbundle structures on r*r(M) or on r*r(M) © es , s > 0 , to isotopy classes of CAT structures on M ; we call this mapping the smoothing ru le. We proceed to prove in § § 4-6 that this smoothing rule is a bijective correspondence provided that M C* N is a homotopy equivalence (which is trivial to arrange once M is known to carry a CAT stru ctu re ). The stable relative existence theorem for normal microbundles makes this bijectivity fairly easy to verify; for the sake of completeness we explain M. Hirsch’s proof of this theorem [Hi^] in Appendix A . (It was first established by R. Lashof and M. Rothenburg in [LRj ] , for a similar purpose.) It remains then to recall, in §8 and §9 , the classification of stable concordance classes of CAT micro­ bundle structures on r*r(M ) by sections of a fibration as described above. Then the full classification theorem is presented in § 10. Hirsch and Mazur initially followed a slightly different route to the classification of CAT structures on M (see [HiMj ] ) , which we quickly see to be equivalent in §7 . It reveals that the isotopy classes of CAT structures on a CAT manifold M naturally form an abelian group. Some features o f our exposition are intended to make the results more readily applicable . We take care to describe how to pass efficiently both from a manifold structure to a bundle section , and from bundle section to a manifold structure . And at every stage we give a relative version . Since we will discuss certain non-manifolds in connection with topological and piecewise bundles ?we adopt the following broad meanings for TOP and PL in this essay . TOP indicates the category of continuous maps of topological spaces ; PL indicates the category of piecewise linear maps of (locally compact) polyhedra . However , DIFF (as always) indicates the category of C°° smooth maps of finite dimensional smooth manifolds .

T58

L. Siebenmann

Essay IV

Our basic conventions about manifolds remain those of Essay I , §2 . Thus manifolds mean metrizable manifolds with boundary (possibly an empty b o u n d ary ), having the same dimension in each com ponent. DIFF manifolds usually do not have comers ; but in a few situations , where they obviously should have , we allow them without mention . For example if T is a DIFF structure on M X I , I = [0 ,1 ] , where M is a TOP manifold with 3M f , then (M X I)p by assumption has corners at 3M X 0 U 3M X 1 of the same sort as X I has . Recall that T is a concordance S 0 to S i if TIM X ( i } = SjX (i> for i=0,l .

159

§ 1. R E C O L L E C T IO N S C O N C E R N IN G M IC R O B U N D L E S

The reader should familiarise himself with the rudiments of Milnor’s notion of microbundle before proceeding ; see [Mi^] . A n-microi p bundle £ over a space X is a diagram £ : X -> E X (X = base ; i = zero section ; E = total space ; p = projection) so that pi = id IX , and in E near each point of i(X) , p looks like the projection X X Rn -> X . Often , without special apology , we will identify X with i(X) and regard X as a subspace of E(£) . According to the Kister-Mazur theorem [Kis] [KuLj ] [SGH j ] there is always a neighborhood U of i(X) in E such that i p lU n £0 : X U > X is a locally trivial bundle with fiber R and zero section i . And £0 is unique up to isomorphism . A micro-isomorphism £x -> £2 ° f n-microbundles over X consists of a neighborhood U of the zero section (X) in the total space E(£x) of £j and a topological imbedding h onto a neighbor­ hood of i2 (X) in E(£2) , making this diagram commute . U

Then £x and £2 are called (micro) isomorphic (= isomorphic in the category of germs of micro-isomorphisms) . If h is an inclusion of a subspace it is called a micro-identity and £1 , £2 are called micro­ identical. This is written £x = £2 . Two micro-isomorphisms h : £1 £2 and h' : £2 -> £3 can be composed to give a micro­ isomorphism h'h : £1 -> £3 . But note that the domain of definition of h'h is in general smaller than that of h .

160

L. Siebenmann

Essay IV

The microbundles defined above are called topological or TOP microbundles . The notion of a PL n-microbundle micro-isomorphism and micro­ identity are defined like the topological entities above but one works in the category of piecewise linear maps of polyhedra (= Hausdorff spaces equipped with a maximal piecewise linearly compatible atlas of charts to locally finite simplicial complexes , cf. [Hu2> § 3 ]) . It is convenient to allow that a topological microbundle is a PL microbundle if it is one after deletion of part of the total space outside of a neighborhood of the zero section . Similarly define DIFF microbundles starting from the category of DIFF manifolds and DIFF maps . If X is a CAT submanifold of Y and £ is a CAT microbundle whose total space E(£) is a neighborhood of X in Y with inherited CAT structure , we say that £ is a CAT normal microbundle to X in Y . (See Appendix A .) The Kister-Mazur theorem mentioned above is valid in all three categories . A stronger version holding for DIFF states that DIFF microbundles are essentially equivalent to vector bundles . However , we retain DIFF microbundles for the sake of uniformity . PROPOSITION 1.1 Every DIFF microbundle £ : M £(£) M over a DIFF manifold M contains a neighborhood E' o f i(M) which can be regarded as a smooth vector bundle with the same zero-section and projection . Furthermore suppose C is closed in M and E'0 is such a vector bundle neighborhood o f the zero section o f £ I U , where U is an open neighborhood o f C . Then E ' can be chosen to coincide (as vector bundle) with E'0 over a neighborhood o f C. A n d if E ” is another choice for E' there is an isotopy h^: E' -> E o f E ' C E through DIFF imbeddings yielding such vector bundle neighborhoods , to a vector bundle isomorphism E ' E ” . Proof o f 1.1 . The uniqueness clause is a well known portion of the uniqueness theorem for DIFF tubular neighborhoods , cf. [Lan] . But it yields the existence assertion by dint of a chart-by-chart construction . ■ An alternative proof is suggested by Milnor [Mi^ , §2.2] . ■

§1.

Recollections concerning microbundles

161

If f : X -»■ Y is a map and £ , rj are CAT n-microbundles over X , Y a microbundle map f : £ -*■ rj covering f can be defined to be a CAT map U -*■ E(r?) of a neighborhood U of X in E(£) that restricts to f on X C E(£) , that respects fibers , and that gives a CAT open embedding to each fiber . (See [Mi^ , § 6 ] .) For example one has a canonical CAT microbundle map f# : f*r? -► i? when f is a CAT map . The fiber product rule defining f*r) shows this is essentially the only example . Indeed , if f is any CAT microbundle map covering f , there is a unique micro-isomorphism \p : £ -* f*r? so that fjj*p - f . Beware that quite a variety of symbols such as f# or

I X (£ I U) , where U is an open neighborhood of C in X , it is possible to choose f to agree^ with f over a neighborhood of C . A sharply relative version ^ states that if C is a closed and a CAT ^ The domains of definition o f f and f' in any fiber may differ nevertheless . $ To prove this sharply relative version , use the more basic Observation : To m ake f agree with given f " over C it suffices to fin d , fo r each p o in t x E C an open neighborhood Vx in X and a C A T micro-isomorphism fx : f I(7 X Vx ) - + I X ( £ \VX ) so that f x and f " agree over I X (Vx C\ C ) . To verify this observation , reread the proof o f the hom otopy theorem in [Mig , § 6] . To then apply the observation to get the given sharply relative version choose V so that £ IVx is trivial and there is a CAT retraction i'x : \ ' x -> V'x n C . Next , as £ IVx is trivial, there exist CAT microbundle maps a and /3 covering I X r x , and themselves retractions : f I (I X Vx )

fx

------- - ----- ►

a 4-

f l ( i x ( V x nc))

I X ( £IVX)

I 0

i x ( £ l v x nc>.

Now there is a canonical CAT micro-isomorphism f making the square commute and clearly f x agrees with f" over I X (Vx n C ) .

162

L. Siebenmann

Essay IV

retract of a neighborhood in X , then given a CAT micro-isomorphism f" : f I(I X C) -»■ I X (£ IC) it is possible to choose f to agree with f" over C . Here is a useful reformulation of the microbundle homotopy theorem , which explains its name . I f f : £ -*• 7 is a CAT micro­ bundle map over f : X -> B say , and F : I X X ->■ B is a CAT homotopy of f , i.e. p2° (0 X f) = Fl (0 X X) , then there exists a CAT microbundle homotopy over F F : I X £ -»• 7 of f , i.e. F is a CAT microbundle map over F with p2° ( 0 X f ) = F l ( 0 X£ ) . Here p2 generically denotes projection to second factor . If F is rel C , then F can be rel C , i.e. be equal to p2o ( l X f) near I X C (the relative version). If F = p2o ( I X f ) on I X C and C is a CAT neighborhood retract in X , then F can coincide with p2° (I X f) over I X C (a sharply relative version) . If £ is a TOP microbundle over a polyhedron X , a PL structure on £ , also called a reduction of £ to PL microbundle , or a PL structure on £ , is simply a PL microbundle £' whose base is the same polyhedron X and whose underlying TOP microbundle is £ . If £ is a TOP microbundle over a DIFF manifold X , we make a similar definition with DIFF in place of PL . A stable CAT structure on £ will mean a CAT structure 17 on £ © es , s > 0 , where es : X - > X X R S->X is the standard trivial microbundle . If rj' is a CAT structure on £ 0 e* , t > s , then we write 77 = 17' if rj © e*-s = r\ ' . This extends the (CAT) micro-identity relation = to an evident equivalence relation = on stable CAT structures on £ . Let £ be a TOP microbundle over a CAT object X (CAT = DIFF or P L ) . A concordance between CAT structures £0 , £1 on £ is a CAT structure 7 on I X £ such that the restriction 7 1i X X is CAT micro-identical to i X £^ for i = 0,1 . (In symbols 7 1i X X = i X £ j , for i = 0,l .) If C is closed in X and there is a neighborhood U of X such that 7 l ( I X U ) = I X (£ol U) , then 7 is called a concordance rel C . Concordance rel C is an equivalence relation , written £0

— £1

rel C .

The microbundle homotopy theorem provides a CAT micro-isomorphism h : 7 -*■ I X £0 that is the identity near I X C . Hence

§1.

Recollections concerning microbundles

163

£0 —£x rel C means that there exists a neighborhood U of X in E (£ i) , and an isotopy h^ : U -*■ E (£ ), 0 < t < 1 , o f the inclusion through micro-isomorphisms of £ to itself to a CAT micro-isomorphism £i £o i further , h^ can fix a neighborhood of C . Compare the Concordance Implies Isotopy Theorem . Next define stable concordance rel C of stable CAT structures £o , £i on £ to mean £0 = ?0 - ?i = Si (~ being CAT concordance rel C) for stabilizations of £0 and f j of . If rj is a stable CAT structure on £ IV for some neighborhood V of C in the base we will write TOP/CAT (£ rel C , r?) for the set o f stable concordance classes rel C o f stable C AT struc­ tures £' on £ that coincide near C with r? in the sense that £' IU = 171U for some neighborhood U of C . If the choice of 77 is evident we omit 17 from the symbol ; and if C = 0 we omit rel C , 77 getting simply TO P/CA T(£). The tangent CAT microbundle r(M) (= Tj^) of a CAT mani­ fold M w ithout boundary is defined by Milnor to be A pi t(M ):M ->■ M X M -*■ M where A(x) = (x, x) and Pi (x, y) = x.

164

Essay IV

§2. A B SO R B IN G THE B O U N D A R Y

Consider a TOP manifold Mm of dimension m , possibly with boundary . Suppose that a CAT structure 2 0 is given on a neighbor­ hood of a closed subset C of M . For m large we aim to settle , first in bundle theoretic terms , then in homotopy theoretic terms , the problem of finding a CAT structure 2 on M that agrees with 2 0 near C , and of classifying all such structures up to concordance rel C if one such exists . The symbol ^ C A T ^ re^ ^ , 2 0 ) will denote the set of concordance classes rel C of such structures . It is abbreviated by omitting Z 0 if the choice of 2 0 is evident (e.g., if M is a CAT m anifold), and by omitting C if C = 0 . The following consequence of the Product Structure Theorem shows that , if m > 6 , it will suffice to accomplish this task when 3M = 0 t PROPOSITION 2.1 Let M' be the open topological manifold obtained from M by attaching to M the collar 3M X !0, °°) identifying x in 3M to (x, 0) in 3M X [ 0 ,° ° ). E xtend 2 0 to a C AT structure 2q on a neighborhood o f C in M ' that is a product along [0, °°) in bM X [0, oo) . Provided that natural bijection

m~>6 (or m = 5 and d M C C ) , there is a

6 : § CAT(M rel C , 2 0)-+ §CAT 5 . ■

166

Essay IV

§3.

M IL N O R 'S C R IT E R IO N FOR T R IA N G U L A T IN G A N D SM O O TH IN G

For the analysis we fix a topological imbedding^ of Mm in a euclidean space Rn , n » m , that is a CAT locally flat imbedding near C ; for simplicity we now regard M as a subspace of Rn . As M is an ANR there exists an open neighborhood N of M in Rn that admits a retraction r : N -*• M onto M (so that r IM = i d ) . We can and do arrange that r is a CAT map to 2 Q on the preimage of a neighborhood o f C . Then r*r(M) restricted to a neighborhood of C in N has a CAT microbundle structure that we call . THEOREM 3.1 Consider the data : M an TOP manifold w ithout boundary o f dimension m > 5 ; C C M a closed su b se t; 2 Q a C AT structure defined on a neighborhood o f C in M ; r : N ^ - M a neighborhood retraction as introduced above ; % the CAT structure on the restriction o f r*r(M) to a neighborhood o f C , derived from 'LQ as above . Suppose that the topological microbundle r*r(M) over N admits after stabilizing a C AT structure £ coinciding near C with %Q . Then M admits a C AT structure 2 coinciding with 2 Q near C. Conversely, i f 2 exists, r *t(M) admits such a C AT structure, even w ithout stabilizing. Proof o f 3.1 . First we dispose of the converse using 3.2 The pull-back rule 7r . Given a CAT structure 2 on M homo­ top r to a CAT map ? : N -*■ • Then , applying the micro­ bundle homotopy theorem to a hom otopy from r to r , obtain a TOP micro-isomorphism from r*r(M) to r*r(M) . Now r*r(M ^) is a CAT microbundle over N (since r is CAT) , which is topologically identical to r* r(M ). Hence the above isomorphism endows ?*t(M) with a CAT microbundle structure £ . ^ It is not necessary to assume that this imbedding is locally flat, or even proper . Hence for n > (m+1 )m such an imbedding is easily deduced from a covering o f Mm by open subsets imbeddable in Rm such that the nerve o f the covering has dimension < m . See [Mu ] , § 2 . 7 ] , or [II, § 1 . 1 ]

§3.

M iln o r's criterion for triangulating or smoothing

167

the above isomorphism endows r*r(M ) with a CAT microbundle structure £ . If C ^ c p and 2 equals 2 Q near C we take some care in making £ agree near C with £Q . Let H : I X N M be a hom otopy from r to r fixing a neighborhood of C . Then H*t(M) is identical to IX r*r(M ) near I X C . Using the microbundle homo­ topy theorem , extend this identification from a neighborhood of I X C to a micro-isomorphism H*r(M) = 1 X r*r(M) over I X N . Over 1 X N it gives a micro-isomorphism from H*r(M)l 1 X N = 1 X r *t(M) to 1 X r* r(M ). Use this micro-isomorphism ?*r(M) -> r*r(M) to define the structure £ on r*r(M) as the image of the structure f* r(M j) on ?*r(M) . The construction 2 +*• £ is called the pull-back rule and is denoted it . ■ It remains to suitably construct a CAT structure on M from a CAT structure on r*r(M) © es , s > 0 , using what we call (even for CAT = PL) : A pi 3.3 The smoothing rule a . Since r(M) is M -» M X M M with A(x) = (x, x) and pj (x, y) = x , the induced microbundle r*r(M) over N has total space Er*r(M) = {(y, r(y), x) E NXMXN } while its projection is (y, r(y), x) +* y e N , and its zero section is i : y +> (y, r(y), r(y)) . There is a natural homeomorphism g : Er*r(M) - > M X N C M X R n given by (y, r(y), x) +* (x, y) which sends i(M) to A(M) C M X N . Composing with the imbedding j : MX Nn -* M X Rn , j(x, y) = (x, y-x) we get an imbedding h = jg : Er*r(M) -> M X Rn that sends i(M) to M X O by i(x) +> (x, 0) . Via h we can think of Er*r(M) as an open subset of M X Rn with i(M) = M X 0 . Clearly h gives a CAT imbedding of a neighborhood of i(C) in E(£q) C Er*r(M) To proceed with the construction suppose that r*r(M) is endowed with a structure of CAT microbundle (w ithout stabilizing) , which agrees near i(C) in Er*r(M) with £Q . Then a neighborhood of i(M) in Er*r(M) has (a fortiori!) a CAT manifold structure ^ This pleasantly explicit description was suggested by M . Brown [Brn^l . Milnor’s description is slightly more general. He uses only the fact that N is a CAT manifold with trivial tangent bundle .

L. Siebenmann

168

Essay IV

which h sends to a CAT manifold structure © on a neighborhood of M X 0 in M X Rn coinciding with 2 Q X Rn near C X 0 . The Product Structure Theorem (local version) produces a concordance of 0 rel C X 0 to a CAT structure that is a product of the form 2 X Rn near M X 0 . This 2 is the wanted structure on M . For the general case suppose that r*r(M) © es over N has a CAT structure %■ As E(r*rM © es) = E(r*rM) X Rs , the imbedding h X (id I Rs) gives us a CAT structure near M X 0 X 0 in M X Rn X Rs from which the Product Structure Theorem produces as above a structure 2 on M . This construction £ +> 2 is called the smoothing rule and denoted o . ■ The proof of 3.1 is now complete . ■ We record the following proposition whose proof is straightforward . PROPOSITION 3.4 (for data of 3. 1). The pull-back rule ir and the smoothing rule o yield two welldefined mappings o TOP/CAT(r*r(M) rel C , £0) §CAT(M rel C > ^ * 7T

Each is natural fo r restriction to open subsets o f M . ■ The meaning of naturality was explained under 2.1 .

§4 . ST A T E M E N T S A BO U T B IJE C T IV IT Y OF TH E SM O O TH IN G R U LE A N D THE PU LL-BACK R U LE .

We now complement Theorem 3.1 by classifying CAT structures on M when one is known to e x is t. CLASSIFICATION THEOREM 4.1 (for data of 3.1) I f M CL+ N is a hom otopy equivalence , the smoothing rule o : TOP/C AT(r*rM

rel C , £Q) -► S CAT^M rel C ’ S o^

is bijective (= one to one and onto) . R em a rk. Once a CAT structure 2 is known on M one can choose M -> Rn to be a CAT imbedding of , thus making trivial the choice of the open set N so that M C_>. N is a hom otopy

§4. Statements about bijectivity

169

equivalence .t THEOREM 4.2 (for same data) . The composition ott o f the pull­ back rule it with the smoothing rule a is always the identity on & C A T ^ re^ ^ ' V ‘ The proofs of 4.1 and 4.2 appear in §6 below . Here is an im portant corollary of 4.1 and 4.2 . THEOREM 4.3 . Suppose M is a CAT manifold w ithout boundary o f dimension > 5 and C C M is closed . Then there is a bijective correspondence oQ : TOP/CAT(r(M) rel C)

§CAT(M rel C)

natural for restriction to open subsets o f M . Its inverse can be described by a pull-back rule like that o f 3.2 . Proof o f Theorem 4.3 from 4.1 and 4.2 . Consider the triangle TOP/CAT(r*r(M) rel C)

°-+

§ CAT(M rel C)

TOP/CAT(r(M) rel C) where the data of 3.1 are so chosen that M Rn is a CAT imbedding and r : N -► M is a CAT deformation retraction . Then the restriction map indicated is easily seen to be a bijection.^ We merely define oQ to make the triangle com m ute . Theorem 4.2 shows a ” 1 can be defined by the following pull­ back r u l e . Let 2 represent x E S ^ ^ j( M rel C) . Find a hom otopy from id IM fixing a neighborhood of C to a CAT map i' : M -► M^; , and using the microbundle hom otopy theorem , derive a micro-isomorphism h : i'*r(M ^) -► t(M ) that is CAT near C . Then ^ However, it is an elementary fact, related to the Kister-Mazur theorem [ Si j 3 ] , that such an N exists whenever M C* N is a topologically locally flat imbedding .

t This requires some care when C f ; indeed , a neighborhood extension property for microbundle maps is wanted . The bijectivity is anyhow implied by Proposition 9.2 below .

170

L. Siebenmann

Essay IV

a “ ‘ (x) is represented by the CAT structure f given to t(M) by h . This interpretation of a ” 1 shows first that oQ is independent of the imbedding M Cl*. Rn and the retraction r and second that a Q is natural for restriction to open subsets . This completes the proof of 4.3 granting 4.1 and 4.2 . ■ 4.4 Supplementary Remarks . (i) If 2 is a CAT structure on M equal the standard one near C the relative simplicial (or smooth) approximation theorem and a pull­ back rule define a natural bijection ip : TOP/CAT(t(M) rel C) TOP/CAT(r(M^) rel C ) . And aQ^p = aQ in view of the pull-back rule defining a ” 1 . Thus 4.3 is meaningful and valid i f M is merely a topological manifold with a C AT structure specified near C . (ii) This suggests that one define a CAT microbundle structure on a TOP microbundle £n over any (paracompact) base space X in a manner independent of any eventual CAT structure on X . Indeed it can be defined as a TOP microbundle map £n -*■ 7 t , there exists a normal microbundle rj to h(I X N) in IX E(f) X Rs that is equal to I X (f © es) , near 0 X N U I X C and that gives a normal CAT microbundle to h (l X N) in 1 X E (f ) q X Rs . Indeed the CAT version of Appendix A first gives 171h( 1 X N) ; then granted an inflation of s the TOP version defines tj on all of h(I X N ) . The microbundle hom otopy theorem now provides a neighborhood U of N in E(J) X Rs and an open imbedding H : I X U -> E( tj) C I X E(f) X Rs that is the identity on 0 X U and a product along I near I X C , and that presents a micro-isomorphism 1 X (J © es) -*• 771h(l X N) . This micro-isomorphism gives the desired new CAT structure on f © es ; the desired concordance (I X U)p of this structure with Uq is just the structure making H a CAT imbedding into E(t?) C I X E (0 © X Rs . ■

Remarks : This brief section contains the one essential intervention of the stable existence and uniqueness theorems for normal microbundles given in Appendix A . The reader may be convinced that this is the right tool to use here by the Observation : Proposition 5.1 can conversely be used to prove the stable relative existence theorem A .l fo r normal TOP microbundles. Indeed as A.l is suitably relative we need only prove A .l when the ambient manifold is covered by one co-ordinate chart ; then 5.1 further reduces A. 1 to the DIFF relative existence theorem for DIFF normal microbundles , which is trivially proved (see end of proof of 1.2 in [III]). In the same vein , one can observe that the classification theorem 4.1 lets one quickly prove 5.1 . Indeed a simple pursuit of definitions suffices , if, in 5.1 , N is an open subset of euclidean space and • f is trivial . But as 5.1 is relative the general version of 5.1 follows . It is reasonable to conclude that the cunning geometry in Appendix

§6.

173

Bijectivity of the smoothing rule

A is the keystone of the classification theorem 4.1 , indeed of this essay ; it should not be neglected .

§6 . B IJE C T IV IT Y OF THE SM O O T H IN G R U LE ; ITS IN V E R S E .

Proof of the classification theorem 4.1 . (i)

a is ontcfi: TOP/CAT(r*r(M) rel C , \ Q)

%C A T(M rel C, XQ) .

Let a CAT structure X representing a given x E § > c a T ^ rel C , X Q) . Then I X R n gives a CAT manifold structure 0 on E(r*r(M)) , using the natural open embedding h : E(r*T(M)) C»

M X Rn

sending i(M) to M X O . Then 5.1 finds an integer s and a concordance of 0 X Rs to 0 ' making r*r(M) © es a CAT microbundle over N representing (say) v E TOP/CAT(r*r(M) rel C , £Q) . Transferring the concordance by h X(id I Rs) into M X Rn+S , we see that the smoothing rule a sends y onto x . ■ o is injective . Let I i ,£ 2 be two CAT structures on r*r(M) © es , s > 0 , agreeing near C in the total space with X Rn+S . Supposing tfCti) ~ °(%2 ) > we are required to show that and | 2 are stably concordant rel i(C) . Supposing cr(£i) = ®(£2) we do clearly get , using h , a conditioned concordance T rel C , from an open neighborhood U of M in E(£ t ) , to U as open CAT sub­ manifold of E(£2) ■ Proposition 5.1 applied over I X N' (where N '= N Cl U) says t h a t , after U is cut down if necessary but not N' = N Cl U , there is a stable concordance of (I X U)p rel (31) X C U I X C , to a CAT structure T' on I X U X R* making I X {r*r(M) © es+ t} I I X N' a CAT microbundle over I X N' . This T' gives a stable concordance f rel C from g j N ' to £2 l N ' . Using a CAT hom otopy of id IN fixing a neighborhood of C to a CAT map f : N -*■N' , one easily derives , via the microbundle (ii)

^ This also follows from 4.2 (proved b e lo w ), but this argument will help the reader to understand the proof of injectivity to follow.

174

L. Siebenmann

Essay IV

homotopy theorem , a stable concordance rel C o f ^ to %2 • Indeed , pulling back by f we get a concordance f*f : f * ^ — f *£2 rel C , but pulling back by the homotopy id IN ^ f rel C we get concordances — f *£j rel C , i = 1, 2 . Here the micro­ bundles involved are all topologically micro-identified to r*r(M) or 1 X r*r(M) via a single application of the relative topological micro­ bundle homotopy theorem to the homotopy (id IN) ~ f rel C . ■ This completes Theorem 4.1 . ■ Remark 6.1. Without the Product Structure Theorem , the classification theorem 4.1 and its proof would remain valid if rel C , 2 Q) were replaced by the set of equivalence classes of CAT structures 2 on M X Rs , s > 0 , coinciding with 2 Q X Rs near C X Rs , under the equivalence O defined by agreeing that S O S X R , and 2 0 2 ' if 2 and 2 ' are concordant rel C X Rs . Proof of Theorem 4.2 : a it = iden tity . This does not require § 5 ; it is for the most part a patient pursuit of definitions . In fact it could well have been presented at the end of §3 ! Given £ representing [£] E TOP/CAT(r*r(M) rel C , £Q) obtained from x E re* C , 2 0) by the pull-back rule , i.e. ir(x) = [£] , we are required to show that a[£] = x . Recall that Er*(rM) is naturally homeomorphic to M X N so that r*r(M) becomes N ^ M X N N where i(y) = ( r ( y ) , y ) . Also the imbedding h : Er*r(M) -> M X Rn is then h(x, y) = (x, y - x ) . To establish a[£] = x amounts to showing that , when £ is regarded as a CAT manifold structure defined on a neighborhood of the zero-section i(N) in E r* r(M ), then there is a neighborhood U of i(N) and a CAT concordance of E(£lU) to h- 1(2 X R s) I U . Following the pull-back rule , let H : I X N -*■ be a homotopy , fixing a neighborhood of C in M , from r to a CAT map r' : N -► • Then E(H* t(M» = {(t, y, H(t, y), x) E I X N X M X M } is naturally identified to IX X N by (t, y, H(t, y), x) +* (t, x, y) . With this identification the zero-section j of H*r(M) is j(t, y) = (t, H(t, y), y) and the projection is the natural one

§6.

175

Bijectivity of the pull-back rule

I X M X N -► I X N . Thus H*r(M) : I X N J-> I X M ^ X N

£ IXN.

Note that H*r(M)l0 X N is 0 X r*r(M) and H*r(M) I 1 X N with total space 1 X X N is the CAT bundle 1X r'*r(M £) • Using the microbundle hom otopy theorem , find a micro­ isomorphism f : I X r*r(M) -*> H* t(M) that is the micro-identity over 0 X N and is the natural micro­ isomorphism over a neighborhood of I X C . Let T be a CAT manifold structure defined near the zero-section in I X Er*r(M) so that f is represented by a CAT imbedding into I X M ^ X N = E(H* t(M)) . Certainly T agrees with 0 X 2 X N and is a product along I near I X i ( C) . The pull-back rule defines % so that T I 1 XM X N gives to 1 X r*r(M) the CAT bundle structure 1X £ . Thus T : 2 X N === E(£) rel i(C) . To conclude it suffices to give a further concordance r : 2 X N =a h_1 (2 X Rn ) rel i(C) , which is in fact easy ; we set T' = J-1 (I X 2 X Rn) where J : I X M X N -> I X M 2 X R n is defined by J(t, x, y) = (t, x, y - g^(x)) with g^. : M -*■ N a homotopy , fixing a neighborhood of C , from MC * N to a CAT map g! : ** N . This completes the proof . ■

176

Essay IV

§7. A P A R A L L E L C L A SSIF IC A T IO N T H E O R EM

When the manifold M has a given CAT structure , an alternative classification of CAT structures is available . Its directness and simplicity are appealing ; what is more , it provides an abelian group structure on §>CAT^M) since Whitney sum makes TOP/CAT(e(M)) an abelian group , e(M) being the trivial bundle over M . THEOREM 7.1 . Let M be a CAT manifold o f dimension > 5 , without boundary. Let C C M be a closed subset and let e(M) be the trivial (zero-dimensional) bundle over M . Then there is a natural bijection o x : TOP/CAT(e(M) rel C)

§ CAT(M rel C) .

The mapping a j , parallel to the smoothing rule of §3 , is defined as follows . I f 0 is a CAT structure on a neighborhood of M X 0 in M X Rs giving es = e(M) a CAT microbundle structure representing xETOP/CA T(M rel C) , then a ^ x ) is represented by any CAT manifold structure 2 on M such t h a t , for some neighbor­ hood U of M X 0 , the structure (2 X Rs) IU is concordant rel C X 0 to 0 1U . It is , to be sure , the Product Structure Theorem that assures us such a 2 exists and has a well-defined class (x) in § CAT(M rel C ), c f. [ I l l , §5.1.1] . Proof o f Theorem 7.1 . A direct application of Proposition 5.1 !

■

In the remainder of this section we relate 7.1 to the classification theorem 4.1 and then discuss an exact sequence of abelian groups entrapping TOP/CAT(e(M )). LEMMA 7.2 . Consider : M a CAT o b je c t; rj a C AT microbundle over M ; £ a TOP microbundle over M ; £(; a C AT structure on the restriction o f £ to a neighborhood o f a closed subset C o f M . In this situation , Whitney sum induces a bijection

§7. A parallel classification theorem

“77®” : TOP/C AT(£ rel C , gQ)

177

TOP/CAT(t7 e g rel C , 7? © gQ) .

Proof o f 7.2: Choosing a CAT microbundle inverse 77' to 17 (using [M ij] , c f . Appendix A) , we get 77'©77 = where is the trivial bundle of dimension k say , over M . An inverse to “17©” is provided by the composition of “17'©” with the natural isomorphism TOP/CAT(t7' © 77 © £ rel C) TOP/CAT(£ rel C) coming from the evident micro-isomorphisms 17' © 17 © £ = e^© £ s £ © e^- . Since we are dealing with stable CAT structures on £ , the verification that this is the inverse of “ 77©” requires a CAT isotopy of id I R^+s to the permutation map (x j xk+s) +* (xk + j x j^ , x j xk ) for s even . ■ We now have a triangle (for the data of 7.1) TOP/CAT(e(M) rel C) “r(M)®”

^ j ^ f f r o m 7.1)

(from 7.2)

TOP/CAT(r(M) rel C)

§ CAT(M rel

^ o ^ (fro m

•

4.1)

whose three sides are bijections . Assertion 7.3 . This triangle is commutative . Proof o f 7.3 : Let KM) be a CAT normal microbundle to M appropriately imbedded in euclidean space Rm+n ; n large ; denote the projection r : E(i>) = N -*■ M and choose an isomorphism KM) © r(M) = em+n - e(M) , to get as in the proof of 7.2 an inverse ‘V©” to “ r(M)©” . Evidently it suffices to check that aj 0 “j>©” = o0 . For this we observe natural CAT isomorphisms of manifolds E(KM) © r(M)) = E(r*r(M)) ^ M X N and identify M with the common zero-section A M . Now a i ° ‘V©” is determined by the rule that assigns to a CAT manifold structure © defined near M = AM XO in M X N X Rs , s > 0 , and standard near C = AC X 0 , a new CAT structure on M obtained via the Product Structure Theorem applied rel C using the trivialized normal microbundle v © r © es = em+n+s to M . On the other hand o 0 is determined similarly , but using the distinct normal microbundle to M having fibers x X N X R s , x G M . Fortunately these two CAT

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normal microbundles to M in M X N X Rs are CAT isotopic , if s is large , by Appendix A . It follows immediately that these two rules give the same map , i.e. oqo “v©” = a as required. ■ To conclude this section , we briefly discuss an exact sequence of abelian groups involving TOP/CAT(e(M)) . Here M denotes a DIFF manifold if CAT = DIFF or polyhedron if CAT = PL . PROPOSITION 7.4 . There is a natural exact sequence o f abelian groups as described below : CAT(M) “ TOP(M) S- TOP/CAT(e(M)) X B CAT(M) ^ B TO P(M ). (A version relative to a closed subset C C M similarly .)

could be described

B CAT(M) is the abelian group (seetM i^ j) of stable isomorphism classes of CAT microbundles over M , the addition coming from Whitney sum of microbundles (now CAT = TOP is allow ed). TOP/CAT(e(M)) also has its addition from Whitney sum . It is an exercise (using CAT isotopics of even permutations on Rn to the identity , as for 7.2) to verify that this addition is well-defined and abelian . CAT(M) is the set of which a typical element is represented by a CAT micro-isomorphism f : em -*■ em of a trivial bundle X0 D em : M -> MX Rm M . Another such g : e11 -*■ e11 represents the same element of CAT(M) if there exists a CAT micro-isomorphism H : I X es -*■ I X es) coincides with 0 X f X (id I Rs-m ) and HI (1 X es) coincides with 1 X g X (id I Rs-n) . We say that h is stably isotopic to h' and write h ~ h' . An abelian group structure in CAT(M) is induced by composition of micro-isomorphisms : thus , given x , x' in CA T(M ), represent them by h , h' : es -*■ es ; then h'h represents x+x' . To verify that this addition is abelian , i.e. h'h ~ hh' , we use Lemma .

h © (id I e11) ~ (id I e11) © h

(n even) .

To prove this lemma one can conjugate using a CAT isotopy of id lR s+n fixing 0 to the permutation (xj , x2 ,..., xs+n) (xs+l

xs+n ’ Xl ’■■■’ xs> ■

m

§7. A parallel classification theorem

179

From this lemma it follows th at hh' ~ h©h' ~ h'®h ~ h'h . Thus CAT(M) is indeed an abelian group . The maps a and 8 are obvious forgetting maps . At the level of representatives , |3 assigns to a TOP micro-isomorphism h : e11 -► e11 the CAT structure £n on e11 that makes h : £n e11 a CAT micro-isomorphism ; and j assigns to a CAT structure £n on e11 the CAT microbundle £n . This completes our description of the sequence of Proposition 7.4 ; it is clearly an exact sequence of group homomorphisms . ■

180

Essay IV

§8. C L A S S IF Y IN G SPACES O B T A IN E D BY E. H. BRO W N'S M ETH O D

In this section and the next we aim to review the classification of reductions of micro-bundles by vertical homotopy classes of sections of suitable fiber spaces . Combined with the results of earlier sections this will yield in § 10 a ‘hom otopy’ classification of CAT manifold structures on any topological manifold of high dimension . The situation is much as for principal bundles with fiber a topo­ logical group (see [H us]) . For any topological group G , the contractible infinite join E = G*G*G*... with the natural right Gaction is a universal numerable principal G-bundle with base E/G called Bq . For any subgroup H C G the restricted action of H on E is again principal , and since E is contractible we dare to write E/H = B j[^ . Now Bq is naturally a quotient of Bjj , indeed we have a bundle quot. (B i G/H B jj > Bq which is none other than the numerable bundle with fiber G/H associated to the universal principal bundle E Bq (read [Hus , p. 7 0 ] ) . Hence reductions (= restrictions) to group H of any numerable principal G-bundle £ over a space X correspond naturally and bijectively to liftings to Bj_j of any fixed classifying map c : X Bq for £ . (Recall that such a lifting of c is equivalent to a section of the t This notation suggests E -►Bjj is universal . Here are two cases where it is : (i) If G -►G/H has local sections (equivalently , is a locally trivial H-bundle) then E = G *G *~ -►Bjj is evidently locally trivial and hence universal (by [Hus , p.5 7 ]), at least for locally trivial principal H-bundles over arbitrary CW complexes . A. Gleason showed that G -►G/H does have local sections if H is a Lie group and is closed in G while G is Hausdorff , see [Ser] . (ii) If G is a complete metric space , and H C G is closed and LC°° , then Michael’s selection theorems [Mic] show that every pull-back o f the principal bundle E Bjj over a CW complex is locally trivial. And it follows (as on p.57 o f [H us]) that E-*Bj_[ is again universal for locally trivial principal H-bundles over CW complexes .

§8.

Classifying spaces obtained by E.H. B row n's rrtethod

181

associated bundle £[G/H] .) If H is a normal closed subgroup of G , then G/H is a topological group and (B is a principal bundle with group G/H . In this case , when one lifting c' : X -*■ Bjj of c is fixed , all other lift­ ings c" : X -► Bjj of c are in bijective correspondence c" A(c" , c') with maps A(c" , c') : X -*> G/H by the rule defining A(c" , c')(x) E G/H by c"(x) = c'(x) • A(c" , c') dot denoting the right action of G/H . This can be further explained by observing that c' provides a section of the induced bundle c* © over X , hence a trivialization of it as G/H bundle ; for sections of c* © are precisely liftings of c . This is the behavior we encounter with reductions of micro­ bundles . In f a c t, reductions of locally trivial topological Rn bundles to vector bundles can be discussed in this framework ; it is really only PL bundles that force us to adopt less concrete methods below . PROPOSITION 8.1 . (CAT = PL or TOP) . There exists a simplicial complex = PL) and a CAT n-microbundle 7 ^ 4 j universal in the following relative sense .

(locally finite i f CA T over B g ^ J(n) that is

(*) L et % be a C AT n-microbundle over an euclidean neighborhood retract X (polyhedron i f C AT = PL) ; let U be an open neighborhood o f a closed set C C X ; and let fQ : % \U -*■ 7^- j y be a CAT microbundle map . Then there exists a microbundle map 1: £ -*■ 7 ^4 j C.

so that f = fQ near

Complement to 8.1 . Incase CAT = DIFF Proposition 8.1 is valid except that , to allow B p j p p ^ to be a DIFF manifold , we define it to be the Grassmannian Gn(Rn+^) of n-planes in Rn+^ , k large , rather than Gn(R°°) which is not a DIFF manifold , and let Tpjjpp be the standard vector bundle that enjoys a weakened universality, namely (*) for DIFF manifolds X of dimension < k . For convenience in what follows we choose Bp)iFF(n) inductively so that k = kn > dim(BDIp p (n_] p + n . ■

182

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PROOF OF PROPOSITION 8.1

Essay IV

(cf. [Wes] [W hi]) .

This takes up the rest of the section . There are semi-simplicial proofs [Mi3 J [RSq] ; however we will establish 8.1 using E. H. Brown’s theory of representability , which has earned a reputation for being widely useful although its results are not always the most precise . Let C be the category of connected simplicial complexes with base point and base point preserving piecewise linear maps . The simplicial complexes are not supposed locally finite^ ; the compactly generated (CW complex) topology is used . A piecewise-linear map f : X -*■ Y of such complexes is a continuous map such that for each simplex (or finite subcomplex) A in X there is some linear sub­ division A' of A such that f maps each simplex of A' linearly into some simplex of Y . If X G C is not locally finite a PL n-microbundle £ over X is a TOP n-microbundle 17 over X together with a PL structure £ IA on 171A for each simplex (or finite subcomplex) A of X ; the PL structures £ I A are supposed to agree wherever they over­ lap . This certainly extends the definition valid for locally finite complexes given in § 1 . A PL n-microbundle £ over X G C with basepoint xQ will be said to be rooted [Mi^ , §7] if it is equipped with a microbundle map b : £ I xQ -► e11 to the trivial bundle en : 0 -*■ Rn -*■ 0 . Two rooted n-microbundles (£ , b) and (£' , b') over X are called equivalent if there exists a PL micro-isomorphism h : £ -*■ £' so that b'h = b near xQ in £ I xQ . Let H(X) be the set of equivalence classes of rooted PL nmicrobundles over X . The standard rule of pull-back makes of H a contravariant functor H : C -»• £ to the category & of pointed sets . It is straightforward to check that H satisfies the classical axioms of E. H. Brown . First some notation : if f : X -> Y in C , the induced map H(f) : H(X) ■*- H(Y) is written f* , and if f is an inclusion we write f*(u) = u lX , u G H(Y) . ^ The use of arbitrary simplicial complexes is a device o f D. White [Whi] that serves to get a maximum of information from E. H. Brown’s method , and with a minimum o f effort .

§8.

Classifying spaces obtained by E.H. Brow n's method

183

(1) Homotopy axiom . I f f , g : X -*■ Y are PL hom otopic as pointed maps then f* = g* as maps H ( X ) < - H ( Y ) . P ro o f. This follows from the microbundle hom otopy theorem (§ 1) , applied (sharply) relative to base point . This relative microbundle hom otopy theorem is valid for arbitrary simplicial complexes , in the form sharply relative to subcomplexes, since one can use the case of a simplex relative to its boundary to obtain the general case via an induction over skeleta . ■ (2) Glueing (or Mayer-Vietoris) axiom . I f X = X t U X 2 with common base point in X 0 = X t O X 2 and x x G H (X x) and x 2 in H (X 2) satisfy x , \ X 0 = x 2 \X 0 , then there exists x E H(X) such that x \ X x = x t and x \ X 2= x 2 . A n d x is unique i f X is the wedge (sum with common base point) X = X i v X 2 . ■ (3) Wedge axiom . The natural map

HWaX a > is a bijection for the wedge pointed com plexes. ■

Va Xa

n H (X a) o f any collection

{Xa } o f

E. H. Brown’s celebrated representation theorem [Brn] [Sp] [Ad] asserts that any functor H : C -*■ & verifying (1) , (2) and (3) is representable in the sense that it admits a ‘classifying’ space B and ‘universal’ element u as follows : (**) There is an object B in C and an element u in H(B) so that Tu : [ X , BJ H (X) , defined by sending f : X -*■ B to f* u = H (f)(u) G H ( X ) , is bijective fo r every X in C . (Here [ X , B] denotes the set o f pointed PL hom otopy classes o f pointed PL maps X -*■ B .) It is more usual for C to be the category CQ of pointed CW complexes and pointed continuous maps . But , given the relative simplicial approximation theorem of [Ze2 ] , Brown’s arguments apply to our PL category , cf. [Whi] ; indeed Brown even axiomatizes the properties of C that are essential [Bm] . Alternatively , since each CW complex is hom otopy equivalent to a simplicial complex and since continuous maps of simplicial complexes can be homotoped to PL maps in a relative way (for both facts use the relative simplicial

L. Siebenmann

184

Essay IV

approximation theorem in a skeletal induction) , one can extend H in an essentially unique way to H :

CQ

fi.

This functor still verifies axioms (1) , (2) and (3) . (Indeed (1) and (3) are evident ; and (2) follows from the fact that any triad (X ; X x , X2) of CW complexes is homotopy equivalent to a simplicial triad , cf . [ I ll, §5.2] .) Clearly B can be replaced by any complex homotopy equivalent in C (or CQ) . Hence B can chosen to be countable because ^ B = H(S^) is countable for all $ ; and then in addition B can be made a locally finite simplicial complex , cf . [IV^] . Then u E H(B) is represented by a ‘universal’ microbundle y . We write B and y now as B p p ^ and 7 p p . Similarly , in fact more easily , one defines BpopfH ) and yT O P • To establish 8.1 it remains only to prove . ASSERTION 8.3 (for CAT = PL or TOP) . L et y ^ A T be the CAT microbundle over B qA j ( n) constructed above . The universality property (**) implies the relative universality property (*) o f 8.1 . The most obvious approach to the proof fails because one finds that a micro-automorphism of £ near C intervenes . Instead we argue roughly that (*) holds for (X , C) = (S^ , 0) , hence also for (B^ , 9B^) , which implies the general case ; the details are given in four steps , which apply to most bundle theories . Step (1) . B and y remain universal in the sense o f (**) i f a new base point b' E B is chosen and y is suitably re-rooted . Proof o f (1) . There is a (pointed) homotopy equivalence (B , b) -> (B , b') homotopic (freely) to id I B , hence covered by a micro-bundle ^ The countability o f H(S^) follows quickly from the fact that , up to isomorphism , there are only countably many simplicial maps f : X -> Y o f finite simplicial complexes . Incidentally , if we were working in the topological category , the ^emavskii local contractibility theorem [Cej would assure that H(S^) is countable , cf. [S ij2 >P-162] . $ Proving that if 7r*B is countable then B is homotopy equivalent to a countable complex can be made an exercise with Brown’s method for representing the functor X +* [X , B] .

§8.

Classifying spaces obtained by E.H. B row n's method

map y -*■ y . From this fact step (1) follows easily .

185

■

Lemma 8.4 . L et f : en -* en be a CAT micro-isomorphism o f the trivial bundle en over R k to itself such that f is the identity on etl I 0 , where 0 = origin o f R k . There exists another CA T micro­ isomorphism f : en -*■ en equal f near R k - 2Bk (where B k = k-ball in R k ) and equal the identity near B k . P roof o f lemma : Indeed f' can be the pullback of f by a CAT map -» that maps 2. 3 k to the origin and fixes R^ - B^ .

Step (2) . Property (*) holds when X - S k , k > 0 , and C is a hemisphere o f S k . Proof o f (2 ). We are given a CAT micro-bundle map f0 : £lU -*■ 7 , U being an open neighborhood of C . Choose a base point xQ E C for , and using step ( 1) let fQ(xQ) (f0 covering f0 ) be the base point for B . By (**) and the homotopy extension property we can find a classifying map f : -*■ B with f = f0 on a neighborhood V of C and a micro-isomorphism y> : £ -> f*7 . Note that , as f *7 1V ± f ^ l V , there is a natural micro-identification ■ f*7 1V which , composed with f*7 -► 7 , yields f0 . And (**) tells us that y> can be chosen equal to p0 on £ I x0 . By Lemma 8.4 we can alter y? near C to * ' : £ - ► f*7 so that

ip' = y> near

C. Then the composed microbundle map

f • | equals

f0 near C .

^

f *Tn rat^

Tn

■

Step (3) . Property (*) holds when X C = W k. Proof o f (3 ). We can assume

is a k-disc D k , k >

is a hemisphere of

0 , and

. Extend

186

L. Siebenmann

Essay IV

f0 : Sk_1 = 9Dk -»■ B over the complementary hemisphere D_

to

get f_ : Dk -*■ B . Identifying f*7 n ISk_^ naturally with £lSk - ^ near Sk - ^ form a microbundle £' over Sk extending £ and f*7 n , together with a microbundle map fo : £' IU' -► y n , U' = U U Dk , extending f0 . Now apply Step (2) .

■

Step ( 4 ) . Property (*) holds in general. Proof o f (4). We can assume X is a simplicial complex . If CAT = TOP , arrange this by passing to a neighborhood in euclidean space having the original X as retract . Triangulate the new X finely and construct the required extension by induction applying step (3) simplex by simplex . ■ This completes the proof of Assertion 8.3 and of Proposition 8.1 .

187

§9. C L A S S IF Y IN G R E D U C T IO N S

The universal microbundle 7 £ AX > (CAT = DIFF or PL) can be regarded as a TOP n-microbundle . As such it admits a map in : TcA T TTOP covering a map j n : BCAX(n) -* BT0P(n) . And by 8.1 j is unique up to hom otopy . Using the well-known device of Serre we now convert j n into a Hurewicz fibration Pn : BCAT(n) BTOP(n) • Thus we define BCAT(n) to be the set ol' commutative squares of continuous maps 0

1 CAT(n)

inclusion

►

I

I

in *

n BTOP(n)

endowed with the compact-open topology . (It can be described as the space of paths in BXQP(n^ each issueing from some specified point of BCAT(n) ■> There is an imbedding in : BCAX(n) -» Bx o p (n ) as constant paths and the initial point map is a deformation retraction BCAT(n) BCAT(n) while the final Point maP Pn : BCAT(n) BTOP(n) is a Hurewicz fibration extending j n : BcAT(n) -> Bx o p (n ) , i.e. pn has the hom otopy lifting property . PROPOSITION 9.1 . L et %n be a TOP n-microbundle over a C AT object X (= D IF F manifold or polyhedron) and let f : £ -*■ J ^ a T over f : X -» B T0P(n) classify £. Then , there is a natural bijective correspondence between concordance classes o f C AT microbundle structures on %n and vertical hom otopy classes o f liftings o f f to P 'cA T (n)' ^ Vertical means that composition with p n gives the constant hom otopy o f f . ) This bijection is denoted Tn ■ TOPn /CATn(£) -* Lift(f)

188

L. Siebenmann

Essay IV

Proof of 9.1 (cf. Browder [Brj , § 4 ]) . To lighten notation here we delete CAT , replace TOP by + and usually suppress n . Here is a rule that defines 7 p*7 + , which we also call G . We will presently explain how to form a microbundle homotopy H : I X £ -*■ p*7+ , H : G — F so that p#F is the given map f : £ -*■ y + ; then if F : £ -► p*7+ is over say F : X -> B' we have pF = f . The rule 17 +*■ F defines y?n . To form H above , first use 8.1 to obtain a TOP microbundle homotopy H+ : I X £ -*■ 7+ , H+ : p^G ^ f , over H+ : I X X -»• B+ , H+ : pG ^ f , say . Then apply the homotopy lifting property of p : B' -»• B+ to obtain a homotopy H : I X X -*■ B' , from G to say F , and lifting H+ (i.e. pH = H+) . Finally note that H : I X X -*■ B' and H+ : I X £ -* 7+ together specify a microbundle homotopy H : I X £ -> p*7+ as required . Observation : The above rule is relative in the following sense : Let t?0 be a CAT structure on £Q = £ IU where U is an open neighborhood in X of a given closed set C C X . Suppose the above construction has been carried out for £Q , r}Q and fQ = f IU , yielding say HQ : G q ^ FQ . Then the construction can be carried out fo r £ , 17 and f yielding say H : G ^ F in such a way that H equals H0 near I X C . ■ Applied to the bundle I X £ (in place of £) with { 0 ,1 } X X as C this observation shows that 0) of f I L equal F0 on K . There are two valid alternatives that are related but not identical : Classical definition . F is represented by a true lifting G of f extending F QlK and defined on K U L ® where L ^ ) is the kskeleton of some PL triangulation r of L making K a sub­ complex . It is essential to agree that another such lifting G ' , defined say on K U , represents the same k-lifting F if there exists a lifting of f I {(I X K) U (I X L)^k ) } extending 0 X G , 1 X G ' and I X (Fq I K ) , where a is a PL triangulation of I X L making I X K a subcomplex and extending 0 X r and 1 X r ' . Modern definition : F is precisely a true lifting F : L -► of f I L into the k-th stage fibration p^ : -*■ B jq p of the Postnikov decomposition of the fibration p : Bp at Bt d p >such that FI K = qj^Q IK , where % p : b CAT

Pk * Bk

* BTOP

is the k-th stage factorization of p . Recall that P^+i = P ^ ^ + i > where : B ^ j -»■ B^ is a fibration with fiber an EilenbergMacLane space KCtt^CTOP/CAT) , k) . 10.10. Interpreting the Postnikov decomposition of p : B y ^ y -^ B y o p • The terminology has just been established above . In case CAT = PL it turns out that TOP/PL ~ K(Z2 , 3) ; thus one chooses Bk = BTOP ^or h < 3 and B^ = Bpy for k > 4 ! In case CAT = DIFF it turns out that B^ is itself a stable classifying space of a manifold theory ; vertical homotopy classes of liftings of f to BJ, correspond (for 9M = and dimM > 5 ) to conditioned concordance classes of k-smooth manifold structures on M . The pseudo-group of k-smooth homeomorphisms of open subsets of Rn , n > 5 , is defined to contain h : U -*■ V if h is C°° nonsingular except perhaps on a closed subset of U having §tanko homotopy dimension < (n - k - 1) (such as the dual of the k-skeleton of some triangulation) , cf. [Ill , Appendix C] . Thus one has an infinite sequence of manifold theories t strung out between the smooth and the topological, corresponding to B j-^y -»• — >B^ -* B ^_j -*■ — »• Bg = ^ See also N. Levitt’s article [Lt] where essentially equivalent theories , independently constructed , are discussed . (The above pseudo-groups are not used).

§10.

Hom otopy classification of manifold structures

199

= Bt o p . Finally we note that PL is essentially one of these theories ; indeed ~ Bpp for 4 < k < 7 simply because the canonical forgetful map B ^ jp p Bpp (from Whitehead’s triangulation theorem) is 7-connected . We shall not attem pt to prove these things here ; instead see [ Si j 5 ] . 10.11. Classification relative to an open su b se t. For the data of 10.1 , consider the problem of discovering and classifying CAT structures on M that coincide with 2 Q on all of the open subset U (rather than just near C) . Agree that two such , 2 ' and 2 " , are equivalent if there exists a CAT concordance F : 2 ' — 2 ” so that TI (U X I) = 2 Q X I ; and denote the set of equivalence classes S c a t (M rel U ,2 q) . Fortunately this problem is isomorphic to an associated problem in dimension m + 1 which is solved by 10.1 . Consider Z = M X I - (M-U) X 0 . Assertion : rel U ,2 Q) = §>c a T ^ rel U X 0 , 2 QX I) . The asserted isomorphism is defined by sending 2 representing x E § C A T ^ re^ U ,2 0 ) to (2 X I) I Z . To show that this map is onto , consider 0 representing y E § c At( Z rel U X 0 , 2 0 X I ) . Apply the Concordance Implies (sm all!) Isotopy Theorem to 0 1(U X I) to get an isotopy h^ of (id IZ) rel U X 0 fixing points outside U X I so that h j ( © ) l ( U X I ) = 2 0 X I . Extend h j 0 to a CAT structure on Z U M X [ 1,°°), that on M X [ 1,°°) is a product I X [ 1,°°) and on U X [0,°°) is a product with [ 0 ,°°) . (For DIFF one needs local collaring theorems here , from [I , Appendix A] .) Translating uniformly to °° in the R factor we clearly obtain a concordance (on Z) of h j 0 rel U X I to the structure (2 X I) IZ . A similar (but relative) use of the Concordance Implies Isotopy Theorem shows this map is injective , which proves the assertion . Finally , note that H*(Z , U X 0) = H*(M , U) for any theory H* — by the projection Z -> M , (use the 5-lemma) . Thus , for purposes o f calculation the problem behaves as i f U werea closed ENR in M . However , beware that F Q obviously may not extend beyond U even if 2 Q does ; in general it will only extend up to hom otopy .

200

L. Siebenmann

Essay IV

10.12 The Homotopy Groups of TOP/CAT . We indicate how this vital determination can best be carried out using our present classification result and surgery. For readers interested in understanding why TOP/PL is not con­ tractible rather than in fully calculating 7r* (TOP/CAT) , we have written Appendix B that reduces the prerequisites to mere handlebody theory and V.A. Rohlin’s signature theorem. For m > 5 , there is a group isomorphism V : 7rm(TO P/CA T)^> ©CAT It is the forgetting map from 7rm(TOP/CAT) = SCAT(Sm) , cf. 10.8 , to the abelian group 0 ^ AT of oriented isomorphism classes of oriented CAT m-manifolds homotopy equivalent to Sm , addition being connected sum [KeM] . Indeed surjectivity of is the (weak) CAT Poincare theorem, which is proved readily by engulfing for m > 5 , cf. [St 2 ] [Lu] [Hu 2 ] [RS 3 ] . Additivity is easy to check. To check injectivity suppose goes to zero in 0 ^ AT ; then is the boundary of a contractible CAT manifold Wm+l, and as Wm+1 is * homeomorphic to Bm+1 by the same engulfing or Smale’s h-cobordism theorem [Mig] [HU2 ] , we see by coning that 2 extends over Bm+1 , whence [2] = 0 in 7rm(TOP/CAT) . The group 0 ^ L is zero for m > 5 . The PL h-cobordism theorem suffices to prove this for m > 6 , via coning. For m = 5 •, one first shows using classical PL-DIFF smoothing theory f and Cerf’s T 4 = 0 [Ce 5 ] , that any PL homotopy 5-sphere M5 admits a Whitehead compatible DIFF structure 0 . Then the equation ©BIFF = o of [KeM] implies that Ma is diffeomorphic to S 5 , whence, by Whitehead’s smooth triangulation uniqueness theorem [W hj][M uj] , M5 is PL homeomorphic to S 5 . The group ©BIFF of smooth homotopy spheres was made famous by Milnor and Kervaire [KeM] who proved that it is finite for m > 5 , and indeed largely determined it. For m = 5, 6 , 7, 8 , 9, 10, 11, its values are 0, 0, Z 28 , Z 2 , Z 2 ®Z2 ®Z2 , Z 6 , Z 992 . See also [Bru] [Brj ] . Whether

7rm ( T O P / C A T ) =

©CAT

for all m , is an intriguing

f See comment in Appendix B (below B.2 ).

§10. The hom otopy groups of T O P / C A T

201

question (especially since an affirmative answer contradicts the classical Poincare conjecture). Nothing is known about @^AT for m = 3 ,4 , (except that 0 ^ L = ®J^IFF by classical smoothing theory [MU3] [Hi3 ] ) . For k < 6 , one exploits homotopy tori to prove that 7rk ( T O P / C A T ) - 7rk(K(Z2 ,3)) , i.e. that it is nonzero only if k = 3 , when it is Z 2 . See [KS j ] . The proof we recommend to readers of this essay is very close to the one presented in §5.3 of Essay V . It is already accessible to readers of this essay, and we shall merely add a few comments here to let the reader adapt that argument efficiently : Work with the 6 -manifold Ik x T 6“k to calculate 7rk(TOP/CAT) . Replace TOPm/CATm by TOP/CAT and when a classification theorem of Essay V is called upon use 10.1 instead. Thereby a map

(#)

7rk(TOP/CAT)

g*(Ik XT6- k rel 9 )

is constructed to the set of those ‘homotopy-CAT’ structures on Ik XT6~k rel 9 that are invariant under passage to standard coverings. This set is 7rk(K(Z 2 ,3 )) by a surgical calculation that is described in [ V, Appendix B] . Then the device of passage to a standard covering of T6“k plus the s-cobordism theorem and the local contractibility Y principle for homeomorphisms [Cej 3 1 [EK] are combined to show that (#) is bijective; the details are in [V, §5.3] . We now indicate how to dispense with the local contractibility principle in the above argument (this would be its first intervention in Essays I and IV ). It is used in [V, § 5.3] to prove injectivity of (#) and again to prove surjectivity. In each case, it reveals that a cer­ tain small self-homeomorphism, a say , of Ik XTn rel 9 is isotopic rel 9 to the identity. 1) For injectivity of (#) one can instead observe that it suffices to prove that a universal cover a : Ik XRn Ik XRn of a is isotopic rel 9 to the identity. Now a is rel 9 (by a suitable choice) and we can extend it by the identity to a self-homeomorphism of Rk XRn = = Rk+n . As a is of bounded distance to id I Rk+n the formula c^(x) = ta:(x/t) , 0 < t < 1 , gives an isotopy with support in Ik XRn from a = c q to the identity, cf. [Kij] .

202

Essay IV

L. Siebenmann

2) For the proof of surjectivity of (#) , a: becomes a CAT automor­ phism of I 2 XT3 rel 9 , and this time we observe that it suffices instead to obtain a topological pseudo-isotopy ( = concordance) rel 9 from a to the identity. It is easy to deduce from the definition of * SQ that a is CAT concordant rel 9 to its standard A3-fold covering ( A any integer > 1 ) . Then a pleasant infinite iteration process pictured in [SijQ, §3] provides the wanted topological concordance from a to id I(I 2 XT3) rel 9 . o

We have established a homotopy equivalence TOP/PL ^ K(Z 2 ,3) • This means thiat the primary obstructions met in calculating PL structure sets are the only ones. More precisely, using 10.8 , 10.9 , 10.1 1 , we see that in the situation of 10.1 the TOP manifold M admits a PL manifold structure equal to 2 0 near C , if and only if a well-defined obstruction A(M) in H4 (M,C;Z2) vanishes. When one such structure is singled out, the pointed set § PL(M rel C ,2 0) is naturally isomorphic to H 3 (M,C;Z2) . Also C may be open in M instead of closed. Knowledge of 7r*(TOP/DIFF) on the other hand, is merely a good first step towards evaluating § Djpp(M rel C ,20) in 10.1 . The next big step is perhaps to describe the homotopy type and H-space stru c­ ture of TOP/DIFF ; this is a subject of current research requiring methods little related to the present essays, see [Sull] [MorgS] [BruM] [MadM] .

203

Appendix A. N O R M A L B U N D L E S

We shall explain M. Hirsch’s proof [H ij] of the relative stable existence and uniqueness theorems for normal microbundles . (If our exposition is at all more accessible than in [Hig] it is only because we have abandoned some of the generality there .) t We adopt the PL or TOP category consistently throughout , and we use the conventions of § 1 concerning microbundles . A normal microbundle to M in W D M is a microbundle £ over M whose total space E = E(£) is a neighborhood of M in W . Two such , £0 and £j , are (sliced) concordant if there is a normal microbundle rj to M X I in W X I , 1= [0, 1] with ^ X k = T?IM X k for k = 0 and 1 , such that for each t E I , the restriction rj IM X t is a normal microbundle to M X t in W X t . We write V '■ to ^ t i • Observe that , by the microbundle hom otopy theorem , one has £0 ^ t i if and only if £0 is isotopic to £t in the sense that there is a neighborhood V of M in E(£0) and an isotopy F : V X I -► W X I (an open embedding fixing M X I with p2 = P2F) from F 0 = inclusion : V C* W to F t a micro-iso­ morphism £0 t i • We write F : £0 £x . As in [I, §2] we can equally well discuss concordance and isotopy relative to a closed subset C C M , i.e. constant on a neighborhood of C in W . We shall encounter composed microbundles . I f £ : X CL* Y -► X and rj : Y CL* Z Y are microbundles , then the composition £ o rj : X CL* Z ^ X , is again a microbundle as a simple pursuit of definitions shows . Data. Mm C Ww denote henceforth manifolds without boundary with M imbedded as a closed subset of W . A closed subset C C M is given , with an open neighborhood U of C in M and a normal microbundle y to U in W . ^ For stronger results consult R. Stem [Ste] ; in particular , provided w - m > 5 + i , (a) existence holds for n = 0 if w > 2 m - i - l , i= 0 ,1,2, and (b) uniqueness holds for n = 0 if w > 2 m - i , i=0,1,2 . For examples of imbeddings for which normal bundles fail to exist or fail to be unique , see [RSq 4] .

Essay IV

L. Siebenmann

204

STABLE EXISTENCE THEOREM A .l. For some integerf n > 0 , M X 0 admits a normal microbundle £ in W X R n so that £ equals y © en near C X 0 in W Y R n. STABLE UNIQUENESS THEOREM A.2. Suppose £0 and £x are normal microbundles to M in W. Then there exists an integerf n > 0 such that £0 © en and £i © en are isotopic normal microbundles to M X 0 in W X R n . The following strongly relative form h o ld s. Suppose £0 coin­ cides with £x near C in W ; let D C M be a given closed set and V an open neighborhood o f D in M . Then there exists an integer n > 0 , an open neighborhood E' o f M Y 0 in W X R n and an isotopy f t : E ' -*■ W X R n , 0 < t < 1 , rel C U ( M - V ) and fixing M from the inclusion / 0 to / x giving (by restriction) a micro-isomorphism (£0 © en)\(C U D) -+ ( \ x © en)\(C U D) . As usual results for manifolds with boundary can be deduced . Uniqueness implies existence . We recall how the existence theorem A.l follows from the uniqueness theorem A.2 . We can assume M is locally flat in W since a construction given in § 3 shows M is locally flat in W X Rm . Then we can use a partition of unity to find a covering of M by < m+1 open sets M0 ,. . .,Mm in M each with a trivial normal microbundle in W . An evident m +1 step application of the strongly relative uniqueness result then constructs £ . A key case of uniqueness : D = Mm C Rm . Consider the two natural tangent microbundles T k ( M) :

M

A

MXM

Pk —

>M ,

k = 1,

2,

where pk(x x , x2) = xk . Identifying M to AM these are two normal microbundles to AM in M X M . Fact 1 . t xM O t 2 M P ro o f: This is at least clear for M = R 1 , where it is a trivial case of the (bi)collaring uniqueness theorem ; it follows for M = Rm by producting ; then for M (open !) in Rm it follows simply by restrictio n . ■ t The proof o f Hirsch presented here does show that the stabilising integer n can be chosen as a function o f m = dimM o n ly . This fact is used in §9.4 .

Appendix A.

Normal bundles

205

Cutting back W if necessary we choose a retraction r : W ->• M . Fact 2 . The homeomorphism h : W X R m -» W X R m , h(x, y ) = (x, y+r(x)) carries M X O onto MM C W X R m , sending the normal microbundle £ ffi em o f M X O in W X R m ( £ being £o or £t ) to the following composed normal microbundle to MM in W X R m : fT,M)opM): M ^

E(£) X M ^

M

where it(x , y ) = p ^ x ) .

Figure A-l Proof: The one total space is contained in the other E(£) X M C E(£) X Rm and the projections agree : (x, y) +* p^(x) ; but h respefcts this projection and carries the one zero section to the other : A(x) = h(x, 0 ) . ■ Fact 3 . A sliced concordance r ): canonically a sliced concordance r k : ( r xM) o( $k XM) O

t

xM

0=

t 2M

( r 2M)o (£k X M ) ,

Proof: Indeed the concordance is

(from Fact 1) yields

k = 0, 1 .

- i?o (£k X M X I) .

■

Fact 4 . (t2M ) ° ( £ X M ) - r 2Wl M . A P2 Proof: The composed microbundle is M -*■ E(£) X M -» M . Conclusion . We now have sliced concordances

■

L. Siebenmann

206

f k : ( r 1M ) o ( | k XM) O r 2Wl M,

Essay IV

k = 0, 1 .

Thus by Fact 2 , £0 ® eIT1 % t i ® em which was the first assertion to prove . Unfortunately it is not immediately clear that £0 © em « © em rel C , since rj : t xM O t 2M is not rel C , which implies fk is not rel C , k = 0, 1 . However £0 = ?i near C does imply that £0 = ?i near AC X X I in W X M X I . Applying the microbundle theorem to and f i respectively we obtain isotopies in W X M ft : r 2WlM «

( r j M ) o (£0 X M)

gt : t 2WIM »

( r j M) o ($, X M)

Since f o = ?i near AC X I we can assume that f^ = gt near AC in W X M . Then h- 1gj.of^1h , 0 < t < 1 , is an isotopy | 0 ® em « £ ! ® em rel C as required to establish the key case of uniqueness. ■ Proof of the uniqueness theorem in general. Given the case just proved , the (stable) isotopy extension result below implies the more general case where D f M is allowed , but Mm C Rm . This strongly relative case then serves to prove the completely general case (Mm C Rm not supposed) by a routine (m +1)-step induction using a covering of M by m +1 coordinate ch a rts. ■ It remains to prove ISOTOPY EXTENSION LEMMA A.3 . (for usual data)+ Let E be an open neighborhood o f C in W , and let E -> W be an isotopy o f E CL* W fixing M n E D C There exists a neighborhood E' o f M X 0 in W X R and an isotopy H t : E' -* W X R o f E 9 C_* W X R fixing all o f MXO and equal f t X R near C X 0 . Proof of A.3 : Consider the isotopy Ft : E X R

-* W X R ,

F t(x, s) = (fr(Sjt)(x), s)

^ M and W need not be manifolds here (we just use normality o f M) ; but if they are manifolds and M is locally flat in W then the stabilisation is known to be unnecessary and Ht could be an ambient isotopy , cf. [EK] .

Appendix A.

Normal bundles

where r : R X I

207

I satisfies :

r(s, t) = t for s > - 1 ;

r(s, t) = 0 for s < - 2 or t = 0

T (E i (-00.11) is hatch'd

Figure A-2 Choose a function p : W -* R equal to 0 near C and equal to 3 near W - E (using normality of W) , and form the homeo­ morphism T:WXR->WXR,

T(x, s) = (x, s - 5 , unique up to PL pseudo-isotopy (= concordance) that one obtains by trivializing (P3XTn) 0 regarded as an s-cobordism rel 3 from 0XI2 XTn to lX I2 XTn . In addition he needs a PL pseudo-isotopy H rel 3 from a to oq , the standard 2n-fold covering of a . It exists because we have constructed a CAT identification rel 3 from (P3XTn) 0 to its standard 2n-fold covering. (The reader may find it amusing to make the construction of a and H more concrete still.) At this point the catastrophe pictured in [Si jq, Fig.2a] inter­ venes to provide a TOP pseudo-isotopy rel 3 from a to idl ( I 2 XTn). From this, one rapidly deduces, for example, that M2 is (topologically!) homeomorphic to S3 XTn and that a TOP manifold W4+n ^ P4 XTn exists although we have observed that there exists no PL manifold of this homotopy type.

Essay V

C L A S SIF IC A T IO N OF S L IC E D F A M IL IE S OF M A N IF O L D ST R U C T U R E S

by

L. Siebenmann

—

d e d ic a te d to

C laude M o r le t

—

216

L. Siebenmann

Essay V

Section headings in Essay V

§ 0.

Introduction

§ 1.

Classifying CAT structures on manifolds by CAT structures on microbundles

§ 2.

CAT structures and classifying spaces

§ 3.

The relative classification theorem

§ 4.

Classifications for manifolds with boundary

§ 5.

The homotopy groups

Appendix A.

r*(TOPm/C ATm)

7

The immersion theoretic method w ithout handle decompositions

Appendix B.

Classification of homotopy tori: the necessary calculations

Appendix C.

Some topological surgery

217

§0. INTRODUCTION The classification we shall establish asserts^ , in its simplest form 2.3 , a hom otopy equivalence CAT(Mm ) =* Lift(f to BcAT(m)) from a space CAT(Mm) of sliced families of CAT (= smooth C°° or piecewise linear ) manifold structures on a topological m-manifold Mm without boundary (m f 4) to a space of liftings to BcAT(m) of any fixed classifying map f : M BxoP(m) ^or tangent microbundle of M . The sliced concordance implies isotopy theorem , proved by some careful engulfing in Essay II , brings this classification theorem within easy reach . The simplest version of it sufficient for our needs below is the BUNDLE THEOREM 0.1. L et M be a metrizable topological mani­ fold with dimM f 4 f dimbM , and let 2 be a C AT manifold structure on A ^ X M such that projection p 2 ■ ’ (A ^ X M )•£ -*■ A ^ is a C AT submersion onto the standard k-simplex ( k > 0). Then p 2 •' ( A ^ X M)*£ -*■ A ^ is a trivial CA T bundle . ^ This result was established as Theorem 1.8 in [II, § 1 ] . A structure 2 as described is none other than a typical k-simplex of the (semisimplicial) space CAT(Mm) . This bundle theorem reveals that if Mm is a CAT manifold , and A u t^ ^ j(M ) denotes a similarly defined semi-simplicial space of CAT automorphisms of M , then irj^(Aut'j’Q p(M ), A u t^ ^ j(M ))

= 7T j,( C A T ( M ) )

, k /> 1 ,

(see 1.5) . Thus the classification theorem is a key tool for comparing the spaces of TOP and CAT automorphisms of M . ^ The present formulation is Morlet’s [Mor^l [Mor^] , it improves on earlier ideas of ours [KSj ] . $ This modest result may be our largest contribution to the final classification theorem ; we worked it out in 1969 in the face o f a widespread belief that it was irrelevant and/or obvious and/or provable for all dimensions , (cf. [Mor^l [R0 2 ] and the 1969 version o f [M or^]) . Such a belief was not so unreasonable since 0.1 is obvious in case M is compact : every proper CAT submersion is a locally trivial bundle .

218

L. Siebenmann

Essay V

Our dedication to Claude Morlet is made in recognition of his im portant and precocious contributions to this classification theory in [M o ^] [Mor^] [M o r^ . Morlet showed us that a nonstable classifi­ cation theorem for sliced families was a much better goal for the handleby-handle classification proceedure we sketched in [KSj] . He observed that the remarkable equations 71^ ^ + j (TOPm/CATm) = = ^(A uto^-pC B 111 rel 9Bm)) (see §3) follow naturally from the sliced theory and the topological Alexander isotopy . Credit for first discovering such an equation goes to Jean Cerf . In f a c t , in 1962 [CeQ] , Cerf derived the case CAT = DIFF from certain technical conjectures [CeQ ; p. 18 , p.23] , all of which were then known for m = 3 through work of Moise and can now be verified readily for m f 4 by use of the version with epsilons of the Sliced Concordance Implies Isotopy Theorem of [ I I , §2.1 ] . Although C erfs equation looks different.it is exactly equivalent - see §3. 4. M. W. Hirsch too had done some early work on sliced classification theorems ; unfortunately , his results were never announced correctly [Hiq , p.75] , cf. [R0 2 ] • R- K. Lashofs work on classification also fits into this c o n te x t. All approaches to the sliced classification theory seem to exploit ideas that first arose in the Smale-Hirsch classification of immersions around 1957-9 [Sm][ Hi j ] . Even Cerf uses them in [CeQ] . Our exposition will follow closely the outline we gave in [KS4 ] , cf. [Sig , §4] , and will exploit the ideas of immersion theory in a way formalized and popularized by M. L. Gromov [Gr] [Ha2 l • (This includes a well-oiled ‘handle-induction’ mechanism replacing that in [KSj ] ) . This technique is eminenty worth learning as it has a great many other applications far and near ; for example the classification of [Si^ , §4] is one of the many results of Haefliger (cf. [HaQ] [Ha j ] [ R ^ ] ) that can be pleasantly reproved and strengthened by this means. Morlet’s original line of argument , cf. M o ^ ] , seems to have been rather less direct than our present one, as it involved spaces of immersions; it is perhaps now fully explained in [BuL] . C. Rourke was working out a direct proof of the sliced classification theorem at the same time we were (see [R0 2 ] ), and we would like to acknowledge a mutually helpful exchange of ideas (through R. Kirby) in summer 1970. Probably all direct proofs will of necessity have much in common.

§0.

Introduction

219

We round out our exposition with a review in § 5 of known results (including our own and Morlet’s) concerning the fiber TOPm/CATm of the forgetful map B£AT(m ) -*■ BjQ p^m ^ . This is the fiber whose homotopy obstructs the liftings mentioned in the classification theorem . We have largely neglected the classical problem of classification of DIFF (= smooth C°°) structures on Mm compatible (in the sense of Whitehead) with a fixed PL (= piecewise linear) structure . Fortunately C .R ourke [R0 2 ] has announced an exposition emphasizing this problem . Our discussion does apply with no essential change to tliis PL-DIFF version up to the end of the first classification theorem 1.4 . (A notable feature is that the restrictions concerning dimension 4 vanish). We adopt the terminology of [ I , §2] , all of it we hope selfexplanatory . When an object of the form I X A X M is considered where I = [0,1] and A is a manifold , it is convenient to write □ (I X A X M) or simply the heiroglyph □ for (I X 3A X M) U (I X A X M ) . The expression rel C means ‘leaving fixed the restriction to some neighborhood of C

220

Essay V

§1 C L A S S IF Y IN G CAT ST R U C T U R E S ON M A N IF O L D S BY CAT ST R U C T U R E S ON M IC R O B U N D L E S 1.1

F o r a TOP manifold without boundary Mm we wish to acquire an understanding of the space CAT(M) of GAT manifold structures on M . It is defined as a complete semi-simplicial set ( = css set = css space = css complex ) t as follows . A typical ksimplex 2 G CAT(M)(k) of CAT(M) is a CAT (= DIFF or PL ) manifold structure on Ak X M such that projection (Ak X M)2 A is a CAT submersion . One often writes 2 : A -► CAT(M) . For any CAT map X : A® -* Ak , there is an induced map X^ : CAT(M)W *- CAT(M)(k) sending 2 to the pullback X#2 of 2 by X. Here (Ak X M )^ pulls back just like a CAT bundle over Ak . For the injective and suijective ordered simplicial maps X : A® -* Ak we get the face and degeneracy operators in CAT(M) . 1.2 Since the question of existence of a CAT structure on M has been extensively discussed elsewhere [Las2 ] [IV] [SijQ , §4] , it would not be unreasonable to suppose that M is CAT (so that CAT(M) is nonempty) . This makes it easy to define a space CAT(r(M)) of CAT structures on the tangent microbundle t(M ) or indeed on any microbundle £ over M . Recall that r(M) has total space E(rM) = M X M , projection p2: M X M -> M (to second factor) , and zero-section 5 :M ->■ M X M , 5(x) = (x,x) . . A CA T structure 2 on a TOP m-microbundle £ : X -*■ E(£) X over a CAT manifold X is by definition a CAT mani­ fold structure 2 on an open neighborhood U in E(£) of the zero section i(X) such that p : Xis a CAT submersion . If in addition i : X -»• is CAT we call 2 a CAT microbundle f For instruction in semi-simplicial topology see [Lam] [May] [R S j] . Let A’ be the category whose objects are the standard simplices Ak C R k + 1 , k = 0 , 1 , 2 , ••• , and whose morphisms are the order preserving simplicial maps among these Ak . Then a css set X is by definition a contravariant functor from A* to the category of sets. We write X : Ak +> X^k^ , and call X^k^ the set o f k-simplices of X .

§1.

Classifying C A T structures by structures on r(M)

221

structure on %. Another such , E' , has the same germ about the zero

section if E = E' on some open neighborhood of i ( X ) . We define a k-simplex of the space CAT(£) [respectively CAT(£)] to be the germ about the zero section , of E , a CAT structure [ respectively a CAT microbundle structure] on the product bundle X £ over X X . (The inclusion CAT(£) C_> CAT(£) is usually and perhaps always a hom otopy equivalence , see § 2.1) . Analogous to the differential in immersion theory is a mapping of css complexes d : CAT(M) CAT(r(M)) defined as follows when M is CAT . For 2GCAT(M)^k'* define dE to be the germ about the zero-section of the CAT structure E X M on E(Ak X rM ) = Ak X M X M . We have expressly chosen as pro­ jection of r(M ) the second factor p2 : MXM -^M , p2(x,y) = y >in order to make this most im portant rule optimally simple. We propose to prove in this section that d is a hom otopy equivalence except perhaps in some cases when m = 4 . Beware that dZ does not lie in the subspace CAT(r(M)) of CAT(r(M)) unless the identity mapping M happens to be CAT . We will clear up this point in §2.1 below . 1.3 A slightly more general “differential” is wanted when the TOP manifold M without boundary has no given CAT manifold structure t What one does is choose a topological embedding M -> N into any CAT manifold N along with a continuous retraction r : N -* M C N . ( As M is an ANR [Sp] , r exists as soon as N is replaced by a small neighborhood of M ) . Consider the pullback t = r*r(M) over N It is P2 t

:M XN (

* N , where 5(y) = (r(y) , y) G M X N .

$ As N is CAT , the space C A T (f) is well defined , and the rule E +> E X N is a map CAT(M) -*■ CA T(f ) ; then passage to germs of structures about 5(M) in M X N defines a complex CATr (M) = = lim (CAT(r IU) ; M C U open in N} and a restriction map CAT(r ) -* CAT f (M) . The new differential ^ The generality is also helpful in checking that the ultim ate classifications of § § 2 , 3 , 4 do not depend on certain choices , like the preferred structure on M in 1.2 .

222

L. Siebenmann

Essay V

d : CAT(M) -»■ CAT r (M) is the composition of these two maps , and clearly coincides with the old differential if M = N . 1.4 FIRST CLASSIFICATION THEOREM . For any TOP manifold M m w ithout boundary , m f 4 , d : CAT(M) -*■ CAT r (M) is a hom otopy equivalence o f css Kan com plexes. ADDENDUM 1.4.1 . I f m = 4 and CAT = P L , d is a hom otopy equivalence provided no com ponent o f M is compact ( c f fLasjJ [Las2 ][S i10> § 4 J ) f . If CAT = PL , one verifies the Kan extension condition for these two complexes by applying the pullback rule to PL retractions A^ -> j (where A^ j is 9A^ minus the interior of the face opposite the ith vertex) . The Kan condition is not so obvious if CAT = DIFF as there is no DIFF retraction A^ -> A^ j , and we shall post­ pone its verification (using 0 . 1) to the very end . The homotopy theory of css Kan complexes is the classical homotopy theory of their geometric realizations (see especially [RS j - I , § 6 ] ) . The reader is urged to exploit this fact constantly to avoid technical difficulties in verifying homotopy equivalences below . For a css complex X the geometric realization IXI is the CW complex formed from the sum JLi A^ X X ^ ) (where X ^ ) is regarded as an k abstract set with discrete topology) by identifying each point (x ,a ) £ A^ X a , a E X ^ ) to (Ax,r) £ A X r , r G X® , whenever A : A^ A^ is an order preserving simplicial map and a =A . Geometrical realization is clearly a functor from css complexes to CW complexes . It is even faithful (injective) if we retain, in the CW complex structure of I Xl , the natural characteristic maps A^ -> I XI , one for each nondegenerate simplex t of X . This permits us to identify css complexes with their geometrical realizations . Any ordered simplicial complex K is thus identified with (the geometrical realization of) the css complex of all order preserving t For closed M4 see the partial results of [LS] ; for CAT = D IFF , cf. 1.6(A). t A degenerate simplex is one of the form A^V for some k> £ .

A : Ak

with

§1.

Classifying C A T structures by structures on r(M )

223

simplicial maps Ak -* K , k = 0,1,2,. . . . A map between css complexes is always understood to be a css map , until the contrary is stated . Here is a useful semi-simplicial approximation lemma from [RS j -1, §5.3 , §6.9] . Any merely continuous map f : K -*■ X from an ordered simplicial complex K to a css Kan complex X , can be continuously homotoped to a css map f' : K X ; further if f is css on a subcomplex L C K , the homotopy can leave f i L fixed . This lemma will give us the liberty to choose convenient representatives for elements of homotopy groups . A css map n : X -► Y is called a Kan fibration if to any com­ mutative diagram A k ,i

Q, Ak

X

l 7T Y

one can add a map A^ -> X preserving commutativity . Observe that X is a Kan complex (i.e. , satisfies the Kan extension condition) pre­ cisely if the constant map X A0 is a Kan fibration . It is a very convenient fact that the geometrical realization of a Kan fibration 7r : X -* Y of Kan complexes is a Serre fibration [Q] ; thus it has the hom otopy lifting property for simplices-and hence for CW complexes . PROOF OF THE CLASSIFICATION 1.4 FOR CAT = PL . Given the Kan complex conditions , arguments conceived for immersion theory , (see especially [HaP] , also [Gr] [Ha2 ] ) permit a proof of the hom otopy equivalence as soon as we verify a few simple facts . (1) The rules U +* CAT(U) and U +> CAT t (U) are contravariantly functorial on inclusions o f open subsets o f M . They convert (i) union o f two into fiber product (ii) disjoint discrete sum (possibly infinite) into cartesian product. ■ Definition : For any subset A C M we use injective limits to define t These rules in fact con stitu te what are called sheaves o f css sets, (see A ppendix A.

Essay V

L. Siebenmann

224

CATm (A) = lim (CAT(U) ; A C U open in M} C A T f (A) = lim {CAT(f IV) ; A C V open in N} Note that we have a natural ‘ differential ’ dA : CATj^(A) -*■ CATr(A) still defined via the rule 2 +> 2 X N . Note that the extended rules A ^ CATj^(A) and A+*CATf ( A) will also enjoy properties (i) & (ii) of (1) , when restricted to closed sets A. (2) I f B is a simplex (linear in some chart o f M ) and A C B is a point then the restriction maps (i) CATm (B) -> CATm (A)

, (ii) CA Tf(B) ^ C A T f ( A )

are hom otopy equivalences. Proof of (2)(i) . There exists a hom otopy h^ , 0 < t < l , of i d | M = hQ such that h^ , 0 < t < l , is an isotopy fixing A with ht (B)CB , while hj crushes B dnto A and induces a homeomorphism M - B ^ M - A . Using this isotopy ht , 0 < t < l , one readily shows that the restriction (i) induces an isomorphism of semi-simplicially defined homotopy groups. Proof of (2)(ii) . The proof is similar but requires instead CAT homotopies in N . More precisely, using relative CAT approximation of continuous functions [Ze2] [Muj ,§4] , one obtains, for prescribed open neighborhoods U C V of A and B respectively in N , a CAT homotopy fj , 0 < t < l , of i d l V ^ f Q fixing A so that f j ( B ) C U . Such homotopies fj. let one show that restriction (ii) likewise induces an isomorphism of semi-simplicially defined homotopy groups. ■ (3) I f A C M is a p o in t, then d : C ATj^(A) hom otopy equivalence.

C A T t (A) is a

Proof of (3) . Indeed one shows that restriction r (CAT r( A) -> CAT( rl A) is a homotopy equivalence by using a CAT deformation of an neighborhood of A to a retraction onto A . Now , (pleasant surprise!) the composition rd : CA T^(A ) C A T ( r l A ) is a true isomorphism of css sets . ■ (4) For any compact pair mappings induced (i)

A C B in M ,the two natural restriction

CATm(B) ^ CATm(A) , (ii) C A Tt(B )

CAT? (A)

are both Kan fibrations. One can deduce the same for any closed pair A C B , by using (1) .

§1. Classifying C A T structures by structures on r(M)

225

Proof of (4). For A and B open (not compact) the sliced concordance extension theorem [ I I , §2.1 ] says precisely that (i) is a Kan fibration since C A Tj^B ) is a Kan complex . The same follows for quite arbitrary pairs A C B in M by taking injective limits . The reader should also deduce fibration (i) for compact A and B from the less refined Bundle Theorem 0.1 with the help of the CAT and TOP isotopy extension theorems . We have here a very essential use of I'll] • As for fibration ( i i ) , given open neighborhoods U C V in N of the closed sets A C B respectively , and a CAT structure 2 on Ak i X t (V) U X r (U) we can find (as CAT = PL) a CAT map r : Ak XV -» Ak jXV U Ak XU respecting projection to V and fixing A n X V union a neighborhood of A X A . Then existence of the pullback r* 2 as a welldefined CAT structure on A ^ X t (V) establishes the fibration (ii) . ■ Given these four properties , the proof of 1.4 , is reduced to homotopy theory by the famous ‘handle induction machine’ of immersion theory [HaP] [Gr] . Here is an outline For A C M , &(A) is the statement that dA : CATj^(A) -*■ C A Tr(A ) is a hom otopy equivalence . It is understood that the simplicial complexes mentioned below are simplicially imbedded in some co-ordinate chart of M . (a) &(A) holds fo r any simplex A . Proof: Combine (3) with (2) . (j3) I f & holds fo r the compacta A , B , A Cl B , it also holds fo r A U B . Proof : Apply (1) and (4) to the commutative square of four inclusions . (Cf. proof of A.2 in Appendix A .) (7 ) &(A) holds fo r any finite simplicial complex A . Proof: Use ( a ) , (|3) and induction on the number of open simplices . (5) 6 (A ) holds fo r any compactum A in a co-ordinate ch a rt. Proof : A is an intersection of finite simplicial complexes . ^ For the student it is eminently worthwhile to fill in the details , and as elegantly as possible .

226

L. Siebenmann

Essay V

(e) &(A) holds fo r an arbitrary compactum A C M . P ro o f: A is a finite union of compacta in co-ordinate charts , a union to which one applies (0) inductively . (f) &(M) holds. Proof: One can assume M connected , so that M is a union of nested compacta A! C A2 C — with Aj C Aj+ j . Define BQ [respectively Bg] to be the disjoint union of the compacta A2j+ ] - A^j [respectively A.2i - A.2j_] ] for i=0,l ,2,... . Now apply the proof of (j3) to the union M = BQ U Be using ( 4) , (e) and (1) . This concludes the proof of 1.4 for CAT = PL .

■

PROOF OF ADDENDUM 1.4.1 for CAT = PL . The reader may omit this proof , as the result is of marginal importance . It does however apply to all dimensions . The appearance of a weak homotopy lifting property and a homo­ topy micro-lifting property (= ‘micro-gibki’ property) in the relevant immersion theoretic method make it convenient to enrich our semisimplicial complexes to become polyhedral quasi-spaces (as in [ Si j q , §4] , cf. Appendix A) . Thus a map P -* CAT*(M) to the enriched space CAT*(M) is by definition a CAT structure 2 on P X M such that first factor projection pi : (P X M )^ -*■ P is a PL sub­ mersion with each fiber a PL manifold . And if £ is a TOP micro­ bundle over a locally compact polyhedron B , a map P -*■ CAT*(£) is a PL structure on the microbundle P X % . Thus our css complexes become contravariant functors from the category of compact polyhedra to the category of sets ; this change is indicated by an added star . We must show that the differential d : CAT*(M)

CAT*?(M)

is a weak homotopy equivalence of polyhedral quasi-spaces , i.e. gives an isomorphism on homotopy groups (defined as for ordinary spaces) . All the basic properties (1)—(4) remain essentially valid in this quasi-space context except the fibration (4)(i) above , established using m f 4 . Note that the Kan fibration property becomes the Serre fibration property (= homotopy lifting property for compact polyhedra). Also weak homotopy equivalence replaces homotopy equivalence . This missing fibration property (4)(i) is replaced here by the following micro-gibki property or ‘homotopy micro-lifting’ property .

§1. Open 4-manifolds

(fi)

227

Given any commutative square OXP

Fq u->

| IXP

* CAT^(B)

f f ->

= (restriction)

CAT* (A)

where P is a compact polyhedron and (B,A) is a compact pair in M , there exists an e > 0 and a map Fe : [0,e]XP -> CAT* (B) extending F 0 so that 7rFe = f | ([0,e]XP)

.

Proof o f ( p ) : This property follows from the PL and TOP (manyparameter) isotopy extension theorems via the ‘union lemma for submer­ sion product charts’ [S ij2 ; 6.9 , 6.10 , 6.15] , applied to charts around parts of the fiber OXPXM of the submersion pj : I XP XM- M , in a form respecting all fibers of the projection p2 to P . More precisely, one obtains such a product chart : [0,8]XPXV-» IXPXM , V an open neighborhood of B in M , such that, representing f by (IXPXU)^ , one can restrict the chart

A^ , 0 < t < 1 , be a CAT homo­ topy of id IAk respecting 3Ak so that r7*(3A^) contains an open neighborhood U of 3A^ .( For CAT = DIFF , rt cannot fix 3 A (pointwise) ; it can be constructed by composing ( k + 1) homotopies that squash a collar of a face 3:A^ into 3.-Ak following the 1 lines radiating from v j ) . Now define a self-map p of I X A X X by p(t,u,x) = (t,r^(u),x) and consider the pullback p*(I X 7 ) over I X Ak X X This bundle inherits from p*(I X 7 ) -* I X 7 -» 7

and g : 7 | ( 3Ak XX) -> 7 ^ AT

a microbundle map p*(I X 7 )! A -*■ 7™ ^ , A = (I X 3Ak U 1 X U) X X , that is CAT wherever this makes sense . By the standard CAT universality property cited this extends rel I X X X to all of p*(I X 7)1 □ to produce a map^ □ (I X A^) -> ® c a T ^ which establishes the hom otopy equivalence a . ■ t The condition that Y be an ENR can be replaced by the condition th at r}m be numerable if we use the more general form o f the Kister-Mazur theorem in [Hoi] together with M ilnor’s classifying space for R m - bundles [Hus] . • In term inology we have occasion to use elsewhere the CAT hom otopy rt of i d l A^ yields a conditioning p*(J X 7 ) o f the simplex 7 : A^ ® CAT(X) ; the conditioned simplex is p*(I X 7 ) I 1 X A^ X X . ^ D eterm ined by a convenient simplicial subdivision o f the prism

I X A^ .

238

Essay V

L. Siebenmann

Continuing the proof of 2.2 , consider now the natural commutative diagram of ‘forgetful’ maps ®CAT(X) 4, ®TOP(X)

fflgsT(x )

(X,BCAT(m)}

^ ©clsp(X)

4K

{X, BTOp(m)}

in which the verticals are known to be Kan fibrations while the horizontals are known to be homotopy equivalences . For the point (t,)= % and 0(£,y>) = f on identifying A®XX = X . The resulting two equivalences between the three css fibers of the three vertical maps, over the points k - (!,) , and f ( in this order ) , are by definition the wanted natural equivalences of Kan complexes CAT(£)

CATcls(£) ^

Lift(f to BCAT(m) ) .

This completes the proof of 2.2 and also of the classification theorem 2.3 .■ PROOF OF COMPLEMENT 2.3.2 . Consider two choices of embedding i : M C* N and retraction r : N -*■M distinguished by subscripts 0 and 1 , yielding equivalences 0A , 0A via 2.3 . (a) Suppose N q C N } and that there is a CAT retraction p : N j -*■ N q so that r j = r0P • Then is is clear that 6 ^ — 6^ .

■

Using (a) we easily reduce the proof of 2.3.2 to the case where N0 , N j are open subsets of Rn , n large , and one has a topological embedding I : I X M C*. N C I X Rn and neighborhood retraction r : N -»• I X M so that i , r both are a product along I near 0 and 1 , and give concordances i : i0 ~ ix and r : r0 ~ rl . (b) In this situation 6% — d \ : CATM(A) -> L i f t f f to BCAT(m) near A ) .

§2.

C A T structures and classifying spaces

239

To prove (b)_we form a diagram C'ATm (A) cf)

4- CAT(r near 1(1 X A)) % qt

-> L ift(IX f to BCAT(m) near IXA) -»• L ift(f to BCAT(m) near A)

Here r is N - 4 MXN N with j(y) = (pr(y),y) , p being the natural projection t(IX M )-^M ; d is the differential Z +*2X N ; 4' is the equivalence of 2.2.1 composed with restriction to I X M . (This equivalence of 2.2.1 being obtained using

5 , see property (4) in proof of 1.4 ; for m = 4 we secure this fibration by assuming H4(M, C; Z2 ) = 0 (for Cech cohomology) , see proof of 1.4.1 and Appendix A . It follows now that there are equivalences of the fibers of r t and r2 over components corresponding under 0C . To fix notations let 2 Q be a CAT structure defined on an open neighborhood UQ of C in M , and let gQ be the lift corresponding to 2 Q under 0uo • Then we write (3.2)

CAT(M rel C ; 2 Q) ® Lift(f to BCAX(m) re lC ;g 0)

for the induced equivalence of the fibers over the germs of 2 Q and gQ respectively . Note that this 0 again arises from a sequence of four perfectly canonical and natural equivalences . We apply this now taking M = N = Rm , m > 5 ; C = Rm - Bm ; 2 Q standard ; f and gQ constant . The left hand side of (3.2) becomes CAT(Rm rel Rm-B m ) = CAT(Bm rel dBm) . The right hand side becomes the singular complex of the space of those maps Bm/9Bm -► -*■ TOPm/CATm sending a neighborhood of the base point (quotient of 9Bm ) to the base point (image of gQ) . Here TOPm/CATm is the fiber of j : BcAT(m) BTOP(m) over the base point (image of f ) . Thus the right hand side is equivalent to the mth loop space S2m(TOPm/CATm) , and we have deduced an equivalence (3.3)

CAT(Bm rel 9Bm) ~ S2m(TOPm/CATm ) .

§3. The relative classification theorem

241

Now the loop space of CAT(Bm rel 3Bm ) pointed by the standard structure is (up to hom otopy) the css group of CAT automorphisms A utCAX(Bm re* 9Bm) because we have the natural Kan fibration^ AutCAX(Bm rel 3) -*• AutX0 P(Bm rel 3)

CAT(Bm rel 3)

where the total space AutXop(Bm rel 3) is contractible by Alexander’s device [ A lj] ^ . Thus we have verified MORLET’S THEOREM 3.4 . AutCAX(Bm rel 3Bm ) ^ J2m + 1(TOPm/CATm)

fo r

m + 4 .

m

Since AutpL(Bm rel 3Bm ) is contractible by the PL version of the Alexander device, one has, for m ¥=■4 , f2m +1(TOPm /PLm ) ^ (point) that is, 7rm+k(TOPm/PLm) = 0 , k > 1 . Historical Remarks : Morlet first obtained and made famous this striking result for PLm/D IFFm in his cours Peccot 1969 [M o ^] • It was probably the raison d’etre for his formulation of the sliced classifi­ cation theorems [M or^HM or^J . Several mathematicians besides us have since taken the trouble to verify it themselves (A. Chenciner , C. Rourke, D. Burghelia, . . . C erfs version of 3.4 for CAT = DIFF in [ C e ^ ^ is (with a slight shift of terminology) (3.4.1) 7ri(AutDiF F (Sm ),0 (m + l)) = 7ri+m+1(A utXOp(Sm ),0 (m +l)). He regards A utcAXSm as a topological group, with C°° topology for CAT = DIFF , and with C° topology for CAT = TOP . Exploiting the natural fibrations A ut£AXSm Sm and 0 (m + l) -*■ Sm , one shows without difficulty that 3.4.1 is equivalent to 3.4 in the equation form (3.4.2)

5Ti(AutDjFF(Bm rel 3Bm )) = 7ri+m + 1(TOPm /D IFFm ) .

^ Geometrically realized , this is a Serre fibration [Q] . $ One can homotop any map of the contractible spaces AutXQp(Bm rel 9) -*■ A CAT(Bm rel 8) to be fiber preserving over CAT(Bm rel 8) . There results a homotopy equivalence of the fibers A u t£^ p (B m rel 8) 12 CAT(Bm rel 8) . ^ For interested readers unable to find [Ce^J , Theorem 4 on p. 365 of [Ce^] closely approaches the case m = 3 .

242

Essay V

§4. C L A S S IF IC A T IO N S FOR M A N IF O L D S WITH B O U N D A R Y

An interesting variant of Morlet’s theorem ([M o ^] [M o ^ ]) comes from generalizing the classification theorem to manifolds with non­ empty boundary Mm , ( m ^ 4 ^ = m —1) . 4.0. For technical convenience we initially single out an open neighbor­ hood W of 3M in M with an open embedding a : W -*■ 3MX[0,°°) so that, for all x in 3M , a(x ) = (x,0) . Also, we form M+ by attaching 3M X(-°°,0] to M identifying 3MX0 to 3M . Write 5 M+ for 3MX(—o°,0] viewed as a subset of M+ ; write W+ = 5M+ U W , and extend a trivially to a + : W+ ->- 3MXR . 3M

> r(9 M )ee1

— ►7xoP®el ""^TTOP

then extend rel SM+ over all r(M +) . Finally, choose a CAT manifold N+ 2) M+ and a retraction r+ : N+ -*■ M+ ; for 9M we insist on taking N' = r^.’ W+ D 9M with the composite retraction N' - 4

W+

9MXR ->• 9M .

With all these choices made, one gets using 2.3 a well-defined hom otopy equivalence, from the diagram 4.1 without CATa (M) , to the fiber product diagram below without L . Lift(f+ to BcAT(m)) (4.2) L ift(9f to BcA T(m -l)) —

| restriction Li ft (f+ to BCAT(m) near 6M+)

Here a * is defined on a k-simplex g : Ak X9M -*■ BCAT(-m_j) on the left by defining a*(g) near Ak X5M+ as v Ak Xa+ . proj v g A XW+ Ak X9MXR A X9M BCAT(m_1) .

L. Siebenmann

244

Essay V

To check that the classifying equivalences d carry a * in (4.1) to a * in (4.2) is straightforward, given r(9M XR) = (r(9M )X R )ee1 . Admittedly one has to check down through the stages of the construction tion of 6 , ( cf. 2.3.1). We then deduce an equivalence CATa (M) =* L , since the vertical maps in both squares are Kan fibrations. By collaring theorems (existence for CAT f and uniqueness for TOP ), the inclu­ sion map is an equivalence CATa (M) Q . CAT(M) to the space of all CAT structures on M (without corners if CAT = DIFF ). And (4.2) is homotopy equivalent by restriction to the simpler fiber product square of Kan fibrations. * Lift(f to BCAT(m))

I

(4.3) Lift(9f to BCAX(m_j))

+ Lift(f | 9M to BCAT(m) ) .

In all, we have an equivalence: (4.4)

CAT(M)

> Lift(f,9f to BcAT(m )>B C A T (m -l))

valid t for dimM 4 =£ dimdM . This classification has a version relative to a closed subset C C M for just the same reasons for which the absolute theorem 2.3 has the one formulated as (3.2) . With notation generalizing that for (3.2) in the obvious way, it gives an equivalence (4.5)

CAT(M rel C ; 2 0) Lift(f,9f to B c A T (m )3 c A T (m -l) rel C ; go,9go)

Applying this to M = [0,oo)XBm- J with C = M - { [0 ,l)X B m-1 } and with f , 9f , g0 , 9gq all constant maps we get (for m ^ 4 ^ m -l ) : (4.6) 7TiAutCAT(IXBm-1 r e in ) = 7ri+m+1(TOPm /CATm ,TOPm _1/CATm_1 ) It says nothing new for CAT = PL . But, for CAT = DIFF , Cerf fF or CAT = DIFF , integration of vector fields may as well be used. For CAT = PL and for TOP exploit [ I, Appendix A] . $And also valid if (i) m = 4 and M connected non-compact, or (ii) m = 5 and 3M has no nonempty compact component. The same proof applies, although for CAT = DIFF we would have to use DIFF quasi-spaces.

§4. Classifications for manifolds with boundary

245

[€ 63] has proved, by studying generic 1-parameter families of smooth real functions, that, for m > 6 and i = 0 , the left side is zero. This adds one dimension to the stability of TO Pj^/D IFF^ implied by the s-cobordism theorem (see 5.2 b elo w ). On the other hand, for i = 1 ,1 . Volodin [Vol] has asserted that the left hand side is Wh3(0) © Z2 , provided m is large ( m > ). K. Igusa (thesis, Princeton Univ., 1976) has confirmed that it is non-zero, admitting a homomorphism onto the advertized Z'i . It is known that K3(Z) = Z48 , (see [Kar] and joint articles of R. Lee and R.H. Szczarba to appear), and it is thought that Wh3(0) will be the order 2 quotient by Z24 •

246

Essay V

§5. THE HOMOTOPY G RO U PS 7r*(TOPm/CATm ) .

5.0. The calculations we review can be summarized as follows: t (1) TOPm /PLm =sK(Z2 ,3) for m > 5 . See 5.5 . (2) TOPm/PLm is contractible for m < 3 . See 5.8 , 5.9 . (3) 7rk+ 1+m(TOPm /DIFFm ) = 7rkA utDIFF(Bm rel 9Bm ) , m See 3.4.

4 .

»k+1+m(TOPm/D IFFm ,TOPm_1/D IFFm_1) ■= xkAutDIFF(Bm->XI r e i n ) , m =£ 4 ¥= m—1 . See 4.6 . (4) 7rk (TOP/DIFF,TOPm /D IFFm ) = 0 , k < m+2 , m > 5 , where TOP/DIFF = U T O P „/D IF F m . See 4 .6 , 5 .2 . m 111 111 (5) 7rm ( T O P / D I F F ) = 0 m , m> 5 , = 7rm K(Z2 ,3) , m < 6 .

See 5.5 .

(6) 7rk (TOP3/D IFF3) = 0 , k < 4 . See 5.8 . (7) TOPm/D IFFm is contractible m < 2 . See 5.9 . 5.1. Recall that TOPm /CATm is by definition the standard homotopy fiber of the forgetful map jm : BcAT(m) BiOP(m) (see 2.0) over a chosen base point. One can arrange that all the squares s B C A T (m )

B C A T (m + l)

s B T O P (m )

B TO P(m + 1)

relating stabilizations s to forgetting maps (verticals) are commutative and present each s as an inclusion. One need only enlarge the spaces by mapping cylinder devices and suitably choose the maps. This done, define B C A T = ^ B C A T (m ) t

and

B TOP = ^ B TO P (m )

Some further information can be found in [BuL]

•

§5. The hom otopy groups of T O P j^ / C A T j^

247

One notes that s : TOPm /CATm -*■ TOPm+ j /CATm+ j is then also an inclusion and that, in analogy with TOPm/CATm , the space TOP/CAT = U (TOPm /CATm ) is precisely the fiber of the standard Serre path fibration -*■ BTOp FIRST STABILITY THEOREM 5.2 . (C A T = DIFF or PL ; m i - 4 , 5 ) For 0 < k < m , t t ^ / T O P J C A T ^ T O P ^ / C A T ^ j ) = 0 . Hence tt^TOP/CAT, TOPm (C ATm ) = 0 fo r i < m + 1 and m > 5 . P roof o f 5.2 . By (4.5) , this group is in bijective correspondence with 7Tc,CAT(IXBk XRn rel □ ) where 1 + k + n = m . But the latter is zero by the Concordance Implies Isotopy Theorem (already in its handle ver­ sion [I , §3]) . Our present heavy machinery could eliminate parts of the proof of [I , §3 ] , cf. proof of 5.3 below. But both the CAT s-cobordism theorem and some topological geometry (involving infinite processes or coverings, and the Alexander isotopy) do seem irreplaceable ingredients. The cases m < 3 will be covered again by the next result. ■ THEOREM 5.3 (C A T = D IFF or PL) [K Sj] . Suppose 4 # m < 6 , and k (M,9M) from a compact CAT m-manifold Mj , such that f } I 9Mj is a CAT isomorphism 9Mj -»• 9M and fj is a CAT embedding near 9Mj . Another such equivalence f2 : (M2 ,9M2) (M,9M) represents the same element of S*(M,9M) if and only if there is a CAT isomorphism h : Mj -*■ M2 so that f2h is homotopic to fj :(M j,9 M j) -*■ (M,9M) fixing a neighborhood of the boundary. [ In case m = 3 , we supplement'this definition by supposing that Mj is Poincare i.e., contains no fake 3-disc (=compact CAT contrac tible 3-manifold not isomorphic to B3 ) . J

248

L. Siebenmann

Essay V

The natural map S(M,3) -»• S*(M,3) sends [2] to the element represented by id : (M ^S M ^) -» (M,3M) . [ For m = 3 , to see that this a valid definition one must know that M£ = (Ik XTn)2 contains no fake CAT 3-disc. A proof is given by [Mac] [HeM] t in fact for the universal covering. We prefer not to appeal to Moise’s proof of the Hauptvermutung in dimension 3 [Mo] . 1 Represent a given element x of tTj^TO P j^/CAT jj,) by a map Ik TOPm /CATm ) sending 3Ik to the base point (image of a classifying map for the standard structure). Then extend to Ik XTm-k by projec­ tion to Ik . T o this map there corresponds by 3.2 a well defined ele­ ment [2]€= g(Ik XTm-k rel 3) represented by a structure 2 . I f x i= 0 , then the image [2]* o f [2 ] in g*(Ik XTm_k rel 3) under the forgetful map §-► §* cannot be zero. To see this, suppose it were, that is, suppose one had a CAT isomorphism h : Ik XTn -*■ (Ik XTn)2 , n = m—k , fixing a neighborhood of 3 and homotopic to the identity rel 3 . Then some standard finite An -fold covering h^ , A > 1 , of .h will be seen to be topologically isotopic to the identity rel 3 . Here h^ fixes 3 and makes commutative the square jk XTn idX(-A) | Ik XTn

(ik x T n)2 x | idX(-X) (Ik XTn)s

where (* X ):T n ->-Tn indicates multiplication by A and 2 ^ is defined to make the right hand map a CAT covering map (An -fold) . For A large one checks that the component of h^ on Tn can approach that of the identity. Further, up to a standard isotopy, h^ has component on Ik arbitrarily near the identity. This standard ‘Alexander’ type isotopy extends h^ by the identity over all of Rk XTn then conjugates by an isotopy that shrinks Ik in Rk radially to towards a point. From the local contractibility theorem [EK] for the f An alternative proof of 5.3 for m = 3 (suggested in [Sig , §5 ] ) neatly avoids this technicality by solving handle problems in coordinate charts where the A lexander‘Shoenflies theorem’ [A^lICe^ ] precludes fake 3-discs. In following the technique of [KSj ] , one should use the Novikov imbedding Tn"*x R Q . Rn in place of the Kirby immersion (Tn-point) Rn . This approach gives our favorite proof of the Hauptvermutung and the triangulation conjecture in dimension 3 . See also the recent article of A.J.S. Hamilton [Hal] , who has succeeded even with the Kirby immersion.

§5. The hom otopy groups of TOPjn/CATjy,

249

group of homeomorphisms of Ik XTn fixed on the boundary 9 , we conclude that h^ is isotopic to the identity rel 9 for X large, which shows [2jJ = 0 in g(Ik XTn rel 9 ) . Since a classifying map for 2^ is the covering idX(*X) followed by that for 2 , we can conclude that x = 0 by restricting the nul-homotopy of the classifying map of 2 X to Ik X(point) C Ik XTn . We have seen that [ 2 ] * lies in the subgroup g*(Ik XTm_k rel 9 ) of g* consisting of those elements that are unchanged by passage to standard Xm -k-fold coverings, \ > \ ; i.e., those [f] G g* , f : M -► Ik XTm~k , such that [f] = [f] whenever one forms a fiber product square M -5—» Ik XTm_k |

|

idX(-X)

M

► Ik XTm_k

Thus the next step is to exploit the im portant surgical calculation of and g* from [HS] and [Wa] . Theorem 5.4 . (C A T = DIFF or P L ) For m < ,3 , one has &*(Ik XTm ~k re lb ) = 0 . For m = 5 or 6 (or m > 5 and C AT = PL ) ,one has a bisection from g * (Ik XTm - k rel 9 ) to H 3 (Ik XTm ~k , 9 ; Z 2) = _ j] 3 -k (jm - k ; z 2 ) . Under this bijection the self-maps o f g* and H 3 respectively, induced by the standard Xw ~k -fold covering, correspond to one a n o th er. Since this Xm-k -fold covering in fact induces multiplication by the integer X3' k in the cohomology group we have the Corollary 5.4.1 . For 4 = £ m K 6 , the set g *(Ik X7m ~k rel 9 ) o f hom otopy smoothings or triangulations invariant under passage to standard Xm ~k-fold coverings is zero unless 3 = k < m , in which case g ^ l b X ^ - b relb ) = Z 2 . ■ Comments on the p ro o f o f 5.4 . (see also Appendices B and C ) . The proof for m > 4 is given in [HS] and [Wa] (cf. [S ig ]) for

250

Essay V

L. Siebenmann

CAT = PL, and the DIFF proof is identical since 7Tj(G/PL) = 7Tj(G/0) for i < 6 . It requires the full strength of non-simply-connected surgery [Wa] and the periodicity in surgery related to producting with CP2 • The source of the Z2 is Rohlin’s theorem that closed smooth almost parallelizable 4-manifolds have signature divisible by 16 (not just 8 ) . The reader should be reminded that the PL proof relies on the PL Poincar£ theorem stating that S*(S5) = 0 . There is still no proof of the latter without Cerf’s difficult proof that T4 = 0 [Ce j ] . I The proof of 5.4 for m = 3 is essentially due to Stallings [St j ] His setting is PL , but there would be no obstacle to giving essentially the same proof for the DIFF category. To convert Stallings’ construction of fiberings over the circle into the result 5.4 one uses the fact that any CAT automorphism of Ik XT2-k , 0 < k < 2 , rel boundary is CAT isotopic rel boundary to the identity. This easily is proved if CAT = PL [Sco] , and the DIFF result follows using example 5.9 below. 1 ■ Returning to the proof of 5.3 recall that we have established an injection (#)

7rk (TOPm/CATm ) -*■ §*(Ik XTm~k rel 3)

.

In view of the vanishing of g* assured by 5.4.1 we know now that tTj^CTOPj^/CATj^) = 0 when 4 ^ m < 6 and either k ^ 3 or m< 3 . In case k = 3 and m = 5 or 6 , the target of the injection (#) is not zero but rather Z2 = §*(I3XTm-k rel 9 ) . The source is naturally isomorphic to g (I3XTm-k rel 9 ) ; indeed, the relative classification theorem 3.2 yields g (I3XTm- 3 rel 9) = [I3XTm- 3/9 ; TOPm /CATm ] = H3(I3 XTm- 3,8; i 3(TOPm/CATm )) = H°(Tm- 3 ;ir3(TOPm /CATm ) = = „ 3(TOPm/CATm ) . For m = 6 , we now show that (#) is surjective. Thus 7r3(TOPm /CATm ) = Z2 for m = 6 , and the same follows for all m > 5 by the stability 5.2 . Represent the nontrivial element of S*(I3XT3 rel 9) by f : (M,9M) (I3 XT3 ,9) . Our aim is to show that f can be a homeo­ morphism. We can apply the CAT s-cobordism theorem to the CAT relative h-cobordism c = (M ; f -1 (OXI2 XT3), f -1 (1 XI2 XT3))

,

recalling that Wh(7rjT n) = 0 . Thus we obtain a CAT isomorphism

§5. The homotopy groups of TOPm/CATm

251

f ': M I3 XT3 equal f near f _1 (Zl) = f _1((lXI2 U IX9I2)XT3). This provides a CAT automorphism 0Xg = f 'f -1 : 0XI2 XT3 -»■ 0XI2 XT3 equal the identity near 0X3I2 XT3 . If g is topologically isotopic rel 3 to the identity we can alter f' near f -1 (0XI2 XT3) to obtain a homeomorphism h : M I3 XT3 equal f near 3M . Then h is necessarily homotopic to f rel 3 , the obstructions being in H*(I3 XT3,3 ;7r#(I3XT3)) = 0 . This would then complete the proof. To establish this , find a topological isotopy rel 3 to the identity from a standard A3-fold covering g^ of g ( A a large integer) , by precisely the argument used in establishing the injectivity of (#) . Now it is clear that g^ = f^fx 1 I 0XI2 XT3 , where f^,f^ : M ^-»I3 XT3 are the corresponding standard A3-fold coverings of f , f ' respectively. Thus we now know that f^ can be replaced by a homeomorphism ^A : ^A l3 X^ 3 homotopic rel 3 to f^ . Fortunately any standard covering f^ : I3 XT3 of f still represents the unique nontrivial element of §* = g* . Thus we have a CAT identification = M so that is homotopic rel 3 to f . The surjectivity of (#) is now established and with it Theorem 5.3 . ■ Remarks: (a) We have presented the basic elements of the above argument before [KS j ] [K ^] [Sig] [Si j q ] . The reader wishing to see variants or more details should consult these references. (b) It is worth noting that, that if we accept the classification theorem of Essay IV rather the results of this essay, then the above argument does prove that ^(TO P/CA T) = irj(K(Z2 ,3)) for i < 6 . A similar remark applies to the next theorem. We are now in a position to draw the threads together. THEOREM 5.5 . I) For m > 5 , TOPm IPLm as K (Z 2 ,3) as TOP/PL . II) TOP/DIFF = TOP/O t has hom otopy groups 7r;- = 7^-AT , 3) for i < 6 and 7r;- = 0 ;- fo r / > 5 . Here @;- is the Kervaire-Milnor group [KeM] o f oriented isomorphism classes D IFF hom otopy spheres spheres under connected sum; and K (Z 2 ,3) is, o f course, the Eilenberg-Maclane space with ttj(K (Z 2 ,3) = Z 2 . t See 2.0 .

252

Essay V

L. Siebenmann

Proof o f II) : In view of 5 .2 ,5 .3 , and [ K i j ] t it remains only to prove 7rm(T0Pm/0 m) = 0 m , m > 5 . Borrowing notation from 5.3 we have a forgetful map S*(Sm rel BH1) -* 0 m , where B™ , Bf? are the northern and southern hemispheres of Sm . It is bijective by the DIFF isotopy uniqueness of oriented imbeddings of Bm into Sm . By §3 , 7rm(TOPm/Om) = J (S m rel B™ ) . But S(Sm rel BlI?) -=*■ S*(Sm rel B™) ; indeed surjectivity results from the Poincare theorem in dimension m (cf. [M ig]) and injectivity results from the Alexander isotopy. Thus irm(TOPm/Onl) = ©m . (We let the reader verify that the bijection established is a group isomorphism.) ■ Proof o f I ) : This result follows from 3.3 , 5.2 and the argument for II) above, in view of the PL Poincard theorem that g*(Sm ) = 0 for m > 6 [S n^] [HU2] • This completes the proof of 5.5 . ■ 5.6 . For a given topological manifold Mm , m > 6 (or m > 5 if 9M = 0 ) , we can now observe that the one obstruction to introducing a PL structure on M is a cohomology class k(M)G H4(M;Z2) , namely the one obstruction to lifting the classifying map f : M -►B'poP(m) c BXoP f°r t (M) to BpL(m) , or equivalently to Bpl . The obstruction to sectioning the fibration (**) K(Z2 ,3) -> BpL Btop is the universal triangulation obstruction k £ H4B top >Z2) (also denoted A ), and k(M) = f*k by naturality of obstructions. We represent k by a map k : Bj q p K(Z2 ,4) and observe that there is a commutative square unique up to homotopy: Bpl —^

AK(Z2 ,4) = (K(Z2 ,4), point)(I’0)

Btop

K(Z2 ,4)

(One can specify k' using any contraction of kj|BpL to the base point.) One observes easily that k' gives a homotopy equivalence on fibers: TOP/PL

K(Z2 ,3) = S2K(Z2 ,4)

,

and so a hom otopy equivalence Lift(f to BpL) Lift(kf to AK(Z2 ,4)). When we single out a fixed lift g : M -* AK(Z2 ,4) we deduce by t Beware that 5.2 failed to prove: 7Tg(TOP/CAT , TOPm /CATm ) = 0 for m > 5 . Cf. [ III, Appendix A] .

,

§5. The homotopy groups of TOPj^/CATm

253

matching common end points of paths, an equivalence Lift(kf to AK(Z2 ,4)) -> K(Z2 ,3)m to the space t of maps of M to K(Z2 ,3) = ^ K (Z 24) for which g also offers a natural inverse equivalence. We can conclude that when Mm is a PL manifold, with m > 6 (or m > 5 and 3M = R 1 . m = 2 : Already in 1926 [Kn] H. Kneser published a pleasant proof based on conformal mapping theory. In 1958 [Sm j] Smale proved that that Autj)iFp(B2 rel 3B2) is contractible, cf. [Ce j ] ; this accomplishes an indirect second proof, via the argument presented for 5.8 . ■ We wish to advertise an elementary proof that is a variant of the reasoning in [Ki^] t . Bjorn Friberg gives a full discussion in [Fr] . We denote by TOP* the space of orientation preserving homeomorphisms of the Gauss plane C R2 fixing the point 1 (not 0 ). Its topology is, of course, the compact-open topology. There is a locally trivial fibration H+ = p-1 (0) TOP2 C - {1} where p selects the image of 0 £ C . Pulling back this fibration over the universal cover of C - ( 1 } , one sees that 0 (2 ) Q. Homeo(R2 ,0) is a homotopy equivalence if and only if the inclusion H+ = p" 1(0) Q. TOP2 is deformable to the constant map. To establish this deformability, consider a typical h e H + . There is an isotopy h ~ h j fixing 0 and 1 and depending continuously on h to a homeomorphism hj such that log Ihj (z) | —log Iz I < — . This involves shuffling circles with center 0 with their images under h . Then there is a wrapping and unwrapping device (see [EK, Appendix] and [S ij2? §4.9] ) , yielding h 2 €= H+ such that h2(ez) = eh2(z) , where e = ex p (l) , and h 2 = h j on a neighborhood of 1 independent of h . The rule h 1 +^ h 2 is again continuous. Lifting h 2 |C* in the universal covering exp : C C -{0} = C* , there results a unique covering h 3 of h 2 |C* with exp°h3 = h2 °exp and h 3 (0) = 0 . This h 3 commutes with translation by the points t Reasoning th at is unfortunately fallacious in dim ensions > 2 , but see [KS^] .

§ 5. The homotopy groups of TOPm/CATm

255

m + 2irin in C with m ,nG Z . Thus |h3( z ) - z | is bounded for zG C , and so the formulae h 3 Q = id IC , h 3 t(z) = th 3(z/t) , 0 < t < 1 , give an isotopy id |C ~ h 3 . Find a homeomorphism

' . It happens to be obvious that t Intuitively a quasi-space is a sort of ‘space’ of which we want to know only the sets of maps to it of certain specified pleasant spaces. The definition is always set up so that projective and injective limits of quasi-spaces exist because they exist in the category of sets. And one wants to allow enough pleasant spaces and maps between them in order to do some homotopy theory for quasi-spaces. Classically a quasi-space Y is defined as a contravariant function from the category C of Hausdorff compacta and continuous maps to the category of sets, which takes union squares (pushout squares) to fiber product squares. Thus if we write Y : X +* [X,Y] for X compact, and X = Xj U X 2 is a union of compacta, then the square [X,Y] [XpY]

I

4

[X2 ,Y] -* [x 1 n x 2 ,Y] of nclusion-induced ‘restriction’ maps is a fiber product square of sets; in other words, for each pair Xj E [Xj,Y] , i = 1,2 mapping to the same element in [Xj Pi X2 ,Y] , there is a unique element x E [X,Y] mapping to both Xj and x 2 . We shall write x | X j = X j , x 1 X 2 = x 2 . This definition suffices for [ Si j q, §4] . However , for § 1.4.1 of this essay we replace C by the category of piece wiselinear maps of compact polyhedra, and in § 1.6(A) we need still another definition. If we replace C by the category of ordered simplicial complexes and order preserving simplicial maps , then the corresponding quasi-spaces are nothing more nor less than css sets (as used in this essay). X This amounts to saying that if Ua , a E J , is a collection of open sets then for any allowable X (see previous note) and any selection o f elements xa E [X , ^(Uq,)] , a E J such that for each pair a , j3 E J one has xa l ( Ua n U 0) = x^KUq, n Up) in [X , (Ua n U^)] such that x |U a = xQ! for each a G J .

260

L. Siebenmann

Essay V

: 3>(A) -*■ (i) has the [weak] t covering homotopy property for maps of a compact o b je c t, we say that the pair (B,A) is [weakly] -flexible or that $ is [weakly] flexible for (B ,A ). Again , if (i) has the covering micro-homotopy property in the sense that some initial interval of any homotopy of a compact object can be lifted , we say (B, A) is $-microflexible (microgibki in Russian) . Recall that in the examples we have in mind , the only immediately provable property was -microflexibility for compact pairs . A cardinal lemma of the immersion theoretic method asserts that if Mm is an open manifold and $ enjoys some extra functoriality^ , (as it often does) then -microflexibility implies ^-flexibility for any simplicial pair of dimension < m in a co-ordinate c h a r t. See [Gr , § 3.3.1 ] [Haj ] ; we will not repeat the proof . The same extra functoriality lets one quickly show , using the elementary engulfing lemma , that in case A C-». B is a finite simplicial expansion the restriction $(B) -* (A) is both a weak fibration and a weak homotopy equivalence . (Compare § 1 .4 , property (2)(a) .) With this much motivation we prove two theorems . THEOREM A .l . Let • U -»■ (A) -*■ . B is an expansion . P ro o f: Apply ( ii) . (7 ) (B, A ) is finite simplicial. Proof : Apply (a) , (j3), and the collapsing lemma . (5) (B, A ) is compact in a co-ordinate chart. Proof : Apply (7 ) and the intersection lemma . (e) (B, A ) is com pact. Proof : Apply (5) and the induction lemma . (77) ( B , A ) is such that B-A has compact closure in M . P ro o f: Apply (e) and the opening remarks about a square of inclusions . (f) The general case. Proof : Apply ( 7?) the filtration lemma and an infinite induction to construct the liftings desired , (cf. last step for A.2) . This completes the proof of A .l . ■ Finally we go through the proof of A.2 in six cases . Here A is understood to be a closed subset of M verifying Hm(A; Z2) = 0 . (a) A is a simplex . Proof : Apply (1) and (2 ). (j3) A is a finite simplicial complex o f dimension < m . Proof : Use (a) , (3) and the preliminary remarks in an induction on the number of simplices . (7 ) A is a finite simplicial complex . Proof : Apply the collapsing lemma and (2) to reduce to case (|3) . (5) A is a compactum in a co-ordinate ch art. P ro o f: Apply the intersection lemma and (7 ) . (e) A is a compactum . Proof : Apply (5) and the induction lemma together with the preliminary remarks . (77) The general case . Proof : The filtration lemma applied to the pair (A, I X X that is a product with I near IX 3 X while F(x) = (f(x),0 ) for x E M , and F(x) = ( f '( x ) ,l ) for x G M ' after the identification 3 W - F _ 1 (1X3X) = M U M ' is made. When m > 6 , and 7i j X is free abelian, every h-cobordism is a product cobordism (cf. [ I,§ 1] ) , so that [ f] = [f' ] then simply means there exists a CAT isomorphism G : M M' with f 'G =« f rel 3 . The first tool in the surgical analysis of S (X ) is a map r to a rather similarly defined set N ( X ) of normal invariants (with our conventions ‘tangential invariants’ would be more appropriate term inology ! ) . An elem ent of N (X) is represented by a pair (f, X is a degree ±1 map of a com pact CAT manifold that is a CAT isomorphism near 3X , and : r(M ) £ is a CAT stable bundle map over f from the tangent (micro-)bundle of M to a CAT bundle £ over X . A nother such pair ( f ', [f,] defines r

: S( X)

-> N ( X )

.

The map r lends itself to com putation because there is a natural bijection first perceived by D. Sullivan (thesis Princeton 1965) A/(X) -* [X rel 3 ,G/CA T] to the 7Tq o f the space of those maps of X to G/CAT carrying 3X to the base point. It arises from a rule to derive from [f,] E N( X) a hom otopy trivialization a : r?k R k of the bundle r?k = £©p(X) so that a is a CAT bundle trivialization near 3X (here v ( X) is the norm al bundle to X in euclidean space and £ is the target of y ) . Since G/CAT classifies hom otopy trivialized CAT bundles, a determines a class [a] E [X rel 3 , G/CAT] . Now the recipe for a . A pproxim ate f : M -> X rel 3 by a CAT imbedding g : M -> E (£ ©r>(X) ) • There is a trivialization of the norm al bundle e to g(M) in this total space such th at the inclusion E (e ) C*. E (£© r'(X )) of total spaces induces, on their stable tangent bundles, the original stable CAT bundle map : r(M ) -* £ up to CAT bundle hom otopy . Stabilizing £© p(X ) once, we can assume (see [III, § 4 ]) that £ © ^(X ) = 77k contains a k-disc bundle 77 j , and that (after a standard move), g(M) lies with trivialized normal bundle in the boundary sphere bundle 5 77 j . B y

266

L. Siebenmann

Essay V.

the (pre-Thom) Pontrjagin construction, one deduces a map E( Sp] )-* S k_1 of degree ±1 on each fiber, and then, by coning, a fiber homotopy equivalence a : 77 -> R k to the trivial bundle over a point. Keeping the construction standard near 3 we find that a is a C A T trivialization near 3X . It is not difficult to show that this rule [f, R k suitably transverse to 0 E R k . Compare [Wa} ,§10 ] .

We can now enunciate PROPOSITION B .l. I f k < 4 , the normal invariant mapping t : SQ(Ik XTn ) [Ik XTn rel 9 , G/CAT] is zero. But the proof requires more basic theory. G/CAT carries a Hopf-space structure representing Whitney sum, and making the functor A ^ [A ,G /C A T ] a contravariant functor to abelian groups, cf. [Adj ] . This H-space structure on G /C A T , with the connectivity of G / C A T ,implies that it does not matter whether the set of homotopy classes [A , G/CAT] is defined using base point preserving maps or arbitrary maps A G/CAT . Incase (Y , 3 Y ) is a C A T submanifold of (X, 3X) with trivial normal bundle e , there is a simple rule to express the restriction map [ X rel 3 , G/CAT] -*

^ [Y rel 3 , G/CAT] in terms of normal maps and transversality. Given [f,^] E E /V(X rel 3) , deform f : M X rel 3 to be transverse to Y (for e ) , deforming if correspondingly. Set N = f_1(Y) , g = f IN , 77 = £ lY . Then r ( M ) lN = r(N ) stably, and thus y gives a C A T stable bundle map \p :t(N )-*7 ? over g . Now [g, \p ] E /V(Y) is the advertised restriction of [f,^ ]E / V (X ) . Wall has constructed by surgery a (usually non-additive) mapping of pointed sets 6 : A/(Xm ) — ► L m(7T1X) to an algebraically defined abelian group L m(7Tj X) that depends on m only modulo 4 . The map 6 depends up to sign on the choice of homology orientation class in H m(X, 3X) . (In the so-called non-orientable case where Wj (X) =£ 0 one must use Wj (X)-twisted integer coefficients for H m(X,3X) ; also L m(7T| X) depends then on wj (X) : Tiq (X) Z 2 as well as on 7rj(X) .) In this situation the main theorem of Wall’s surgery states that the kernel of 6 in A/(X) is the image r(S (X )) , provided m = dim X > 5 . We shall make extensive use of Wall’s geometric periodicity theorem [Waj , § 9 . 9 ] asserting that producting with P8 = (CP2)2 , or with any (CP2)k , does

not alter surgery obstructions, i.e. the square A/(X)

px

J « ( P 8XX)

-fl_

Lm(7r j X)

I Lm. 8( i , X )

267

Appendix B. The necessary calculations

is commutative. (See [Morgj][Ran] for more general P .) The service rendered by periodicity in this appendix is the technical one of lifting our calculations into high dimensions out of the way of notorious difficulties met by surgery and handlebody theory in dimensions 3 , 4 , and 5 , cf. [Si8] . As starting material, we shall need information about 6 for it|(X) = 0 . The group L m(0) is 0 , Z 2 , 0 , Z for m = l , 2 , 3 , 4 respectively, see [Br2 ][Ser3][M iH ]. Applied to [f,cp] G A/(Xm) , f : M m X , m=0 mod 4 , the map 6 yields 1/8 of the signature of the intersection form of M restricted to the kernel of f* : H *(M ) H *(X ) ; for m = 2 mod 4 , the map 6 yields the A rf invariant (from ^ ) of this form (for Z 2 coefficients), cf. [RSu] . We shall gradually need to know that A/(Sk ) = 7rk (G/CAT) (C A T = D IF F or PL) is 0 for k=0 , and is isomorphic to L k (0) for k = 1,2,3,4 , while d is bijective for k = 2 and is multiplication by 2 for k = 4 . For C A T = D IF F these results are quickly deduced by using the fibration G /D IFF B0 BG and the basic information 7rk O = Z 2 , Z 2 ,0 , Z for k = 0 , l ,2 , 3 and 7rk (G) = Z 2 , Z 2 , Z 2 , Z 24 ,0 for k = 0, 1,2,3,4 , cf. [Hus] [Ser2 ] [M iK] . The reader should pause to do this most basic calculation weaving in the following facts : (a) w2 comes from BG , (b) the 2-torus, with standard trivialization of its tangent bundle, represents an element x o f' 7T2(G/DIFF) with A rf invariant 0(x) =£ 0 , (c) is onto [Toda] , equivalently any spin 3-manifold is spin nul-cobordant, cf. [Lick] , (d) the least signature of a closed D IF F spin 4-manifold is 16 (R o h lin ’s th eorem ) [MiK] , cf [IV, Appendix B] . The same result for C A T = PL is deduced from the D IF F calculation and a map {p : /VDIFF(Sk ) /VPL(Sk ) , k < 4 , constructed by Whitehead’s triangulation theorems. This \p turns out to be bijective for k < 4 by classical smoothing theory; compare [IV, Appendix B] . Fortunately Cerf’s difficult result T4 = 0 [Ce 5 ] is not needed, the injectivity for k=4 being assured by signatures. Also triangulation and smoothing of bundles is not strictly necessary; it is enough to triangulate and smooth total spaces and use some normal bundle theorems in [IV, Appendix A] . For C A T = P L , the map 6 : 7rk (G/CAT) L k (0) is bijective for all k > 5 . One uses the PL Poincare theorem (beware k = 5 [IV, Appendix B] ) , plus (for surjectivity) Milnor’s plumbing [B ^ ] and (for injectivity) the main theorem of surgery. For C A T = TOP , it turns out (after our calculation of ^ (T O P / C A T ) ) that 6 ; 7rk (G/TOP) -► L k (0) is bijective for all k , cf. [ S q o > § 5, § 13] , and C.l below.

The calculation of [IkXTn rel 9 , G/CAT] is quite elementary. Let H be any Hopf-space, such as G/CAT , for which the rule Y [ Y,H] is a functor to abelian groups. Applying this functor to the Puppe cofibration sequence X/9 -> (XXT1) ^ - * (XXl)/9 -> 2(X /9) -> 2(XXT* / 9) -> ... , we get an exact sequence of abelian groups [X re ld ,H ] +2- [XXT1 rel 3 , H] J - [XXI rel 3 , H]

...

.

L. Siebenmann

268

Essay V

Note that a is a retraction since (XXT1)/9 retracts canonically onto X/9 . For a similar reason 7 is zero and so (S is injective, and we have a canonical decomposition (t)

[XXT1 rel 9 , H]

- [X rel 9 , H] ® [XXI rel 9 , H]

It is easily checked that the ‘transfer’ map in [XXT1 rel 9 , H] induced by multiplication by a positive integer X in T 1 respects the summands on the right, fixing the first while giving multiplication by X in the second. We use ( t) n times in succession to establish (B.2)

[Ik XTn rel 9 , H ]

- © f1?) [Ik+i rel 9 , H] = f (") ^k+i(H) = f ( i) [ S k+i , H]

After s applications there are 2s summands each of which will split into two at the next stage. The first application for example gives [IkXTn rel 9 , H]

= [IkXTn_1 rel 9 , H] © [Ik+1XTn_1 rel 9 , H ] .

Inspection shows that each one of these 2n = ? ( ? ) summands i [Sk+1 , H] corresponds naturally to a standard subtorus T1of Tn , and that the composite map (IkXTn)/9 (Ik XT^)/9 Sk+1 induces the injection of this [Sk+1 , H] . Here a comes from pro­ jection Tn -> T* and 0 from collapsing the (i - l)-skeleton of V . Passage to the standard X-fold covering along any one factor T 1 of Tn corresponds to the map on the right in (B.2) that is multiplication by X in any summand [Sk+1,H] corresponding to a subtorus T1 containing this T 1 and to multiplication by 1 in the remaining summands. Hence the subgroup, called [IkXTn rel 9 , H] 0 invariant in (B.2) under passage to standard finite coverings is just the summand [Sk , H] . After all these preliminaries we can give Proof of Proposition B.l We have observed, for H = G/CAT , that the restriction map [Ik XTn rel 9 , H ] 0 -* [Ik rel 9 , H] = [Sk ,H] is an isomorphism, and also that 6 : [Sk , H] -* 1^(0) is injective, k < 4 . It follows,

Appendix B. Classification of hom otopy tori

269

using geometric periodicity and the discussion of normal maps, that this injection d can now be expressed as a composition /V0(IkXTn)

/V(PXlkXTn)

N (PXlk )

Lk (0)

where P = (CP2)2 , and p is restriction (expressed via transversality). Consider now [f, PXlk is a hom otopy equivalence of fibers. This implies 0[g,\M = = 0 E Lg+^O) = Lk(0) , and we conclude, in view of our prelimi­ nary remarks, that 0[f, Z.(Ik XTn ) . k+ri > 5 ,Js the sum © (?)K 1+k+i , o f which the subgroup invariant under transfer is K 1+k .

272

L. Siebenmann

Essay V

In the Sullivan-Wall sequence (B.3) for X = Ik XTn , this is equally the image of d or the kernel of r . Also the transfer in the sets N , L and 5 clearly correspond under the maps and actions in (B.3) . Thus, combining B.l and B.5 , we get THEOREM B.6 . For CAT = DIFF or PL , we have S0(Ik XTn rel 9) = K 1+k = = 7rk(K(Z2 ,3)) provided k 5 , that is used to prove the Stable Homeomorphism Theorem [Ki j ] . The basics of non-simply connected surgery are essen­ tial, including the geometric periodicity theorem [W aj,§5.8, §6.5] . Thus most of the first 112 pages of [Waj] must in the end be under­ stood.*^ The one algebraic result about L-groups required is L](0) = 0 , for which a pleasant proof is given in [B ^ ] . To be sure, handlebody theory is needed (as in Essay I ), plus Farrell’s fibering theorem [Fa] based on [Si] ] . The PL result here requires no classical smoothing theory. Either the PL or the DIFF result suffices to prove the Stable Homeomorphism Theorem as in [Ki] ] . (ii) Concerning the prerequisites for proving SQ(Ik XTn ) = = 7rk(K(Z2 ,3)) , k < 4 , we can say much the same for CAT = DIFF except that it is the calculation of Lk+1(0) that is needed. When k = 3 Rohlin’s Theorem [MiK] is vital. For CAT = PL , we must add classical smoothing theory (at present 200 pages of arduous reading: we suggest [MU3 ] ,then [Hig] , then [HiM2>I] ) plus T] = 0 for i < 3 (we suggest [V, §3.4, §5.8] or [C e^ pp. 127-131 ] or again a recent improvement of the latter by A. Douady [Lau, Appendix] ). f U nfortunately for beginners, Wall’s exposition is couched in the language of (n+1 )-ads. In learning this language note that [0 ,°°)n = Rg is naturally a (n+1 )-ad (the i-th of n subsets being defined by vanishing of the i-th co-ordinate); it can play a role of 'm odel’ analogous to that of the standard geometric n-simplex in semisimplicial work.

273

Appendix C. SO M E T O PO LO G ICA L S U R G E R Y

In contrast to Appendix B which carries out calculations vital to these essays, the present appendix is optional material that exploits what has gone before to enlarge the reader’s understanding of surgery as it relates to topological manifolds. In particular it relies all on the surgical apparatus set out in Appendix B, and on all the tools for the TOP version of surgery that have been forged in Essay III. Staying initially in the elementary line of argument of Appendix B , we complete the classification of DIFF , PL , or TOP homotopy tori. A delightful isomorphism 0 : 7rk (G/TOP) = Lk (0) , k > 0 , lets one quickly calculate that STOP(Ik XTn) = 0 , k+n > 5 ; then structure theory (essay IV or V) reveals that SCAT(Ik XTn) = [ 2 kTn , TOP/CAT] , k+n > 5 . (Recall that these homotopy-CAT structure sets are rel boundary. ) This done, we adopt F. Quinn’s semi-simplicial formulation [Qnj 1 of Wall’s surgery, and proceed to explain some far-reaching consequences of the isomorphism ^ (G /T O P ) = L^O ) . To begin, we derive the periodicity of Sullivan and Casson G/TOP ^ £24(G/TOP) from the periodicity Lk (7r) = Lk+4(7r) of algebraic surgery obstruction groups. Then, having paid due attention to fundamental group, we can observe a corresponding periodicity STOP(X) = STOP(I4 XX) in homotopy-TOP structure sets (rel boundary), where X is any compact TOP manifold of dimension > 5 . This makes the whole TOP Sullivan-Wall exact sequence satisfy periodicity. We close with a few applications of these ideas. (i) STOP(Ik XTn ) = 0 for k > 5 , by induction on n via a simple splitting argument; hence ST0 P(Ik XTn ) = 0 , k+n > 5 . And this new proof applies immediately to significant generalizations by Wall and Quinn that let one calculate many groups L^Cn) . (ii) A PL analog of all the foregoing yields a rapid, if ultra sophisti­ cated, PL proof ab initio that SPL(Ik XTn ) = [ 2 kTn ,K (Z2 ,3)] , k+n > 5 . (It must have much in common with A. Casson’s unpub­ lished semi-simplicial one dating from 1967-8.)

274

L. Siebenmann

Essay V

(iii) We mention a long standing general program involving fibrations, to reduce DIFF surgery problems to the study of the map G/DIFF -> G/TOP and to the parallel TOP surgery problems, which enjoy periodicity. We begin with THEOREM C.l . The surgery map 0k : 7rk (G/TOP) -►Lk (0) is an isomorphism for all k > 1 . Proof of C.l . For

k> 5

, argue as for PL in Appendix B .

For k < 4 the fibration TOP/CAT G/CAT G/TOP together with the equation 7rk (TOP/CAT) = 7rk (K(Z2 ,3)) ( k < 4 , and CAT = DIFF or PL , see [IV, § 10.11 ] [V, §5] ) show that 0k is an isomorphism for TOP since (Appendix B) it is one for CAT . Coming to k = 4 , there is even some trouble defining 04 , since there is (still !) no TOP microbundle transversality theorem when the expected preimage would have dimension 4 . (See [III, § 1 ] ; onecan in fact use M. Scharlemann’s transversality theorem [Sch] .) Avoiding this problem, we define 04 to make the following square commute, where P = (CP2)2 • ^ tt4(G/TOP)

= [S4 ,G/TOP] —*-> L4(0) PX I [PXS4 ,G/TOP]

II ^ L 12(0) .

The exact sequence 0 -*■7r4(G/CAT) 7t4(G/TOP) -► -*■ir3 (TOP/CAT) -*■ 0 leaves just two possibilities: either 04 is an isomorphism, or 7r4(G/TOP) is Z © Z2 and has the same image 2Z in L4(0) = Z as 7r4(G/CAT) . We thus complete the proof by show­ ing 04 is onto. The discussion in Appendix B (TOP version) identifies 04 to the geometric forgetting map Q' :

/v£OP(IXX)

----- ►Z-5OP(X)

a

L4(0)

,

where X = I3XTn , with 3+n = 6 (say), and subscripted‘oh’ con­ tinues to indicate a (sub)set invariant under transfer to standard finite

275

Appendix C. Some topological surgery

coverings. The surjectivity required now follows from the commuta­ tive diagram of forgetting maps with exact rows (see B.3 ). /VTOP(IXX)0- ^ - * t

Z.5°P(X)

S ™ P(X)

~ t

I

,

“ I

ze ro

.

/VgAT(IXX) ------- > Z.qAT(X)

\ (by [5,Pr oo f o f 5.3], I c f [ I V , §10.11] )

-SqAT(X) = Z2 (by B.6)

To verify the exactness observe that the square at left is a summand of the similar square without subscripted o h ’s. ■ For an alternative proof see [SijQ, §13.4 ] . The classification of Appendix B is enlarged and clarified by THEOREM C.2 . ( with notations from Appendix B) (i)

STOP(Ik XTn ) = 0 J o r k+n > 5 .

(ii) SCAT(Ik XTn) = [Ik XTn rel d , TOP/CAT] = © (?) 7rk+i(TOP/CAT)

,

fo r CAT = DIFF or PL , and k+n S* 5 . Proof of C.2 part (i) : TOP hom otopy tori are trivial. Imitating Appendix B , we apply the Sullivan-Wall exact sequence (TOP version) : (B.3) N (IXX) ^ L(X) S ( X ) - ^ /V ( X ) -^ Z-mOrjX) , with X = Ik XTn , m = k+n > 5 . The analysis above B.5 shows that the homomorphism d' decomposes as a sum of 2n surgery maps dj : 7Tj(G/TOP) Lj(0) . By C.l each dj is an isomorphism, so d r is an isomorphism. At this point S(X) = 0 amounts to showing that d has zero kernel. For k > 1 , and k+n > 6 , we have a proof in hand since d factorises (see below B.3 ) as /V(Ik XTn ) - £ - * U Ik_1XTn)

L m(Zn) ,

while (as above) d f is an isomorphism. If k > 1 , and k+n = 5 , the same argument works via the detour of producting with T 1 . ■ There remains the case k = 0 , which is troublesome because

276

L. Siebenmann

Essay V

: /V(Tn) -►Ln(Zn) is not additive. (For example the composed map [T8 ,G/TOP] = A/(T8) -► Lg(Z8 ) -»• L8(0) given by signature is not additive as the Hirzebruch genus for signature shows; the non zero cup products in T8 cause the trouble). 6

J. Levine found the following elementary proof that : S(Tn) -*■ /V(Tn) has image zero, see [HS2 ] - It entirely obviates discussion of 6 . t

For each of the (j1) subtori T1 of Tn we have a composed map /V(Tn) /V(PXTn) p-+ N{PXT*) Z-i(O) where p is restric­ tion. Summing these we get a map a : /V(Tn) -*■ ©i(”)Lj(0) , which could in fact be identified to 6 . Avoiding even this identification we propose to show that o r = 0 and then that a - 1(0) = 0 . To show that ot = 0 , consider any standard subtorus T1 and the corresponding component S(Tn) -»• Lj(0) of o r , which is the composition S(Tn) N (Tn) N (PXTn) p-+ N ( ? X V ) ^ d —►LjfZ1) Lj(0) . Farrell’s fibering theorem shows that the image of any [f] G S(Tn) is represented in /V(PXT*) by a normal map that is a homotopy equivalence, and therefore goes to zero in LjCZ1) . To show that o- 1(0) = 0 , we induct upwards on n (J. Levine’s trick). If a(x) = 0 , inductive hypothesis assures that the restriction of x for each standard Tn_1 C Tn is zero. Then the analysis of [Tn ,G/TOP] (see B.2 ) shows that x lies in the injected subgroup 7rn(G/TOP) on which a is clearly the injective surgery map 7rn(G/TOP) -»• Ln(0) . This completes the proof that r is zero, and therewith the proof of Theorem C.2 . ■ Proof of C.2 part (ii) from part (i) . Since the set [Ik XTn rel 3 , TOP/CAT] has been shown in §5 (or in Essay [IV] ) to be in bijective correspondence with the set S(Ik XTn) of concordance classes rel boundary of CAT manifold structures on Ik XTn , we have only to prove: LEMMA C .3. Let X be a compact C AT manifold such that 5TOP(X) = 0 = ST0 P(IXX) . Then the forgetting map §(X )-* -*■ S CAT(X) is bijective. Proof of lemma . Surjectivity clearly follows from ST0 P(X) = 0 ; injectivity from STOP(IXX) = 0 . ■

Appendix C.

277

P E R IO D IC IT Y IN TO PO LO G ICA L S U R G E R Y

This is one of the distinguishing features of the topological cate­ gory as one can see by reading [S ij0 , § 14, § 15] [MorgL] [CapS] . THEOREM C.4 . There exists a ‘p eriodicity ’ map I I : G/TOP -»• f i 4 (G/TOP) giving a hom otopy equivalence to the identity component o f i?,4(G/TOP) , and a new (hom otopy everything \) t H-space structure on G/TOP , such that fo r any compact TOP manifold Xm ( with twisted orientation class) the resulting square (see below B .l fo r 6 ) [X rel 3 , G/TOP] (□)

n

L ^ ttjX)

35

[I4 XX rel 8 , G/TOP]

Lm +^X )

is a commutative square o f group homomorphisms. Remarks. a) The proof as we explain it below is a typical application of F. Quinn’s semi-simplicial formulation of Wall’s surgery [Qn j ] . b) The periodicity and H-space structure we shall describe in fact coincide with those defined rather differently by D. Sullivan using his Characteristic Variety Theorem [Sullj ] . Further, J. Morgan is able to verify the commutativity and additivity of the square (□) using his unpublished generalization of the Characteristic Variety Theorem for non simply connected manifolds. (I am grateful to him for explaining to me this line of proof.) c) It should be clear that this result can be of great assistance in calcuf All H-space structures we shall meet are homotopy associative and h om o­ topy commutative. What is more, I believe that one could interpret H-space stri cture henceforth to mean hom otopy everything H-space structure, which is a much more complicated and refined sort of structure, [BoV], cf. [Seg] [May 2 ] ; but perhaps little would be gained thereby.

L. Siebenmann

278

Essay V

la ting with the Sullivan-Wall surgery exact sequence. For example we could have used it to abbreviate the classification of TOP homotopy tori above (obviating J. Levine’s argument). d) A distinct periodicity T: G / T O P £ 2 4(G/TOP) , (called the Casson-Sullivan periodicity in [SijQ, §14] ) , has become well known through C. Rourkes’s exposition [ R o q ] (which presents a correspond­ ing PL «ear-periodicity). Beware that T fails in general to make the square (□) of C.4 commute. For an example close to our heart, take X = T8 and observe using Hirzebruch L-genus that even the square [T8 ,G/TOP]

L8(0)

r [I4 XT8 ,G/TOP]

L 12(0)

is non-commutative. The trouble arises because the construction of T involves Whitney sum; incidentally we have already observed that the upper 6 is not additive when Whitney sum is used in [T8 ,G/TOP] . In short, n has superior properties. In what follows, m is a fixed integer > 0 , and 7r is a fixed finitely presented group equipped with a fixed 'o rientation’ hom om orphism w : 7r -> Z 2 . The sym ­ bols X, M, W, etc. stand for (varying) com pact TOP manifolds each equipped with an orientation class in Wj-twisted integral homology. We shall use the s-version of surgery involving simple rather than ordinary hom otopy equivalence; so the L-groups appearing are the Ls-groups [W a j, p .249] Thus we rely on the result of [III, §5] that every (X, 9X) is a simple Poincare pair. There is no pressing reason for switching from the slightly simpler h-version we have used so far, but the result of C.4 is clearly a little stronger in the s-version. We shall need to use num erous semi-simplicial sets that lack degeneracy opera­ tions; these are called A-sets, see [ R S j, I] . The proof of C.4 will follow Q uinn’s article [Q n j] rather closely. We break it into seven steps. 1) Lm ( 7r) is defined to be the A-set of which a typical zero-simplex is a referenced normal map f, ? sending (Ak X3+X )' to the base point; here prime indicates a standard ordered triangulation which the reader can supply. The right hand square com m utes strictly and strictly respects H-space m ultiplication. Thus, applying 7r0 to the outer rectangle, we get a com m u­ tative square. A straight forward pursuit of definitions reveals that this is our square (□) .

Appendix C. Periodicity in topological surgery

281

In case X+ is not triangulable, let the space underlying X+ be a stable normal disc bundle to X in a euclidean space Rn , and let 3+X be its restriction over the boundary 3X . The pair (X+,3+X) is readily given a triangulation using the Pro­ duct Structure Theorem of Essay I , (cf. [Ill, §4, §5] ) . This restores meaning to the left hand square. But unfortunately the assembly procedure no longer carries us into Lm+giTTjX) and L ^ ^ C ^ X ) . To remedy this simply, one can follow up the assembly procedure by an induc­ tive TOP m icrobundle transversality procedure, as now described. (Compare the more sure-footed but tiresome arguments to resolve such technical difficulties in §2 .) Discussing the upper o first, observe that assembly canonically creates for each k-simplex of A(X+,Lg(0)) an assembled normal map f , K(7Tj (X), 1) so as to land up in L g ^ C ^ jX ) as desired. Doing the same (in relative fashion) for the lower map o , we make the right hand square com m utative; a com m utes with H-space m ultiplication at least up to hom otopy. Then applying 7Tq to the outside hom otopy com m utative rectangle we again get the square (□) ; this is amplified by step 8 below. ■ Our first target, the p r o o f o f the periodicity theorem C.4 has been reached. We now restate it in a sharper semi-simplicial form and go on to establish the geo­ m etric periodicity S(X) = S (I 4 XX) , d im X > 5 . ) The outer rectangle of (7) is naturally hom otopy equivalent to the hom otopy com m utative square

8

(G/TOP)(x / a> — (8)

n

Lm+8(7rl x > XCP2

f2 4 (G/TOP)( x /a ) The bottom Q for example is prescribed on a typical k-simplex as follows; regard it as a map Ak XI4 X X -* G/TOP ; use this map to pull back the universal bundle; cross with P 8 ; apply transversality to get a normal map to Ak XI4 XXm XP8 rel Ak X3(I 4 XXXP) ; and lastly regard this as a simplex of L ^ ^ C ^ X ) using the obvious reference map to Ak XK(7r jX ,l) . 9)

When dim X - m > 5 , the square ( 8 ) is naturally equivalent to a square N (X )

(9) N(I 4 XX)

Lm+4(7rl X )

Essay V

L. Siebenmann

282

N(X) is the A-set o f normal maps to X (rel 3) : aty p ica l k-simplex is a normal map to Ak XX rel Ak X3X, (of course given in blocks, one over each Ak XX , so that face maps are defined by restriction). The maps Q just add the standard refer­ ence map to Ak XK(7r 1 X ,l) . The equivalence with ( 8 ) is given by a hom otopy com m utative diagram :

n

n °

(9) w

u

XP

w

where the outer rectangle is ( 8 ) and a is defined using TOP transversality.

■

10) S ( X ) N ( X ) L m( 7r 1 X) is a hom otopy fibration for dim X = m > 5 , called the structure fibration. S(X) is here the A s e t o f hom otopy-TO P structures on X rel 3 defined in analogy with N(X) so that 7r 0 S(X) = S(X) and more generally 7rkS(X)) = «S(Ik XX) . The map r is defined by adding structure to con­ vert simple equivalences into normal maps (cf. Appendix B ) . The fibration property is proved by considering the Serre fiber F(X) of 6 : N (X )-> L|T1 ( 7T1 X) where L' is the sub A-set of L consisting of all simplices for which the reference map is a 1-equivalence (on all faces). Using [W a], §9] , one shows that = Lm by inclusion for m > 5 . There is a natural map S (X ) -> F(X) which one readily shows to be a hom otopy equivalence, see [Wa ], § 9 ], by using the obstruction-free tt~tt surgery theorem of [Waj ,§ 4 ] . It is not difficult to show that the long exact hom otopy sequence of the struc­ ture fibration is the Sullivan-Wall structure sequence. What we have said here in (10) applies to D IFF and PL m anifolds as well, and even to Poincare spaces, cf. [J] [LLM] [Q n 2 ] • 11) There is a periodicity n :S(X) S(I 4 XX) =* £24 S(X) of hom otopy-TO P structure A-sets for dim X > 5 . It is defined to be the induced hom otopy equi­ valence of the hom otopy fibers of the maps 6 in the square (9) . Thus we have a hom otopy com m utative diagram S(XU

(ID

n S(I 4 XX)

T

N(X)

n

-

Lm(*lX) (9)

-

XCP9

0

► N(I 4 X X )- 2 > L m + 4 (ir 1 X)

To specify II : S(X )-* S(I 4 XX) in a preferred hom otopy class, we need to choose q hom otopy making square (9) com m ute; this is best built in Lm+ 1 2 ( 7r 1 X) using TOP transversality, cf. (7) . To then verify that the hom otopy class of fl : S(X) S(I 4 XX) is thereby welldefined, independent of the choices involved, we can rely on the extendibility of its construction as we have specified it. Thus, having specified constructions of this FI for XXO and again for XXI (call the results flo and fli ) , we can extend to a

Appendix C. Periodicity in topological surgery

283

construction of an equivalence II : S(X) -*■ S(14 XX) where X is the triad XX(I;0,1) , cf. [Q n j] . We conclude that n 0 — n j using the four equivalences by restriction S(X) ~ S(XXi) , and S(I 4 XX) S(I 4 XXXi) ,i = 0 , l .

Here are some salient conclusions that require no semi-simplicial language. PERIODICITY THEOREM FOR STRUCTURES C.5 . For any compact TOP manifold Xm , m > 5 , the Sullivan-Wall long exact structure sequence is a long exact sequence o f abelian groups, and it is canonically isomorphic to the one fo r I4 XXm . In particular S(X) = S(I4 XX) . As Wall remarks [W aj, § 10] , there is no sign that for DIFF or PL manifolds (in place of TOP ) the structure sets should all naturally be groups. The periodicity S(X) = S(I4 XX) should also exist for noncom­ pact TOP manifolds Xm , m > 5 , cf. [Mau2 ] .

The periodicity C.5 is surely an attractive result; but my treat­ ment of it evokes a rueful song: Maybe I ’m d o in ’ it wrong. It ju st d o n ’t move me the way th a t it should ... Som etim es I throw off a good one. Least I think it is. No, I know it is ... Maybe I ’m d o in ’ it wrong ... R andy Newman

284

Essay V

L. Siebenmann

A PPLIC A T IO N S OF P E R IO D IC IT Y

I)

Generalized Klein Bottles. This is a reedition of [Waj , § 15B]

with an improved proof.

A group G is poly-Z of rank n if it has a filtration 0 = Gq C G j C ... C Gn = G so that G^ is normal in G^+i with quo­ tient G^+i/Gk = Z (infinite cyclic), 0 < k < n . If t E G generates G/Gn-i , the inner automorphism 0(x) = t_1xt of Gn_j can be realized by a homotopy equivalence O' : K(Gn_j ,1) K(Gn.j ,1) and one observes that the mapping torus of d f is a K(G,1) . A generalized Klein bottle is a closed manifold X s K(G,1) with G poly-Z . For homological reasons d im X ^ ran k G . The Klein bottle is the simplest one that is not a torus. For any poly-Z group G , Farrell and Hsiang [FH 2] have shown that Wh(G) = 0 . Also, we can see by induction on rank that K(G,1) has finite homotopy type. This will let us use Farrell’s fibration theorem [Fa] (cf. [IV, Appendix B.3] ). HANDLEBODY LEMMA C.6 . Let Xm , be a compact CAT( =PL or TOP) manifold with boundary, hom otopy equivalent to K(G,1) where G is poly-Z o f rank n . Suppose m ~ n > 6 and 7^(9 X) = irfX) by inclusion, i = 0 , 1 . Then the set S(Xm) o f hom otopy CAT structures on X m rel d is zero. Proof of Lemma C.6 by induction on n . The map g i X - ^ T 1 corresponding to G G/Gn_i = 7r 1T 1 has homotopy fiber K(Gn.j ,1) , and so Farrell’s fibration theorem (see [III, §3] for its TOP version) tells us that g can be deformed first on the boundary then in the interior to become a CAT bundle projection with fiber X \ ^ K(Gn_i,l) over O E T 1 say, so that 7r j 9X| = 7r 1Xi by inclusion. Remark: The necessary fact that the homotopy fiber of g l 9 X above has finite type follows from use of Poincare duality, with Z[Gn_j ] coefficients, in the infinite cyclic covering. At the cost of assuming that

Appendix C.

Klein bottles and L-groups

285

m ^ rank G , one can see that both X and 9X split because X is a regular neighborhood of a split K(G,1) , e.g. of one that is a mapping torus as was described. This last argument adapts best to the generaliz­ ations mentioned below. Consider now any [f] E S (X m ) , f : (Mm ,3) (X,3) . Applying Farrell’s theorem rel 3 to the map gf , we can deform gf rel 3 to a CAT bundle projection with fiber M^1"1 and deform f rel 3 to respect fibers. Then f is split at an equivalence fj : Mj Xj , repre­ senting [ fj] E S (X i) zero by induction. We can thus further deform f fixing 3 so that in addition fj is an isomorphism. Cutting X and M open along Xj and Mj , we get from f a well defined homotopy equivalence f0 : (M™^) (X™,3) representing [f0 ] E S ( X 0 ) = 0 , which again is zero by induction. Then we can finally deform f fixing (3 M )U M | to an isomorphism. Thus [f] = 0 . The induction starts with S(Bm) = 0 , m > 5 . This is the one part of the argument that fails for DIFF manifolds. ■ UNIQUENESS THEOREM C.7 [W a^ § 15B] .

Let Xm , m > 5 , be a compact TOP manifold that is a K(G,1) with G poly-Z ( rank < m ) . Then ST0 P(X) = 0 . Proof of C.7 : S(X) = S(I8XX) by TOP periodicity C.5 . And 5(18XX) = 0 by C.6 . ■

Remark. Wall’s proof is rather different (more computational), although at the crucial point he envisages the use of periodicity in G/TOP (indeed C.4 exactly fits his needs). Is there a proof not strongly dependent on periodicity? Wall suggests there is one [W aj, p. 2291 . EXISTENCE THEOREM C.8 .

Given any poly-Z group G o f rank m , there exists a closed TOP manifold Xm ^ K ( G ,l) . Proof of C.8 (Wall’s argument): Consider the filtration Gq C Gj C ... C Gm = G for G and suppose we have a TOP mani­ fold Xn ^ K (G n ,l) , n > m . Then we have a hom otopy equivalence f : X n -*X n whose mapping torus T(f) is a K(Gn+1,l) . For n ^ 4

286

L. Siebenmann

Essay V

we can deform f to a homeomorphism and set T(f) = Xn+1 . This de­ formation is possible for n = 3 by [ St j ] [Neu] (as X3 will be irredu­ cible by construction). For n > 5 it is possible by C .l . Thus the only remaining problem is to construct X5 . As S(X4 XT1) = 0 , we can build M6 ^ K (G 5 ,1)XT1 as above, then fiber it over T 1 with fiber X5^K (G 5,1) . ■ STRUCTURE THEOREM C.9 .

I f ir is poly-Z o f rank m then d : [K(7r,l) ,£2k(G/TOP)] ^ Lm+k(7r’w l) is an isomorphism for k > 0 , where Wj is the orientation homomorphism for K(7r,l) . ^

Proof : Immediate from C .l , C.8 and the Sullivan-Wall structure sequence. ■

GENERALIZATIONS C.10 (by F. Quinn [Qnj

using [C ap j?2,31 )

(i) Let g be Waldhausen’s simplest class of ‘accessible’ finitely presented torsion-free groups, namely the one generated from the trivial group by successive construction of free product with amalgamation and and its one-sided analog ( = HNN extension) . These include poly-Z groups, free groups, classical knot groups. Theorem: In the s- or the h-surgery theory, for any TOP manifold Nm ^ K ( 7r,l) , m > 5 , with n E g , the group S(X) is 2-primary. [Hints: One can imitate the proof above for the poly-Z case using, in place of Farrell’s result, the strongest splitting theorem of S. Capped [Capj 2] , which lets one split up to h-cobordism provided two 2-primary obstructions vanish (the more mysterious one comes from Cappell’s UNil functor and may indeed be non-zero [Cap3 ] ) . For technical convenience in extending the handlebody lemma C.6 , one can initially suppose that m dim K(7T,1) and that X is a pro­ duct with [0,1] and suitably split t . This factor [0,1] of X per­ mits geometric doubling in S (X ) , which eventually kills Cappell’s 2-primary obstructions, and also any distinction between the h- and s-theories. In a number of important special cases, e.g. surface groups t See the remark in the handl ebody lemma C.6 . Since Wh(7r) is not known to be zero for w £ g , one must add the elementary observation (for the h- or the s-theory), that S(X) = S(X') if X' is X with an h-cobordism added along 3X .

Appendix C. P L hom otopy tori by Casson's approach

287

and classical fibered knot groups, a closer examinations reveals that S(X) = 0 since Cappell’s obstructions and Whitehead groups vanish, see [Cap! ] [Wald] . (ii) We conclude that 9 is an isomorphism mod 2-primary groups from [Xm rel 9 , G/TOP] to Lm (7TjX) , whenever X s K(tt,1) , it E g . And it is a strict isomorphism in the more favorable cases mentioned. (iii) According to Sullivan [Sullj] the group [Xm rel 9 , G/TOP ] is isomorphic mod 2-primary groups to KO°(Xm rel 9) , and isomorphic rationally to © (H4k(X,9;Q) ; ; k G Z } , all this for X connected and 9X =£ 0 . (iv) Incase w 1Xrn = 0 , and Xm is chosen in Rm , one has an (extraordinary) Poincare duality isomorphism, from each of these three (graduation 0 ) cohomology groups, to the corresponding m-th (extraordinary) homology group of X . Thus we have (writing 7T1X = 7T ) : 9 : Hm(X ;G/TOP) - Lm(7r) + 9 : KOm(X)'® Z [ 1/2] - LmOr) ® Z[ 1/2] 0 : ® Hm+4k(x ;Q) Lm(7r)®Q . By (ii) the last two are isomorphisms if X = K(7r,l) with 7t E g . For the non-orientable case (with wj : tt Z2 non zero), see [Qnj 2 ] • II) PL hom otopy tori by Casson’s approach. One can carry out the discussion of periodicity for PL manifolds in place of TOP , because 6 : 7rk (G/PL) Lk (0) is an isomorphism at least for k > 4 . Indeed periodicity was first exploited by Sullivan and Casson in the PL context. We can pretend it is 1967 . The one, notorious failure of 0 to be an isomorphism (it is Z X 2 >Z for k = 4 ) prevents the PL periodicity maps from being true homotopy equivalences; in fact II : G/PL £24(G/PL) is easily seen to have fiber K(Z2 ,3) . We propose to use the PL version of diagram (11) (preceding C.5 ), with X a compact PL manifold of dimension > 5 . Applied t

The spect rum for G / T OP is o f course the periodic where Q k = S2k(G / TOP) .

a 3 M 2 M ] , ...

^-spectrum

£24 ,

L. Siebenmann

288

Essay V

twice, it gives a homotopy commutative diagram with rows homotopy fibrations: S(X)

1

------ ►(G/PL)(x / 9> — -

n« S(I8 X X )

L m( * l X )

-

X(CP2)2

-r — ►f i 8(G /P L )W 9) - ^ L L m+8(7T,X)

Here n 8 = (£24FI) on . Note that the homotopy fiber of n 8 : G/PL -*■ -►£28(G/PL) is just K(Z2 ,3) . Also note that the left hand square is a (homotopy) fiber product square since an equivalence is induced from the fiber £2Lm of r to the fiber £2Lm+8 of r ' , namely the looping of X (CP2)2 in the mapping of Puppe fibration sequences. It follows that the fibers o f the two maps n 8 are hom otopy equivalent. The fiber of the II8 at left is S(X) for X = K(G,1) with G poly-Z : indeed S(I8+ kXX) = 0 , k > 0 , by the (PL) handlebody lemma C.6 , thus S(I8 XX) is contractible, and so the fiber is S(X) itself! The fiber of the n 8 at center is K(Z2 Therefore we have a natural homotopy equivalence S (X )~ K(Z2 ,3 )W 9) (for dimX > 5 , X ~ K (G ,1 ) , G poly-Z ) , whence S(Ik XX) as H3-*(X,9 ;Z2) , k > 0 . This elegant calculation must be a close approximation to A. Casson’s unpublished semi-simplicial classification of PL homotopy tori dating from 1967-68 . (It is not clear whether he took advantage of the handlebody lemma C.6 ). I have not given it greater emphasis because of the sophistication required and equally because it would seem to deny access to TOP manifolds to those equipped with no more than DIFF techniques. Ill) Related problems . Contemplate the following homotopy commutative diagram for X a compact CAT m-manifold, m > 5 .

Appendix C. Connections with structures and triangulations

ST/C(X) -CU

I SC(X) ST(X)

“

289

(TOP/CAT)(x / 9)

i

--------►(G/CAT) ST(X) and one requires only the TOP s-cobordism theorem to show that it is a homotopy equivalence. Since the lower left square is a fiber product square (argue as in II) we can obtain a ‘classifying’ hom otopy equivalence 7 as indicated. One can show that, on arc components 7r0 , it is precisely the classifi­ cation of [IV, §10] (for X compact). To deal also with the existence problem for CAT structures one can modify the middle column (use spaces of liftings !). This inserts the classification of [IV] into a larger context. Although the proof of it provided thereby is not really easier than that in [IV] , there are analogous situations where this approach involving surgery seems the most workable one. This is the case for the problem of classifying up to concordance arbitrary triangulations (perhaps not PL homogeneous) of a given TOP manifold. For this we refer to forthcoming articles by T. Matumoto and D. Galewski & R. Stern. For a given CAT manifold a primary surgical problem is to understand the vertical fibration on the left with base space ST(X) . This has been accomplished (as Quinn notes [Qn j ] ) when CAT = PL and X is a torus (or PL generalized Klein bottle) ; indeed S r/c (X) ^ Sc (X )^ K (Z 2 ,3)x . In general, note that the vertical homo­ topy fibration on the left is the pull-back by r of the middle fibration. But this r is the fiber of 8 . Hence, if one can fully analyze the middle vertical homotopy fibration and 8 (or 7 ), the job will be done. Much of Sullivan’s work on surgery [Sullj 2 ] has been directed to understanding the middle fibration beginning with an exhaustive study of G/TOP , (see also [Qn3 ] [MadM] ) .

Annex A. Reprinted with permission, from Annals of Mathematics, Vol. 89, No. 3, May 1969, pp. 575-582.

Stable homeomorphisms and the annulus conjecture* By Robion C. Kirby A homeomorphism h of R n to R n is stable if it can be w ritten as a finite composition of homeomorphisms, each of which is somewhere the identity, th a t is, h = h,h 2 • • • hr and h{ \ Ui = identity for each i where Uf is open in R’. Stable Homeomorphism Conjecture, shc* : All orientation preserving homeomorphisms of R n are stable. Stable homeomorphisms are particularly interesting because (see [3]) SHC* => AC„, and AC* for all k sS n shc„ where AC* is the A n n u lu s Conjecture, AC* : Let f , g : S n~l —* R n be disjoint, locally flat imbeddings w ith f ( S n~l) inside the bounded component of R n — g(Sn~l). Then the closed region A bounded by / ( S n-1) and g(Sn~l) is homeomorphic to S'"-1 x [0, 1]. Numerous attem pts on these conjectures have been made ; for example, it is known th a t an orientation preserving homeomorphism is stable if it is differentiable a t one point [10] [12], if it can be approximated by a pl home-, omorphism [6], or if it is (n — 2)-stable [4]. “ S ta b le ” versions of AC* are known ; A x [0,1) is homeomorphic to S n~l x I x [0, 1), A x R is S n_l x I x R, and A x S h is S n_1 x I x S k if k is odd (see [7] and [13]). A counter-example to AC, would provide a non-triangulable -re-manifold [3]. Here we reduce these conjectures to the following problem in pl theory. Let T n be the cartesian product of n circles. H auptverm utung fo r Tori, iit„ : Let T n and t n be homeomorphic pl ?i-manifolds. Then T " and r" are pl homeomorphic. I f n ^ 6, then h t„ = shc*. {Added December 1, 1968. I t can now be shown th a t SHC* is true for n =£ 4. If n < 3, this is a classical result. Theorem 1 also holds for n = 5, since Wall [19, p. 67] has shown th a t an end which is homeomorphic to S 4 x R is also pl homeomorphic to S 4 x R. In the proof of Theorem 1, a homeomorphism/ : T n —► r n is constructed. If / : f ’ --» f" is any covering of / : T* —>vn, then clearly f is stable if and only if Theorem 1.

* P artially supported bv NSF G rant GP

292

ROBION C. KIRBY

/ is stable. Using only the fact th a t / is a simple homotopy equivalence, W all's non-simply connected surgery techniques [15] provide an “ obstruction ” in H*(Tn; Z 2) to finding a pl homeomorphism between T n and z n. It is Siebenm ann’s idea to investigate the behavior of this obstruction under lifting / : T n —» z n to a 2w-fold cover ; he suggested th a t the obstruction would be­ come zero. Wall [16] and Hsiang and Shaneson [17] have proved th a t this is the c ase; th a t is, if z n is the 2”-fold cover of a homotopy torus z n, n ^ 5, then z n is pl homeomorphic to T n{ — T n). Therefore, following the proof of Theorem 1, / : T n-+ zn is stable, so / is stable, and thus shcwholds for n ^ 4. Hence the annulus conjecture ACn holds for n ^ 4.) {Added A p ril 15, 1969. Siebenmann has found a beautiful and surprising counter-example which leads to non-existence and non-uniqueness of triangu­ lations of manifolds. In particular h t„ is false for n ^ 5, so it is necessary to take the 2n-fold covers, as above. One may then use the fact th a t f: T n—>zn is homotopic to a pl homeomorphism to show th a t / : T —>zn was actually isotopic to a pl homeomorphism. Thus, although there are homeomorphisms between T n and another pl manifold which are not even homotopic to pl homeomorphisms, they cannot be constructed as in Theorem 1. Details will appear in a forthcoming paper by Siebenmann and the author. See also R. C. Kirby and L. C. Siebenmann, On the triangulation o f m anifolds and the H auptverm utung, to appear in Bull. Amer. Math. Soc.) Let X (M n) denote the space (with the compact-open topology) of orienta­ tion preserving homeomorphisms of an oriented stable ^-manifold M, and let S3C(ikP) denote the subspace of stable homeomorphisms. SX(i2n) is both open and closed in 3C(22n). Since a stable homeomorphism of R n is isotopic to the identity, we have

T heorem 2.

the Corollary.

S3C(Rn) is exactly the component o f the id e n tity in 3C(Rn).

Corollary. A homeomorphism o f R n is stable i f and only i f it is iso­ topic to the id en tity.

I f M n is a stable m a n ifo ld , then S3C(ikP) contains the id en tity component of 3C( Mn). In general this does not imply th a t the identity component is arcwise connected (as it does for M n = R n or S n)> b ut arcwise connectivity does follow from the remarkable result of Cernavskii [5] th a t X(ikfn) is locally contractible if M n is compact and closed or M n = R n. From the techniques in this paper, we have an easy proof of the last case. T heorem 3.

STABLE HOMEOMORPHISMS

293

X (i2n) is locally contractible. We now give some definitions, then a few elementary propositions, the crucial lemma, and finally the proofs of Theorems 1 — 4 in succession. The following definitions may be found in Brown and Gluck [3], a good source for m aterial on stable homeomorphisms. A homeomorphism h between open subsets U and V of R n is called stable if each point x e U has a neigh­ borhood Wx a U such th a t h | Wx extends to a stable homeomorphism of R n. Then we may define stable manifolds and stable homeomorphisms between stable manifolds in the same way as is usually done in the p l and differential categories. W henever it makes sense, we assume th a t a stable structure on a manifold is inherited from the p l or differential structure. Home­ omorphisms will always be assumed to preserve orientation. T heorem 4.

P rop osition 1. A homeomorphism o f R n is stable i f it agrees w ith a stable homeomorphism on some open set. P rop osition 2. Let h e 3C(Rn) and suppose there exists a constant M > 0 so that |h(x) — x | ^ M f o r all x e R n. Then h is stable. P roof. This is Lemma 5 of [6]. L etting r B n be the w-ball of radius r, we may consider 5D n = i(5Bn) as a subset of T n, via some fixed differentiable imbedding i: 5B n —> T n. P rop osition 3. There exists an im m ersion a: T n — D n —>R n. P r o o f. Since T n — D n is open and has a trivial tangent bundle, this

follows from [8, Th. 4. 7]. P roposition 4. I f A is an n x n m a trix o f integers w ith determ inant one , then there exists a diffeomorphism f: T n —> T n such that /* = A where /*: :r1(Tfw, Q — iz,(Tn, t0). P r o o f. A can be w ritten as a product of elementary matrices w ith

integer entries, and these can be represented by diffeomorphisms. P roposition 5. A homeomorphism o f a connected stable m anifold is stable i f its restriction to some open set is stable.

For the proof, see [3, p. 35] P r o p o sitio n 6 . Let f: S 71”1 x [ — 1,1] —>R n be an imbedding which contains S n~l in its interior. Then f |S n~l x 0 extends canonically to an imbedding o f B n in R n. P roof. This is shown in [9]. However, there is a simple pro o f; one

ju st re-proves the necessary p art of [2] in a canonical way. This sort of canonical construction is done carefully in the proof of Theorem 1 of [11]. The key to the paper is the following observation.

294

ROBION C. KIRBY

L emma . Every homeomorphism o f T n is stable. P r o o f. Let e: R n — > T n be the usual covering map defined by

e(x l9 •••,&*) = (e2~iXl, • ••, e2' iXn) and let t 0 = (1, 1, • • •, 1) = e(0, • • •, 0). e fixes a differential and hence stable structure on T n. Let h be a homeomorphism of T n>and assume at first th a t h(t0) — t 0 and h*: 7r1(T,n, t0) —>7il( T n1 1 0) is the identity m atrix, h lifts to a homeomorphism h: R n R n so th a t the following diagram commutes. Rn- ^ R n

rp n

_______ ^

n

Since I n — [0, 1] x • • • x [0, 1] is compact, M = sup {| h(x) — x | | x e I n] exists. The condition h * = identity implies th a t h fixes all lattice points w ith integer coordinates. Thus h moves any other unit ^-cube w ith vertices in this lattice in the “ same ” way it moves I n; in particular | h(x) — x \ < M for all x e R n. By Proposition 2, h is stable, e provides the coordinate patches on T ny so h is stable because e~lhe | e~l (patch) extends to the stable homeomorphism h for all patches. Given any homeomorphism h of T n, we may compose w ith a diffeomorphism g so th a t gh(t0) = t0. If A — (gh)f, then Proposition 4 provides a diffeomorphism / w ith f ^ = A ~ (gh)z\ so (fgh)* = identity. We proved above th a t f g h was stable so h — g~lf =l(fgh) is the product of stable home­ omorphisms and therefore stable. Let g be a homeomorphism of R n. ga induces a new differentiable structure on T n — D n, and we call this differential manifold T n —D n. We have the following commutative diagram, P roof

of

Theorem 1.

a

Rn

ga

—^

Rn

a and ga are differentiable and therefore stable, so g is stable if and only if the identity is stable (use Proposition 1). Since T n — D n has one end, which is homeomorphic to S n~l x B, and n 6, there is no difficulty in adding a differentiable boundary [1]. Since

STABLE HOMEOMORPHISMS

295

the boundary is clearly a homotopy (n — l)-sphere, we can take a C '-triangulation and use the p l /i-cobordism theorem to see th a t the boundary is a p l (n — l)-sphere. To be precise, there is a proper p l imbedding /S: S ’1"1 x [0, 1 )—■*T n — D ”, and we add the boundary by taking the union T n — D ” U p *S“=1 x [0,1] over the map /3. Finally we add B n to this union, via the identity map on the boundaries, to obtain a closed p l manifold r n. We can assume th a t 0 be chosen so th a t N (rB n, e) c3C(D") consists of stable homeomorphisms. Then there exists a d < 0 such th a t if h e N ( j ( r B n), 8 ) c DC(M"), then hj{ 2 rB") ( z j ( R n) j~ lh j | 2r B n e N (rB n, e). We may isotope j~ lhj | 2r B n to a homeomorphism H of R n w ith H = j~ lh j on r B n and therefore H e N (rB n, e) c DL(D"). Thus H is stable and so j~ 'h j \ 2rB n is stable. By Proposition 5, h is stable, and hence N ( j ( r B n), d) is our required neighborhood of the identity. P roof of T h eo r em

4. We will observe th a t Theorem 2 can be proved in a “ canonical” fashion ; th a t is, if varies continuously in DC(Rn), then H varies continuously in 3fC(T“). F irst note th a t ;K(7?”) may be contracted onto DC0(Rn), the homeomorphisms fixing the origin. The immersion t a : T n — D ” —*R n can be chosen so th a t ae = id on 1/4D". Pick a compact set C and e > 0 as in the proof of Theorem 2 and let h e N(C, s). h lifts canonically to h: T n — 2D" —►T" — D". Since £(int5D " — 2D”) contains d4D”, it follows from Proposition 6 th a t h(dSDn) bounds a canonical «-ball in 4D". Then h | T ” — 3D" extends by coning to H: T n T n. Clearly H (t0) = t 0 and ii* = identity so H lifts uniquely to a home­ omorphism g: R n —> •R n, w ith | g(x) - x ■< constant for all x 6 R, (see the lemma). We have the commutative diagram P roof of T h eo r em

STABLE HOMEOMORPHISMS

297

9

R

T n — 3D n

* T n — 2D

a

a

R

h

R

Since e(l/4B n) fi 4D n = 0 and ae = id on 1/4B n, it follows th a t g = h on 1/4B n. The construction of g being canonical means th a t the map f : DC0(Rn) —>:X0(Rn), defined by -f(h) = g, is continuous. Let P t: R n —>R n, 1 6 [0 ,1], be the isotopy w ith P„ — h and — g defined by Pt(x) = g |

--- • ^p- 1/i((l -

if t < 1, and Px = g .

Let Qt: R n—>Rn, £ e [ 0 , 1] be the isotopy w ith Q„ = g and QL = identity defined by if t < 1, and Qx = identity. Now let h ,: R n —>R n, t e [0,1] be defined by if 0 ^ t ^ 1/2 if 1/2 fg it sS 1 . It can be verified th a t ht is an isotopy of h to the identity which varies continuously w ith respect to h. Then H t: N(C, e) —►3L0(72B), t e [0, 1] defined by H t(h) = ht is a contraction of N(C, s) to the identity where H t (identity) = identity for all t e [0, 1], This proof can be easily modified to show th a t if a neighborhood V of the identity in DC0(i2n) is given, then C and e may be chosen so th a t N(C, e) contracts to the identity and the contraction takes place in V. To see this, pick r > 0 and 8 so th a t N (r B n, 8 ) c V. Then we may re-define a and e so th a t ae = identity on r B n. If h e N (rB n, 8 ), then P t e N (rB n, 8 ), and if e is chosen small enough (with respect to 8 ), then h e N ( r B n,s) implies th a t Qt e N (rB n, 8 ). Therefore N (rB n, e) contracts in V. U

n iv e r sit y of

In st it u t e

for

Ca l ifo r n ia , L A

dvanced

os

Study

A

ngeles, and

298

ROBION C. KIRBY R

eferences

[1]

W. B r o w d e r , J. Levine, and G. R. Livesay, F inding a boundary fo r an open manifold, Amer. J. Math. 87 (1965), 1017-1028.

[2 ]

M

orton

B row n,

A proof o f the generalized Schoenflies theorem, B u ll. Amer. M a th . Soc.

6 6 (1960), 74-76.

[ 3 ] ----------- and Herman Gluck, Stable structures on m anifolds, I, n, 79 (1964), 1-58.

iii,

Ann. of Math.

A. V. C e r n a v s k i i , The k-stability of homeomorphisms and the union o f cells, Soviet Math. 9 (1968), 729-732. [5] , Local contractibility of groups of homeomorphisms of a manifold, to appear. [ 6 ] E. H. C o n n e l l , A pproxim ating stable homeomorphisms by piecewise linear ones', Ann. of Math. 78 (1963), 326-338. [ 7 ] A. C. C o n n o r , A stable solution to the annulus conjecture, Notices Amer. M ath. Soc. 13 (1966), 620, No. 66 T-338. [ 8 ] M o r r i s W. H i r s c h , On embedding differentiable m anifolds in euclidean space, Ann. of Math. 73 (1961), 566-571. [ 9 ] W i l l i a m H u e b s c h and M a r s t o n M o r s e , The dependence o f the Schoenflies extension on an accessory parameter (the topological case), Proc. N at. Acad. Sci. 50 (1963), 1036-1037. [10] R. C. K i r b y , On the annulus conjecture, Proc. Amer. Math. Soc. 17 (1966), 178-185. [11] J. M. K i s t e r , Microbundles are fiber bundles, Ann. of Math. 80 (1964), 190-199. [12] W. A. L a B a c h , Note on the annulus conjecture, Proc. Amer. Math. Soc. 18 (1967), 1079. [13] L. S i e b e n m a n n , Pseudo-annuli and invertible cobordisms, to appear. [14]-- --------- , A total Whitehead torsion obstruction to fibering over the circle, to appear. [15] C. T. C. W a l l , Surgery on compact manifolds, to appear. [16]-- --------- , On homotopy tori and the annulus theorem, to appear. [17] W. C. H s i a n g and J. L. S h a n e s o n , Fake tori, the annulus conjecture, and the conjectures o f Kirby, to appear. [18] J. L. S h a n e s o n , Embeddings w ith codimension two o f spheres in spheres and h-cobordisms o f S 1 X S 3, Bull. Amer. Math. Soc. 74 (1968), 972-974. [19] C. T. C. W a l l , On bundles over a sphere w ith a fibre euclidean space, Fund. Math. LXI (1967), 57-72. (Received October 29, 1968) [4 ]

Annex B. Reprinted with permission, from Bulletin of the American Mathematical Society July, 1969, Vol. 75, No. 4, pp. 742-749.

ON THE TRIANGULATION OF MANIFOLDS AND THE HAUPTVERMUTUNG BY R. C. K I R B Y 1 A N D L. C. S I E B E N M A N N 2

Communicated by William Browder, December 26, 1968

1. T h e first a u t h o r 's so lution of the stab le h o m e o m o rp h ism c o n ­ je c tu re [5] leads n a tu r a lly to a new m e th o d for deciding w h e th e r or n o t e v e ry topological m anifold of high dim en sio n su p p o r ts a piecewise linear m anifold s tr u c t u r e (tria n g u la tio n problem ) t h a t is essentially u n iq u e ( H a u p tv e r m u tu n g ) cf. Sullivan [ 1 4 ] . A t this tim e a single obstacle re m a in s3— n a m e ly to decide w h e th e r the h o m o to p y g ro u p 7t3( T O P / P L ) is 0 or Z 2. T h e positive results we o b ta in in spite of this obstacle are, in brief, these four: a n y (m etrizable) topological m a n i­ fold M of dim ension ^ 6 is triangulable, i.e. h om eo m o rp h ic to a piecewise linear ( = P L ) manifold, provided H 4(M ; Z 2) = 0; a h o m e o ­ m o rp h ism h : M 1 - + M 2 of P L m anifolds of dim ension ^ 6 is isotopic to a P L h o m e o m o rp h ism p ro vided H Z( M ; Z 2) = 0 ; a n y c o m p a c t t o p o ­ logical m anifold has th e h o m o to p y ty p e of a finite com plex (with no p ro v is o ) ; a n y (topological) h o m eo m o rp h ism of c o m p a c t P L m anifolds is a simple h o m o to p y equ ivalence (again w ith no proviso). R. L ash of a n d M. R o th e n b e r g h a v e proved some of th e results of this paper, [9 ] a n d [ l0 ] . O u r work is in d e p e n d e n t of [10]; on the o th e r h a n d , L a s h o f s p a p e r [9] was helpful to us in t h a t it showed the relevance of Lees' im m ersion th e o re m [l 1 ] to o u r work a n d rein­ forced o u r suspicions t h a t th e Classification theorem below was correct. W e h a ve divided o u r m ain re su lt into a Classification theorem an d a Structure theorem. (I) C l a s s i f i c a t i o n t h e o r e m . Let M m he a n y topological m anifold of dim ension m ^ 6 (or ^ 5 i f the boundary d M is empty). There is a natural one-to-one correspondence between isotopy classes of P L struc­ tures on M and equivalence classes of stable reductions of the tangent microbundle r ( M ) of M to P L microbundle.

(T h e re are good re la tiv e versions of this classification. See [7] a n d proofs in §2.) E xplanations. T w o P L s tr u c tu re s S a n d S ' on M , each defined b y a P L c o m p a tib le a tla s of c h a rts , are said to be isotopic if there exists a 1 Partially supported by N SF Grant G P 6530. 2 Partially supported by N SF Grant G P 7952X. 3 See note added in proof at end of article.

300

ON THE TRIANGULATION OF MANIFOLDS

topological isotop y h ty of 1 m s o t h a t hi is a P L h o m e o m o r­ phism of ( M , 2 ) w ith (M , 2 ') . If M has a m etric d a n d c is a co n tin u o u s fu nc tio n M —>(0, oo), th e n h t is called a n e-isotopy prov id ed d(x, h t(x)) < e(x) for all x £ A f a n d all / £ [0, 1 ]. T o (I) we can a d d : Isotopic P L structures are e-isotopic fo r a ny e. By n -m icro bun dle one can, b y th e K is te r - M a z u r theorem , u n d e r ­ st a n d sim ply a locally-pro duct b un dle w ith fiber E u clid ean n-space R nj a n d zero-section. If £ is th e n a T O P ( = topological) m icrobundle, over a locally finite simplicial com plex X j w ith fiber R n, a re d u c tio n of £ to P L m icro bu ndle is given b y a tria n g u la tio n of £ as a P L m icro ­ b u n d le over X . T w o such tria n g u la tio n s of £ give e q u iv a le n t r e d u c ­ tions if the id e n tity of £ is bu n d le isotopic to a P L isom orphism from th e one P L m icrobu nd le s tr u c tu r e to the other. T h e notion of stab le re d u c tio n differs in allowing a d d itio n of a trivial b u n d le a t a n y m o m e n t. Since M is n o t a priori trian gu lable, one should, to define red uction s, first pull ba c k r ( M ) to a h o m o to p y -e q u iv a le n t simplicial complex. T h e to ta l space of a no rm al m icro bu nd le of MVJ {collar on d M } in R m+k (k large) is co nv en ien t. T h is te c h n ica lity obscures, b u t does n o t d e stro y , th e p le a s a n t p rop erties of th e n otion of reduction. If T O P m/ P L m is th e fiber of th e m a p , B P L m—» B T O P m of classifying spaces for m icrobundles, define T O P / P L as the telescope of th e sequence T O P i / P L i —>TOP2/ P L 2—>TOP3/ P L 3- ^ • • • arising from sta b iliz a tio n of bundles. (II) Structure theorem (P ar tia lly announced tn [s]). 7rk( T O P / P L ) is 0 i f i ^ 3 and Z 2 or 0 i f i ^ 3 . Also Trk( T O P m/ P L m) = T T k (T O P /P L ) by stabilization, fo r k < m , m ^ 5 . W h e n it becam e kno w n th a t , w ith W all, we had show n t h a t 7t*;(T O P/PL) is 0 for &5^3 a n d g Z 2 for &= 3, we [8] a n d L ashof a n d R o th e n b e r g [10] in d e p e n d e n tly noticed t h a t L ees’ im m ersion th e o ­ rem [ l l ] gives th e corresp on ding n o n sta b le results above. T h is was of critical im p o rta n c e to L a sh o f’s tria n g u la tio n th eo rem [»]■ T h e equ ivalence classes of sta b le re d u ctio n s of t ( M ) can be p u t in one-to-one correspondence w ith vertical h o m o to p y classes of sections of a b u n d le over M w ith fiber T O P / P L , n am e ly th e pull-back b y a classifying m a p M —* B t op for r ( M ) of the fibration T O P / P L —> 5pl —»23top. C o m b in in g (I) a n d (II) we find (1) There is ju s t one well-defined obstruction in H 4(M ; ttz( T O P / P L ) ) to im posing a P L structure on M . (2) Given one P L structure on M the isotopy classes of P L structures on M are in (1-1)-correspondence with the elements of H * ( M ; ir * ( T O P /P L ) ) .

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As a p p lic a tio n s of (I) alone consider: (a) T h e to ta l space E of a n y no rm a l &-disc b u n d le [3] of M u in Rn+k^ 6, is trian g u lab le as a P L m anifold, since t (E) is trivial. (b) If A: E —>Ef is a h o m e o m o rp h ism of parallelizable P L w -m a nifolds, th ere exists a topological disc-bundle a u t o m o r p h is m a: E X D * —>EXD* over E (s large) so t h a t ( h X i d ) o a : E X D * —> E ' X D 8 is topologically isotopic to a P L ho m e o m o rp h ism . P r o o f o f (b). T h e P L re d u c tio n of t ( E ) given by h is classified b y an e lem en t y £ [ E , T O P m/ P L m]. Since r ( E ) a n d t(jE') a re trivial bundles, y comes from x £ [ E , T O P w]. R e p re s e n t —x by a n a u t o ­ m orph ism j8: E X R ^ —^ E X R ™ of the trivial R m bundle. T h e n , u p to b u n d le isotopy 0 X 1 # e x te n d s [3] to a disc b u n d le a u to m o r p h is m a: E X D m+l—*E X D m+\ where R m+l = i n t D m+l. T h e P L re d u c tio n s of r ( E X D m+l) given by (h X i d ) o a a n d b y id\ E X D m+l are s ta b ly th e sa m e ; so (b) follows from (I). T h e se seem ingly in n o c e n t o b se rv a tio n s read ily affirm tw o im p o r­ t a n t c o n jec tu res (cf. [13]). ( I I I ) F i n i t e n e s s o f c o m p a c t t y p e s . Every compact topological manifold has the homotopy type of a finite complex— even i f it he nontriangulable.

(IV) T o p o l o g i c a l i n v a r i a n c e o f t o r s i o n s . Every topological manifold M has a well-defined simple homotopy type4, namely the type of its normal disc bundles triangulated as P L manifolds. I n particular, if h: M —+M’ is a homeomorophism of compact connected P L man i fo ld s, the Whitehead torsion r(h) £W h(7TiM ) of h is zero. 2. W e now sketch th e proof of (I) a n d (II). I m p o r t a n t ele m e nts of it were a n n o u n c e d in [7], [fi]. A n i m p o r t a n t role is played b y Lees’ re ce n t classification th e o re m for topological im m ersions in c od im e n ­ sion zero [ 11 ], a n d by W a ll’s su rg ery of n o nsim p ly con n ec te d m a n i­ folds [l S ] ; we suspe c t t h a t one or b o th could be elim in a te d from the proof of (I), b u t th e y are essential in th e proof of (II). In this regard see th e w eak er tria n g u la tio n th eo re m s of L ashof [9] a n d Lees [ 11 ] proved before (I). H a n d l e s t r a i g h t e n i n g p r o b l e m P (h) . Consider a homeomorphism h: B k X R n—>Vm, k + n = m t (where B k —standard P L k-ball i n R k) onto a P L manifold V so that h \ d B k X R n is a P L homeomorphism. Can one f i nd a topological isotopy h t : B k X R n—^ V n, of h = h Q such that (1) h i \ B k X B n is PL ,

(2) ht — h , 0 ^ t ^ 1, on d B k X R n^ J B k X (R n — r B n) f o r some r ? 4 This makes good sense even when M is noncompact, cf. [13, p. 74].

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O ne should th in k of B k X R n as an open P L han dle with core B k X 0 . T h e analysis of P ( h) is based on the ma i n diagram below, which for &= 0, originated in [S]. M

ain diagram

m —» + d indicates b o u n d a ry , B k —s t a n d a r d P L &-ball in R k. D n = a. P L n-ba.ll w ith in tD " = R n = euclidean n-space, so t h a t b o u n d e d m a p s R n—>Rn extend b y the id e n tity to D n. T n = n-toru s, the w-fold p r o d u c t of circles; D 0 = a P L n-ball, col­ lared in T n. A rra n g e t h a t h is P L ne a r 0 B k X R n a n d successively c o n s tr u c t e, a , IT, g, g' (when possible), H y H \ f . E x p la na ti on s . a, e a n d the inclusions of B k X 2 B n are chosen so t h a t th e triangles a t left c o m m u te ; B k X ( T n — D 0) w ith th e un iq u e P L s tr u c tu re m a king ha P L is b y definition [Bk X { T n — D q) ]'\ W is P L a n d “caps it off”, g e xte nds id. Fin d in g gr is the T o r u s p r o b l e m Q(g). To extend g \ d B k X T n to a P L homeomorphism homotopic to g. Supposing gf solves Q(g), form H a n d 'squ eeze’ to allow extension to H f ; then d e fin e / b y engulfing to exten d the n a tu r a l P L identifica­ tion H ' ( B k X B 71) —>h(Bk X B n). Since / em b e d s th e ball B k X D n, for a n y isotopy H I of H ' fixing d ( = b o u n d a ry ), th ere exists an isotopy h t of h fixing d a n d B k X R n —h ~1f ( B k X D n) so th a t, for each t t h t ^ f H l on B k X B n. T h u s , if H i is the A lexander iso to py of H ' to t h e iden­ tity , h t solves P ( h ) \ \

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P

r o p o s it io n

1.

303

I f g is derived from h as above, Q(g) is s o lv a bl e ^P ( h)

is solvable. P r o o f . I t rem ains to show I X V, fixing d , from h to a P L h o m e o m o r­ phism . One can ex ten d th e c o n stru c tio n of g from h to ge t from the isotopy h a h o m e o m o rp h ism g: l X B k X T n- > W ' with g| | ( ) } X B k X T n = gy t h a t is P L 011 d m inu s {0 } X int#* X T n. A p p ly in g th e .v-cobordism theo rem to W ' one gets a solution of Q(g). ■ G iven th e p roblem P ( h) consider the ta n g e n t b u n d le m a p /z*: r ( B k X R n)-~^r( Vm). B oth b undles are P L trivial. R e stric tin g over B k X jo} in the source a n d projecting to th e fiber in the t a r g e t we get a m a p ( B k, d B k)-~> (TO Pw, P L m) (to be u n d e rs to o d semisim plicially [12, § 2]).C all its c la ss d (A )e 7 r* (T O P w, P L TO) ^ 7 r * ( T O P « / P L m). D e n o te this g ro u p b y II* (w).

P r o p o s i t i o n 2. P ( h) is solvableRm, P L on d , such t h a t if V m is B k X R n w ith th e P L s t r u c tu r e mak> ing h 0 P L , th e n h = id\ B k X R n—* Vm has d(h) = x . m

F o r a n y solution h t of P (h ) , th e in duced ta n g e n t bu n d le m a p gives a m a p / : l X B k—>TOPm sending l X d B k\ J \ 1 J X B k into P L W. T h u s , given two solutions h tl hi of P{h) we can piece to g e th e r / , / ' to get a ‘difference’ class 5(ht} h i ) £7r*+i ( T O P m, P L m) =7r/e+i(;w). P r o p o s i t i o n 3. Let h t solve P ( h) f o r h: B k X R n—>Um, k < m . Given a ny y i n n * +i(m + 1) there exists a solution hi of P ( h) such that s5(ht, h i ) = y (s denotes stabilization which by (II) is an isomorphism). I n d i c a t i o n o f p r o o f . O ne can reduce to the special case where h t is th e id e n tity solution of P ( i d \ B k X R n). P rop osition 2' provides a prob lem P(/zi), hi: B k+1 X R n—>V m+l w ith d ( h i ) = y , which yields a to ru s problem Q(gi), gi: B k+lX T n—* W m+l. Im pose on B k+1X T n — l X B k X T n th e P L s tr u c tu r e 2 m a k in g g\ PL . 2) is s t a n d a r d on d. A p p ly th e s-cobordism th e o re m to (I; 0, l ) X B k X T n w ith s tr u c tu r e

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S to derive a P L a u to m o rp h is m d of {1} X B k X T n t h a t fixes d. Solve P ( i d \ B k X R n) using 0 (for g' of the m ain d ia g ra m ) in place of id, a n d call th e solution h [ . T h e n sd(id, h i ) —y> P r o o f o f (II). (a) For m > k > 3 there is an elementary proof that n*(m)=0. I t is a n in d u c tio n on n —m ~ k ^ \ exploiting th e m ain dia gra m . Also, surg ery can be used as in (b). (b) n*(m)=0 f o r 5 and k = 0, 1, 2. R e p re se n t x ^ I h i m ) b y a problem P( h) a n d pass to th e to ru s problem Q(g). T h e r e is a n o b ­ s tru c tio n [g] £ i / 3~ * ( r n; Z 2) to solving Q(g). F o r [g] to exist, g need only be a h o m o to p y equivalence w hich is a P L hom e o m o rp h ism on d , a n d , stric tly speaking, [g] is th e o b s tru c tio n to h o m o to p in g g (m odulo d) to a P L h o m eo m o rp hism . N o te t h a t to solve P (h ) it suffices to solve Q(g) w here g is th e covering of g for (2Z )nC Z n

TTi(T n).

T o see this ju s t a d d a new tier o ver g in th e m ain d ia g ra m ! N ow [ g ] = £ * [ g ] . B u t p* kills PP~k( T n\ Z 2). T h u s P (h ) is solvable a n d x = 0. (Also Q(g) is solvable b y Pro po sition 1, so [g] was a lre a d y 0!) (c) n 3( w ) C ^ 2 f o r m ^ S . In th e a b o v e a r g u m e n t p ^ —id. B u t b y inspection of definitions x>—►[g] £ H ° ( T n; Z 2) is add itiv e . (d) St abil ity : s: n 3(w)—>n3(w + 1) is an i s o mor phi sm , w^ 5. T h a t 5 is o n to follows b y th e d e sc e n t a r g u m e n t for (a). T o pro ve 5 is injec­ tive we check th a t, if P (h ) gives Q(g) with su rg ery o b stru c tio n y £ Z 2, th e n P ( h X l i t ) also gives y (E Z 2. P rio r to th is work, C. T . C. W all, W. C. H siang, a n d J. Shan eso n (jointly), a n d A. Casson u nd e rsto o d th e classification of h o m o to p y tori (the case when k = 0 a b o v e ).5 W h e n ou r specific q ue stion s (see [6] a n d [8]) were posed, W all [16] a n d H sia n g a n d S h a n e so n [4], in d e p e n d e n tly verified t h a t [ g ] = £ * [ g ] a n d e x te n d ed th e ir w ork to th e cases k ^ O . I t seems to us t h a t th e fibration th e o re m of F arrell [ l] rew orked in [2] a n d [15]) plays a n essential role in th e c o n s tr u c ­ tion a n d use of [g]. W e t h a n k W . B row der for a v e ry tr a n s p a r e n t definition of [g]. Propo sition s 1, 2, 2', 3 a n d the a b o v e s ta b ility p e r m it us to p rov e

(I). Uniqueness ( H a u p tv e r m u t u n g ) . C onsider a n u n b o u n d e d T O P m anifold M w ith tw o P L s tru c tu r e s S, 2 ' giving re d u c tio n s p, p' of 6 C asson a p p a ren tly h ad exam in ed the general case

k ^ 0 [ 16 ].

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t ( M)

to P L m ic ro b u n d le t h a t are (stab ly ) related b y a d e fo rm a tio n a of red uction s. T a k e a h a n d le d eco m p ositio n of M , a n d suppose t h a t, for a su b h a n d le b o d y M 0, one has found an isotop y h u of 1m w ith h91Mo being PL , which realizes a | Mo. If H is a ^-h an dle a tta c h e d to M 0, th e o b s tru c tio n in 7r*(T O P /P L ) (P ropo sitio n 2) to prolonging h t fixing Mo to an iso to py h tt 0 ^ / ^ 5 + l, w ith h8± \ \ M 0V J// being P L , can be identified with the o b s tr u c tio n to e x ten d in g over M 0KJH the d e fo rm a tio n d e te r m in e d b y h t \ Mo betw een th e re d u c tio n s given b y 2 a n d 2'. I t is zero because a | M o ^ J H exists. P ro po sition 3 allows us to choose h u 0 ^ / ^ 5 + l, so as to realize a \ M o ^ J H . T h is indicates how to c o n s tr u c t in d u c tiv e ly the desired isotopy. B y using small handles, one can m a k e this isotopy small. In a relativ e form, which this a r g u m e n t also yields, the case for open m anifolds ta k e s care of m anifolds w ith b o u n d a ry . Existence (T ria n g u la tio n ). Once it is s ta te d in a re lative form (see [7]) we can assu m e t h a t d M = 0 an d (by passing to c h a r ts of a locally finite cover) t h a t M is triang ulable. A gain we use a h a n d le ind uctio n, a n d sp re a d the P L s tr u c tu r e h a n d le by handle. Pro po sition 2' ta k e s the a b o v e role of P roposition 3, so this a r g u m e n t for existence is sim pler th a n t h a t for uniqu eness above. A d d e d i n P r o o f . W e h a v e p roved t h a t 7r3 ( T O P / P L ) is Z 2 n o t 0, see [ l 7 ] . H e re is a n a r g u m e n t in outline. S u rg ery provides a h o m o ­ to p y eq uivalence g \ B 1X T n—>W, n ? t 5 , t h a t is a P L h o m e o m o rp h ism on d a n d has nonzero in v a ria n t [ ^ ] G F ( r n; Z 2). (H ere a n d below c o m p are proof of II.) U sing th e 5-cobordism the ore m (cf. Pro p o sitio n 3) one derives from g a P L a u to m o rp h is m h of T n, well-defined by [g] up to P L pseudo -isotopv (concordance). L e t gx, X = 1, 3, 5, • • - be th e s ta n d a r d Xn-fold covering of g a n d let h\ be a Xn-fold covering of h derived from g\. F o r X large, h\ can be a r b itr a r ily close to id\ T n; hence is topologically isotopic to id\ T n b y a result of C ernavskii ( D o k la d y 1968) p rov a ble [6] by a m e th o d of [5]. T h u s if 7 r* (T O P /P L ) were zero, Ax would be a t least P L pseudo-isotopic to id \ T n. B u t it is not, since [gX] = [g] X-0 as X is odd. T h e re fo re 7r3( T O P / P L ) = Z 2. T h is discovery leads to m a n y s trik in g conclusions (e.g. see N otices A m er. M a th . Soc., J u n e 1969). T h e se will be discussed fully in a p a p e r d e v o te d to a careful d e v e lo p m e n t of o u r results. R eferences 1. F. T. Farrell, Thesis, Yale University, New Haven, Conn., 1967. 2. F. T. Farrell and W. C. Hsiang, Manifolds with 7u = Z X aU (to appear). 3. M. W. Hirsch, On non-linear cell-hundles, Ann. of M ath (2) 84 (1966), 373-385.

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4. W. C. Hsiang and J. L. Shaneson, Fake tori, the annulus conjecture and the con­ jectures of K irb y , Proc. N at. Acad. Sci. U.S.A. (to appear). 5. R. C. Kirby, Stable homeomorphisms and the annulus conjecture, Ann. of M ath. (2) 90 (1969). 6. ----------, Announcement distributed with preprint of [5], 1968. 7. R. C. Kirby and L. C. Siebenmann, A triangulation theorem, N otices Amer. M ath. Soc. 16 (1969), 433. 8. R. C. Kirby, L. C. Siebenmann and C. T. C. Wall, The annulus conjecture and trianp,illation, N otices Amer. M ath. Soc. 16 (1969), 432. 9. R. K. Lashof, Lees' immersion theorem and the triangulation of manifolds, Bull. Amer. Math. Soc. 75 (1969), 535-538. 10. R. K. Lashof and M. Rothenberg, Triangulation of manifolds. I, II, Bull. Amer. M ath. Soc. 75 (1969), 750-754, 755-757. 11. J . A. Lees, Immersions and surgeries of topological manifolds, Bull. Amer. M ath. Soc. 75 (1969), 529-534. 12. C. P. Rourke and B. J. Sanderson, On the homotopy theory of L-scts (to appear). 13. L. C. Siebenmann, Ot! the homotopy type of compact topological manifolds, Bull. Amer. M ath. Soc. 74 (1968), 738-742. 14. D. P. Sullivan, On the kauptvermutung for manifolds, Bull. Amer. M ath. Soc. 63 (1967), 598-600. 15. C. T. C. Wall, Surgery on compact numifolds, preprint, University of Liverpool. 16 -----------1Qn homotopy tori and the annulus theorem, Proc. London M ath. Soc. (to appear). 17. R. C. Kirby and L. C. Siebenmann, For manifolds the Ilauptvermutung and the triangulation conjecture are false, N otices Amer. Math. Soc. 16 (1969), 695. U n iv e r s it y

of

I n s t it u t e

C a l if o r n ia , L o s A

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Annex C. Reprinted with permission, from Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, ^diteur, Paris 6e, 1971, Volume 2, pp. 133-163.

TOPOLOGICAL MANIFOLDS* by L .C . S IE B E N M A N N

0. Introduction. Homeomorphisms — topological isomorphisms — have repeatedly turned up in theorems o f a strikingly conceptual character. For example : (1) (19th century). There are continuously many non-isomorphic compact Riemann surfaces, but, up to homeomorphism, only one o f each genus. (2) (B. Mazur 1959). Every smoothly embedded (n — l)-sphere in euclidean H-space R n bounds a topological «-ball. (3) (R. Thom and J. Mather, recent work). Among smooth maps o f one compact smooth manifold to another the topologically stable ones form a dense open set. In these examples and many others, homeomorphisms serve to reveal basic rela­ tionships by conveniently erasing some finer distinctions. In this important role, PL (= piecewise-linear)(**) homeomorphisms o f simplicial complexes have until recently been favored because homeomorphisms in general seemed intractable. However, PL homeomorphisms have limitations, some o f them obvious ; to illustrate, the smooth, non-singular self-homeomorphism / : R R o f the line given by f ( x ) = x + — exp (— l/x 2) sin ( 1 /x ) can in no way be regarded 4 as a PL self-homeomorphism since it has infinitely many isolated fixed points near the origin. Developments that have intervened since 1966 fortunately have vastly increased our understanding o f homeomorphisms and o f their natural home, the category of (finite dimensional) topological manifolds(***). I will describe just a few o f them below. One can expect that mathematicians will consequently come to use freely the notions o f homeomorphism and topological manifold untroubled by the frus­ trating difficulties that worried their early history.

(*) This report is based on theorems concerning homeomorphisms and topological mani­ folds [44] [45] [46] [46 A] developed with R.C. Kirby as a sequel to [42]. I have reviewed some contiguous material and included a collection of examples related to my observation that 7r3(TOP/PL) =£ 0. My oral report was largely devoted to results now adequately descri­ bed in [81], [82].

(**) A continuous map / : X -+ Y of (locally finite) simplicial complexes is called PL if there exists a simplicial complex X' and a homeomorphism s : X' -►X such that s and fs each map each simplex of X' (affine) linearly into some simplex. (***) In some situations one can comfortably go beyond manifolds [82]. Also, there has been dramatic progress with infinite dimensional topological manifolds (see [48]).

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1. History. A topological (= TOP) m-manifold M m (with boundary) is a metrizable topo­ logical space in which each point has an open neighborhood U that admits an open embedding (called a chart ) / : £ / - ► R+ = {(Xj, . . . , x m) E R m\x l > 0}, giving a homeomorphism U « / ( ( / ) . From Poincare’s day until the last decade, the lack o f techniques for working with homeomorphisms in euclidean space R m (m large) forced topologists to restrict attention to manifolds M m equipped with atlases o f charts f a : Ua-* R ™, U Ua = M , (a varing in some index set), in which the maps (where defined) are especially tractable, for example all DIFF (infinitely differentiable), or all PL (piecewise linear). Maximal such atlases are called respectively DIFF or PL manifold struc­ tures. Poincar6 , for one, was emphatic about the importance o f the naked homeomorphism — when writing philosophically [ 6 8 , §§ 1 , 2 ] — yet his memoirs treat DIFF or PL manifolds only. Until 1956 the study of TOP manifolds as such was restricted to sporadic attempts to prove existence o f a PL atlas (= triangulation conjecture) and its essential uniqueness (= Hauptvermutung ). For m = 2, Rado proved existence, 1924 [70] (Kerekjarto’s classification 1923 [38] implied uniqueness up to isomor­ phism). Form = 3, Moise proved existence and uniqueness, 1952 [6 2 ],cf.amisproof o f Furch 1924 [21]. A PL manifold is easily shown to be PL homeomorphic to a simplicial complex that is a so-called combinatorial manifold [37]. So the triangulation conjecture is that any TOP manifold M m admits a homeomorphism h : M -►N to a combinatorial manifold. The Hauptvermutung conjectures that if h and t i : M -* N ’ are two such, then the homeomorphism h'hT1 : N -+ N* can be replaced by a PL homeo­ morphism g : N N'. One might reasonably demand that g be topologically isotopic to h ’h~l , or again homotopic to it. These variants of the Hauptvermutung will reap­ pear in §5 and § 15. The Hauptvermutung was first formultated in print by Steinitz 1907 (see [85]). Around 1930, after homology groups had been proved to be topological invariants without it, H. Kneser and J.W. Alexander began to advertise the Hauptvermutung for its own sake, and the triangulation conjecture as well [47] [2]. Only a misproof o f Noebling [ 6 6 ] (for any m) ensued in the 1930’s. Soberingly delicate proofs o f triangulability o f DIFF manifolds by Cairns and Whitehead appeared instead. Milnor’s proof (1956) that some ‘well-known’ S 3 bundles overS 4 are homeomor­ phic to S 7 but not DIFF isomorphic to S 7 strongly revived interest. It was very rele­ vant ; indeed homotopy theory sees the failure o f the Hauptvermutung (1969) as quite analogous. The latter gives the first nonzero homotopy group 7r3 (TOP/O) = Z 2 o f TOP/O ; Milnor’s exotic 7-spheres form the second tt7 (TOP/O) = Z28. In the early 1960’s, intense efforts by many mathematicians to unlock the geo­ metric secrets of topological manifolds brought a few unqualified successes : for example the generalized Shoenflies theorem was proved by M. Brown [7] ; the tangent microbundle was developed by Milnor [60] ; the topological Poincare conjecture in dimensions > 5 was proved by M.H.A. Newman [65].

309

TOPOLOGICAL MANIFOLDS

V

Of fundamental importance to TOP manifolds were Cemavskii’s proof in 1968 that the homeomorphism group o f a compact manifold is locally contractible [10] [11], and Kirby’s proof in 1968 o f the stable homeomorphism conjecture with the help o f surgery [42]. Key geometric techniques were involved — a meshing idea in the former, a particularly artful torus furling and unfurling idea(*) in the latter. The disproof o f the Hauptvermutung and the triangulation conjecture I sketch below uses neither, but was conceived using both. (See [44] [44 B] [46 A] for alternatives). 2. Failure of the Hauptvermutung and the triangulation conjecture. This section presents the most elementary disproof I know. I constructed it for the Arbeitstagung, Bonn, 1969. In this discussion B n = [— 1 , 1]n C R n is the standard PL ball ; and the sphere 1 = dBn is the boundary o f B n. T n = R n/Z n is the standard PL torus,the «-fold product o f circles. The closed interval [0 , 1] is denoted I.

S n~

As starting material we take a certain PL automorphism a o f B 2 x T n, n > 3 , fixing boundary that is constructed to have two special properties ( 1 ) and ( 2 ) be­ low. The existance o f a was established by Wall, Hsiang and Shaneson, and Casson in 1968 using sophisticated surgical techniques o f Wall (see [35] [95]). A rather naive construction is given in [80, § 5 ], which manages to avoid surgery obstruction groups entirely. To establish (1) and (2) it requires only the 5 -cobordism theorem and some unobstructed surgery with boundary, that works from the affine locus Q4 : z \ + z \ + z \ = 1 in C3. This Q4 coincides with Milnor’s E s plumbing of dimension 4 ; it has signature 8 and a collar neighborhood o f infinity M 3 x R, where M 3 = SO (3)M 5 is Poincare’s homology 3-sphere, cf. [61, § 9.8]. (1) The automorphism (3 induced by a on the quotient T 2+n o f B 2 x Tn(} x Tn extending the stan­ dard equivalences T*+n 21 0 x T* x T n and T(J3) ~ 1 x T 3 x T n. The symbol # in­ dicates (interior) connected sum [41 ]. (2) For any standard covering map p :B 2 x T n B 2 x T n the covering automor­ phism o f a fixing boundary is PL pseudo-isotopic to a fixing boundary. ( Co­ vering means that potl = ap). In other words, there exists a PL automorphism H o f ( / ; 0 , l ) x ^ 2 x r fixing I x bB2 x T n such that H \0 x B 2 x T n = 0 x a and H\ \ x B 2 x T n = 1 x aj . (*) Novikov first exploited a torus furling idea in 1965 to prove the topological inva­ riance of rational Pontrjagin classes [67]. And this led to Sullivan’s partial proof of the Hauptvermutung [88]. Kirby’s unfurling of the torus was a fresh idea that proved revo­ lutionary. (**) This is the key property. It explains the exoticity o f T(/3) — (see end o f argument), and the property (2) —(almost, see [80, § 5]).

310

L.C. SIEBENMANN

In (2) choose p to be the 2"-fold covering derived from scalar multiplication by 2 in R n . (Any integer > 1 • would do as well as 2.) Let a 0( = f t ) , a 1 )a 2 , . . . be the sequence o f automorphisms o f B 2 x T n fixing boundary such that ak+l covers otk , i.e. pa*+1 = otkp. Similarly define H 0(= H ) , H x , H 2, . . . and note that is a PL concordance fixing boundary from a k to afc+1. Next define a PL auto­ morphism H ' o f [ 0 , 1 ) x 5 2 x T n by making H '\[a k , ak+x] x B 2 x T n 9 where 1 ak ~ * — correspond to H k under the (oriented) linear map o f [ak ,a k+1] onto [ 0 , 1 ] = /. We extend H ' by the identity to [0 , 1) x R 2 x T n . Define another self-homeomorphism H " o f [0 , 1) x B 2 x T n by H " = where < p(t,x , y ) = ( f , (1 - t ) x , y )

.

Finally extend H " by the identity to a bijection H " : / x B2 x Tn

/ x B2 x Tn

It is also continuous, hence a homeomorphism. To prove this, consider a sequence q l , q 2, • • • o f points converging to q = (t0 , x 0 , y 0) in I x B 2 x T n.Convergence H ' ( q j) H ”(q ) is evident except when t0 = 1 , x 0 = 0. In thelatter case it is easy to check that p 1H n( q j ) - ^ p l H ,,(q) = 1 and p 2H " (q j) -+ p 2H "(q ) = 0 as j -►«>, where p f , i = 1 , 2 , 3 is projection to the i-th factor o f I x B 2 x T n . It is not as obvious that p sH n(q j) -+ p 3H "(q ) = y 0 . To see this, let H k \ I x B 2 x R n -* / x B 2 x R n be the universal covering of H k fixing I x dB2 x R n. Now sup ( | p 3z — p 3Hk z\

;

z G [0 , 1] x B 2 x i?"} = Dk

is finite, being realized on the compactum / x B 2 x / " . And, as H k is clearly where 6n( t , x , y ) = ( t , x , 2ny), we have D k = ^ k D 0. Now

is

> the maximum distance o f p$Hk from p 3, for the quotient metric on T n = R n! Z n ; so 0 implies p 3H " ( q j) -►p 3H "(q ) = y Qi as j -+ . As the homeomorphism H " is the identity on I x 3 5 2 x T n it yields a selfhomeomorphism g o f the quotient / x T 2 x T ” = / x T2+" . And as g 10 x r 2+n = 0 x 0

,

and g | 1 x r 2+" = identity, g gives a homeomorphism h o f T (0) onto T (id) = T X x T 2+n = T 3+n by the rule sending points (t , z) to g~x ( t , z) — hence (0 , z) to (0 , j3_1 (z)) and (1 , z) to (1 , z ) The hom eom orphism h : T 3+n » T(j3) belies the Hauptvermutung. Further, (1) offers a certain PL cobordism (W ; T 3+", T(j3)). Identifying r 3+" in W to T (P ) under h we get a closed topological manifold

TOPOLOGICAL MANIFOLDS

x 4+n ^

311

{ T 1 x T 3 # Q U oo }x Tn

( ^ indicating homotopy equivalence). If it had a PL manifold structure the fibering theorem of Farrell [19] (or the author’s thesis) would produce a PL 4-manifold X 4 with w x( X 4) = w 2 ( X 4) = 0 and signature o ( X 4) = crCS1 x 7 3 # Q U °°) = a(Q U 5. Lashof [50] gave the first proof that was valid for m — 4. A stronger and technically more difficult result is sketched in [63] [45]. It asserts a weak homotopy equivalence o f a “ sliced concordance” variant o f CATm{Mm) with L ift(r to ^ Cat(ot))‘ This *s va^ without the openness restriction if m =£ 4. For open M m (any m), it too can be given a proof involving a micro-gibki property and Gromov’s procedure. 5. The product structure theorem. T h e o r e m 5.1 {Product structure theorem). — L e t M m be a TOP manifold, C a closed subset o f M and o 0 a CAT (= DIFF or PL) structure on a neighborhood o f C in M. L e t be a CAT structure on M x R s equal o0 x R s near C x R s. Provide that m > 5 and bM C C.

Then M has a CAT structure o equal a0 near C. A n d there exists a TOP isotopy (as small as we please) h t : M a x R s -►(M x R s)£ , 0 < t < 1, o f h 0 = identity , fixing a neighborhood o f C x R s, to a CAT isomorphism h x. It will appear presently that this result is the key to TOP handlebody theory and transversality. The idea behind such applications is to reduce TOP lemmas to their DIFF analogues/ (*) Alternatively, for our purpose, -#r(G) can be the ordered sim plicial com p lex having one d-sim plex for each equipped bundle over the standard d-sim plex that has total space in som e R n C R~. (* * ) The forgetful map \p : # r(pLm) Z?r(Pim) a sim plicial com p lex, then define

-®pL(m) *s m orQ delicate to define. O ne can m ake sim plex by sim plex.

316

L.C. SIEBENMANN

It seems highly desirable, therefore, to prove 5.1 as much as possible by pure geo­ metry, without passing through a haze o f formalism like that in § 4. This is done in [46]. Here is a quick sketch o f proof intended to advertise [46]. First, one uses the CAT s-cobordism theorem (no surgery !) and the handlestraightening method of [44] to prove — without meeting obstructions — T h e o r e m 5.2 ( Concordance implies isotopy ). — Given M and C as in 5.1, con­ sider a CAT structure T on M x I equal a 0 x I near C x i , and let V \ M x 0 be cal­ led a x 0. (T is called a concordance o f a rel C).

There exists a TOP isotopy (as small as we please) h t : M a x / (M x / ) E, 0 < t < 1, o f h Q = identity, fixing M x 0 and a neighborhood o f C x /, to a CAT isomorphism h x. Granting this result, the Product Structure Theorem is deduced as follows. In view o f the relative form o f 5.2 we can assume M = R m. Also we can assume s = 1 (induct on s !). Thirdly, it suffices to build a concordance T (= structure on M x R 3 x I ) from o x R s to E rel C x R s. For, applying 5.2 to the concordance T we get the wanted isotopy. What remains to be proved can be accomplished quite elegantly. Consider Figure^ 5-a.

Cross hatching indicates co-incidence with £. Double vertical hatching in­

dicates where the structure is a product along/?.

Figure 5a

TOPOLOGICAL MANIFOLDS

317

We want a concordance rel C x R from 2 to a x R . First note it suffices to build 2 2 with the properties indicated. Indeed 2 2 admits standard (sliced) concor­ dances rel C x R to a x R and to 2 . The one to a x R comes from sliding R over itself onto (0 , «>). The region o f coincidence with a x R becomes total by a sort o f window-blind effect. The concordance to 2 comes from sliding R over itself onto (— o °, — 1). (Hint : The structure picked up from 2 2 at the end o f the slide is the same as that picked up from 2). It remains to construct 2 2. Since M x R = R m+1, we can find a concordance (not rel C x R ) from 2 to the standard structure, using the STABLE homeomor­ phism theorem(*) [42]. Now 5.2 applied to the concordance gives 2 , , which is still standard near M x [0 , «>). Finally an application o f 5.2 to 2 , |TV x [— 1 , 0], where A is a small neighborhood o f C, yields 2 2. The change in 2j|A f x 0 (which is standard) on N x 0 offered by 5.2 is extended productwise over M x [0 ,«>). This completes the sketch. It is convenient to recall here for later use one o f the central results o f [44]. Recall that TOPm/PLm is the fiber o f the forgetful map B Ph(m) ^ B TO¥(m). And TOP/PL is the fiber the similar map o f stable classifying spaces B ?L -+ B TOV. Si­ milarly one defines TOPm/DIFFm = TOPm/Om and TOP/DIFF = TOP/O. T heorem

5.3(**) (Structure theorem).— TOP/PL ~ K (Z2, 3) and 7r*(TOPm/CATm) = 7rfc(TOP/CAT)

fo r k < m and m > 5. Here CAT = PL or DIFF. Since 7rfc(Om) = n k ( 0 ) for k < m , we deduce that 7rk(TOP , TOPm) = 0 for k < m > 5, a weak stability for TOPm. Consider the second statement o f 5.3 first. Theorem 4.4 says that itk (TOPm/CATm) = ir0(CATm(5 fc x R m- k) ) = S ^ for k 5. Secondly, 5.1 implies S™ = S™ +1 = S™*2 = • • • >m > k. Hence *t (TOPm/CATm) = irk (TOP/CAT). We now know that n k (TOP/PL) is the set o f isotopy classes o f PL structures on S l: if k > 5. The latter is zero by the PL Poincare theorem o f Smale [84], combined with the stable homeomorphism theorem [42] and the Alexander isotopy. Simi­ larly one gets 7rfc(TOP/DIFF) = 0 k for k > 5. Recall 0 5 = 0 6 = 0 [41]. The equality tt*(TOP/PL) = tt*(TOP/DIFF) = ttk ( K ( Z 2, 3)) for k < 5 can be deduced with ease from local contractibility o f homeomorphism groups and the surgical classification [35] [95], by H 3( T S ; Z2), o f homotopy 5-tori. See [43] [46 A] for details. Combining the above with 4.4 one has a result o f [44].

(*) Without this we get only a theorem about compatible CAT structures on STABLE manifolds (of Brown and Gluck [8]). (**) For a sharper result see [63] [45], and references therein.

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C l a ssific a t io n th eo rem 5 .4 . — For m > 5 a TOP m anifold M m (w ithout boun­ dary) adm its a PL m anifold structure i f f an obstruction A (M ) in H A( M \Z 2) vanishes. When a PL structure 2 on M is given, others are classified (up to concor­ dance or isotopy) by elements o f H 3(M ; Z2).

Com plem ent. — Since 7rk(TOP/DIFF) = tt* (TOP/PL) for k < 7 (see above calcu­ lation), the same holds for DIFF in low dimensions. Finally we have a look at low dimensional homotopy groups involving G = lim {Gn \n > 0} where Gn is the space of degreed maps,?"-1 S n~ l . Recall that irn G = ttn+kS k, k large. G/CAT is the fiber of a forgetful map B CAT B G, where B G is a stable classifying space for spherical fibrations (see [15], [29]).

The left hand commutative diagram o f natural maps is determined on the right. Only 7r3TOP is unknown(*). So the exactness properties evident on the left leave no choice. Also 3 must map a generator of 7r4 G/TOP = Z to (1 2 ,1 ) in Z ® Z/2Z = tt3TOP. The calculation with PL in place of O is the same (and follows since 7rf(P L /0) = r f = 0 for / < 6). 6. Simple homotopy theory [44] [46 A]. The main point is that every compact TOP manifold M (with boundary bM) has a preferred simple homotopy type and that two plausible ways to define it are equivalent. Specifically, a handle decomposition o f M or a combinatorial triangulation o f a normal disc-bundle to M give the same simple type. The second definition is always available. Simply embed M in R n , n large, with normal closed disc-bundle E [31]. Theorem 5.1 then provides a small ho­ meomorphism o f R n so that h (b E ), and hence h (E ), is a PL submanifold.

(*) That 7r4 G/TOP is Z (not Z © Z2) is best proved by keeping track of some normal invariants in disproving the Hauptvermutung, see [46 A]. Alternatively, see 13.4 below.

TOPOLOGICAL MANIFOLDS

319

Working with either of these definitions, one can see that the preferred simple type o f M and that o f the boundary dM make o f (M , bM ) a finite Poincare duality space in the sense o f Wall [95], a fact vital for TOP surgery. The Product structure theorem 5.1 makes quite unnecessary the bundle theoretic nonsense used in [44] (cf. [63]) to establish preferred simple types. 7. Handlebody theory (statements in [44 C] [45], proofs in [46 A]). 7.1. The main result is that handle decompositions exist in dimension > 6. Here is the idea o f proof for a closed manifold M m, m > 6. Cover M m by finitely many compacta A J t . . . , A k , each A f contained in a co-ordinate chart U{ « R m. Suppose for an inductive construction that we have built a handlebody H C M containing ^4,U . . . CMI+1, / > 0. The Product Structure Theorem shows that H n Ug can be a PL (or DIFF) m-submanifold o f Ug after we adjust the PL (or DIFF) structure on Uv Then we can successively add finitely many handles onto H in Ut to get a handlebody H ' co n ta in in g ^ U . . . U ^ . After k steps we have a handle decomposition o f M. A TOP Morse function on M m implies a TOP handle decomposition (the con­ verse is trivial) ; to see this one uses the TOP isotopy extension theorem to prove that a TOP Morse function without critical points is a bundle projection. (See [12] [82, 6.14] for proof in detail). Topological handlebody theory as conceived o f by Smale now works on the model o f the PL or DIFF theory (either). For the sake of those familiar with either, I des­ cribe simple ways o f obtaining transversality and separation (by Whitney’s method) o f attaching spheres and dual spheres in a level surface. Lem m a 7 .2 . (Transversality). — L et g : R m -► R m , m > 5 , be a STABLE h o ­ meomorphism. In R m, consider R p x 0 and 0 x R q , p -I- q = m, with ‘ideal’ transverse intersection at the origin. There exists an e-isotopy o f g to h : R m R m such that h (R p x 0) is transverse to 0 x R q is the follow ing strong sense. Near each po in t x G h ~ 1(0 x R q) n R p x 0, h differs fro m a translation by at m ost a hom eom orphism o f R m respecting both R p * 0 and 0 x R q.

Furthermore, i f C is a given closed subset o f R m and g satisfies the strong trans­ versality condition on h above fo r points x o f R m near C, then h can equal g near C P roof o f 7.2 — For the first statement e/2 isotop g to diffeomorphism g using Ed Connell’s theorem [14] (or the Concordance-implies-epsilon-isotopy theorem 5.2), then e/2 isotop g using standard DIFF techniques to a homeomorphism t i which will serve as h if C = 0 . The further statement is deduced from the first using the flexibility o f homeo­ morphisms. Find a closed neighborhood C' o f C near which g is still transverse such that the frontier C' misses g- 1 (0 x R q) O (R p x 0) — which near C is a discrete collection o f points. Next, find a closed neighborhood D o f C* also missing g~ \ 0 x R q) O (R p x 0), and 6 : R m -+ (0 , o o ) so that d ( g x , 0 x R q) < 8 (jc) for x in D n (R p x 0). If e : R m “►((), 5 note that any oriented TOP manifold M n is oriented cobordant to a simply connected one M' by a finite sequence o f 0 and 1-dimensional surgeries. But, i o m = 5 ; Z2) = H l ( Ml ;Z 2) = 0 s o M' is smoothable. Hence R s = 0 . For n = 6 we prove P r o p o s it io n

morphism.

11.2. — The characteristic num ber Aw2 : £2froP-* Z2 is an iso­

TOPOLOGICAL MANIFOLDS

323

Proof. — It is clearly non-zero on any non-smoothable manifold Af6 ^ CP3, since w2(Af6) = w2(CP3) ¥= 0, and we will show that such a Af6 exists in 15.7 below. Since £2^° = 0 it remains to prove that Aw2 is injective. Suppose Aw2 (Af6) = 0 for oriented Af6. As we have observed, we can assume Af is simply connected. Consider the Poincare dual D A o f A = A (t M ) in H 2( M6 ; Z 2) = H 2( M6 ;Z)'® Z2 = tt2(Af6) ® Z2 and observe that it can be represented by a locally flatly embedded 2-sphere S C Af6. (Hints : Use [24], or find an immersion of S 2 x R 4 [52] and use the idea of Lemma 7 .1 ) .

Note that A|(Af — S ) = A (M — S ) is zero because A [ x ] = x »DA (the Z2 in­ tersection number) for all x £ H 2(M ;Z 2). Thus Af — S is smoothable. A neighborhood o f S is smoothed, there being no obstruction to this ; and S is made a DIFF submanifold o f it. Let N be an open DIFF tubular neighborhood o f S. Now 0 = Aw2 [Af] = w2 [DA] = w2 [S] means that w 2( t M ) \ S is zero. Hence N = S 2 x R 4. Killing S by surgery we produce Af', oriented cobordant to Af, so that, writing Af0 — M — S 2 x 5 4 , we have Af' = M 0 + B 3 x S 3 (union with boun­ daries identified). Now Af' is smoothable since Af0 is and there is no further obstruc­ tion. As £2^° = 0, Proposition 11.2, is established. P r o p o s it io n 1 1 .3 . —

The characteristic num ber (@ A ) w 2 : £22TOP-► Z 2 is injec­

tive, where @ = S q 1. P roof o f 11.3. ~ We show the (0A) w 2 [Af] = 0 implies Af7 is a boundary. Just as for 11.2, we can assumeM is simply connected. Then 7r2Af = H 2(M ; Z)and we can kill any element o f the kernel o f w2 : H 2(M ; Z ) -* Z2 , by surgery on 2-spheres in Af. Killing the entire kernel we arrange that w2 is injective. We have 0 = (J3A) w 2 [M] = w2 [Dj3A]. So the Poincare dual DP A of j3A is zero as w2 is injective. Now |3A = 0 means A is reduced integral ; indeed /3 is the Bockstein 5 : H 4 (M ;Z 2) ^ H 5( M , Z ) followed by reduction mod 2. But H 5 (Af ; Z ) = H 5 (Af ; Z2), since H 2 (Af ; Z ) = H 2 (Af ; Z2) (both isomorphisms by reduction). Thus |3A = 0 implies 5A = 0, which means A is reduced integral. Hence D A is reduced integral. Since the Hurewicz map 7r.Af -+ H 3(M ; Z ) is onto, DA is represented by an embedded 3-sphere S. Follo­ wing the argument for dimension 6 and recalling 7r20 = 0, we can do surgery on S to obtain a smoothable manifold.

324

L.C. SIEBENMANN

12. Unoriented cobordism (*). Recalling calculations of £2? and table i_________|

4

from Thom [91] we get the following | 5

|

6

7

o

o

o

Z2

R2 < Z 2

Z2

Z2 ® Z2 ® Z2

Z2

z 2 ®z 2

z 2 ®z 2 ®« 7

af3 z z2 jjSTO P/jjSo

=R'


5, with H 3( Mm, Z2) = 0. (Sullivan had a slightly stronger result [88]). Now precisely the same construction produces a periodicity map motopy commutative square

tt'

in a ho­

G/PL —” £24 G/PL (14.1)

I n 4* G/TOP i

£24 G/TOP

The construction uses TOP versions o f simply connected surgery and transver­ sality. Recalling that the fiber o f sp is K (Z2 , 3) we see that Sl4ip is a homotopy equi­ valence. Hence 1r' must be a homotopy equivalence. Thus (tf')-1 ° (£24 5 is homotopic to a PL homeomorphism if H 3(M ; Z2)/Image H 3(M ; Z ) = 0, or equivalently if H 4(M ; Z) has no 2-torsion [88]. Here [M , G/PL] is geometrically interpreted as a group o f normal invariants, represented by suitably equipped degree 1 maps / : M* -► M o f PL manifolds to M , cf. [95]. The relevant theorem o f Sullivan is : (15.2) hence

The Postnikov K-invariants o f G/PL, except fo r the first, are all odd ;

(G/PL)(2) = {K (Z2 ; 2) x ggq2 K (Z(2), 4)> x K (Z2 , 6) x K (Z(2), 8) x K (Z2 , 10) x K (Z(2), 12) x . . . , w/iere Z,0* = Z I — >— >— >••• 1 is Z with — fo r all od d primes p adjoined. This is K2) p [3 5 7 J one o f the chief results of Sullivan’s thesis 1966 [87]. For expositions o f it see [72] [13] [74] [89].

(*) Section 15 (indeed §§10-16) discusses corollaries of ir3(TOP/PL) = Z2 collected in spring 1969. For further information along these lines, the reader should see work of Hollingsworth and Morgan (1970) and S. Morita (1971) (added in proof).

(**) The localisation at 2, A (2) = A ® Z(2) of a space A will occur below, only for countable .//-spaces A such that, for countable finite dimensional complexes X , [X , A ] is an abelian group (usually a group of some sort of stable bundles under Whitney sum). Thus E.H. Brown’s representation theorem offers a space A ^ and map^4 “*>4(2) so that [X >A] ® Z(2) = [X For a more comprehensive treatment of localisation see [89]. The space A « Q is defined similarly.

TOPOLOGICAL MANIFOLDS

329

Sullivan’s argument adapts to prove (15.3)

The Postnikov K-invariants o f G/TOP are all o d d ; Hence

(G/TOP)(2) = K ( Z 2 , 2) x K ( Z (2) , 4 ) x K ( Z 2, 6) x K ( Z (2), 8) x . . . Indeed his argument needs only the facts that (1) TOP surgery works, (2) the signature map Z = n4k(G/TOP) -►Z is x 8 (even for k — 1, by 13.4), and (3) the Arf invariant map Z2 = 7r4ik+2 (G/TOP) Z 2 is an isomorphism. Alternatively (15.2) =* (15.3) if we use f t 4(G/PL) ^ G/TOP from §14. R em ark 15.4. It is easy to see directly that the 4-stage of G/TOP must be K (Z2 ,2 ) x K {Z , 4). For the only other possibility is the 4 stage of G/PL with K-invariant 6Sq2 in H 5( K( Z 2, 2) ,Z ) = Z2. Then the fibration K ( Z 2, 3) = TOP/PL GL/PL -* G/TOP would be impossible. (Hint : Look at the induced map o f 4 stages and consider the transgression onto 5Sq2 ). This remark suffices for many calculations in dimension < 6 On the other hand it is not clear to me that (15.2) =►(15.3) without geometry in TOP. P roof that

Kernel

TOP/PL G/PL the image o f [X , 12G/TOP] in [X , TOP/PL] = H 3( X ; Z2), consists o f the reduced integral cohomology classes. Clearly this is the image o f [X , 12 (G/TOP)(2)] under 12 (G/TOP)(2)

(TOP/PL)(2) = TOP/PL. Now / (2) is integral reduction on the

factor AT(Z(2) ,3 ) o f 12(G/TOP)(2) because 7r4(G/TOP) -►7r3(TOP/PL) is onto, and it is clearly zero on other factors. The result follows. The argument comes from [13] [72]. P roof that Coker(*) = {Image H * ( X ;Z ) + Sq2H 2( X ; Z 2)}. The following lemma is needed. Its proof is postponed to the end. L emma 1 5 .5 .

— The triangulation obstruction A : B TOP -►K (Z2 , 4 ) is an H-map.

Write : A -►B for y : G/PL G/TOP and let 5, and write $ CAT0W), CAT = PL or TOP, for the set o f ^-cobordism classes o f closed CAT m -manifolds M ' equipped with a homo­ topy equivalence / : M f M. (See [95] for details). There is an exact sequence of pointed sets (extending to the left) : . . . ^ [ZAf , G/CAT]

L m+1(n , w x) - $ CAT( M ) - ^ [M , G/CAT]

L m{ it, w ,)

.

M (above)as It is due to Sullivan and Wall [95]. The map v equips each / : Af' a CAT normal invariant. Exactness at § CAT (M ) is relative to an action o f L m+l( ir , w x ) on it. Here L k ( ir , w t) is the surgery group of Wall in dimension k for fundamental group 7r = irxM and for orientation map Wj = Wj (Af) ; ir Z2 . There is a generali­ sation for manifolds with boundary. Since the PL sequence maps naturally to the TOP sequence, our knowledge of the kernel and cokemel of [M , G/PL]

[M , G/TOP]

331

TOPOLOGICAL MANIFOLDS

will give a lot of information about 3, have PL structure. Such a PL structure is unique up to isotopy, since H 3 (CP„ , Z 2) = 0. ] 6. Manifolds homotopy equivalent to real projective space P n. After sketching the general situation, we will have a look at an explicit example o f failure o f the Hauptvermutung in dimension 5. From [54] [94] we recall that, for n > 4, (16.1) [Pn , G/PL] = Z4 © J

*/(G/PL) • Z2

.

1= 6

This follows easily from (15.2). For G/TOP the calculation is only simpler. One gets (16.2) [Pn, G/TOP] = 2

7rf (G/TOP) • Z2

.

1-2

Calculation of ^ PL0P”) = / „ (16 3)

is non-trivial [54] [94]. One gets (for

1)

/ 4 \ G/PL] - [P*, G/TOP] = Z2 ® Z2

,

which sends Z4 onto Z2 = 7r2 G/TOP. R em ark 16.4. — When two distinct elements of ^PL(/>”) ,n > 5, are topologically the same, we know already from 15.1 that their PL normal invariants are distinct since H 3(Pn ;Z 2) is not reduced integral. This facilitates detection o f examples.

332

L.C. SIEBENMANN

Consider the fixed point free involution T on the Brieskom-Pham sphere in Cm+1

V ? - 1 : zd 0+ A + A +

•••*«

=0

, |z|=l

,

given by T (z0, z x , . . . , z m) = (z0, — z x, . . . , — z m). Here d and m must be odd positive integers, m > 3, in order that £ 2m_1 really be topologically a sphere [ 61 ]. As T is a fixed point free involution the orbit space = £ 2m_1 j j [s a DIFF manifold. And using obstruction theory one finds there is just one oriented equivalence n ^m_1 (Recall P°° = K (Z2 , 1 )). Its class in ^>CATCP2m_1) clearly determines the involution up to equivariant CAT isomorphism and con­ versely. Theorem 16.5. — The manifolds 11^, d odd, fall into fo u r diffeom orphism classes according as d = 1 ,3 , 5 ,7 m od 8, and into two hom eom orphism classes accor­ ding as d = ± 1 , ± 3 m od 8. 11^ is diffeom orphic to P 5. Rem ark 16.6. — With Whitehead C 1 triangulations, the manifolds 11^ have a PL isomorphism classification that coincides with the DIFF classification (§5, [9] [64]). Hence we have here rather explicit counterexamples to the Hauptver­ mutung. One can check that they don’t depend on Sullivan’s complete analysis of (G/PL)(2). The easily calculated 4-stage suffices. Nor do they depend on topological surgery. Pr o b l e m .

— Give an explicit homeomorphism P s « II7 .

Rem ark 16.7. — Giffen states [23] that (with Whitehead C 1 triangulations) the manifolds U 2dm~1 , m = 5 , 7 , 9 , . . . fall into just four PL isomorphism classes d = 1 , 3 , 5 , 7 mod 8. In view of theorem 16.4., these classes are already distin­ guished by the restriction of the normal invariant to P 5 (which is that of n^). So Giffen’s statement implies that the homeomorphism classification is d = ± 1 , ± 3 mod 8. P roof o f 16.5.

(**)

The first means of detecting exotic involutions on 5 5, was found by Hirsch and Milnor 1963 [32]. They constructed explicit(*) involutions (Afjr-i >ft-)> r an integer > 0, on Milnor’s original homotopy 7-spheres, and found invariant spheres 2r_i 3 ^ 2r-i D 2r—i • They observed that the class o f in T2g /2 T 2g is an invariant of the DIFF involution (Af2r_, ,j3r) — (consider the suspension opera­ tion to retrieve (M2r_ 1 ,0 ) and use T6 = 0). Now the class of M \r_ x in Z2g = T7 is r(r — l)/2 according to Eells and Kuiper [18], which is odd iff r = 2 or 3 mod 4. So this argument shows (M2r_, , /3r) is an exotic involution if r = 2 or 3 mod 4 . Fortunately the involution (M s2r_ x ,/3r) has been identified with the involution

(^2r +l >T).

(*) j3r is the antipodal map on the fibers of the orthogonal 3-sphere bundle M\ r_ j . (**) See major correction added on pg. 337.

333

TOPOLOGICAL MANIFOLDS

There were two steps. In 1963 certain examples (X s , a r) o f involutions were given by Bredon, which Yang [101] explicitly identified with W ^r-i >ft-)* Bredon’s involutions extend to 0 (3 ) actions, otr being the antipodal involution in 0 (3 ). And for any reflection cc in 0 ( 3 ) , ot has fixed point set diffeomorphic to L 3(2r + 1 , 1 ) : + z \ + z \ = 0 ; \z | = 1. This property is clearly shared by (S^r+i ,7"), and Hirzebruch used this fact to identify (E^r+i , T ) to (X, ar ) [33, §4] [34], The Hirsch-Milnor information now says that II* is DIFF exotic if cl = 5,7 mod 8. Next we give a TOP invariant for n d in Z2 . Consider the normal invariant vd of Ud in [Ps, G/ 0] = Z4 . Its restriction vd \P2 to P 2 is a TOP invariant because [ P \ G/O] = [P2 , G/TOP] = Z2 . Now Giffen [22] shows that vd \P2 is the Arf invariant in Z2 of the framed fiber of the torus knot zjj + z 2 = 0 , |z0|2 + |z x\2 = 1 in S 3 C C2 . This turns out to be 0 for d = ± 1 mod 8 and 1 for d = ± 3 mod 8, (Levine [53], cf. [61, § 8]). We have now shown that the diffeomorphism and homeomorphism classifica­ tions of the manifolds 11^ are at least as fine as asserted. But there can be at most the four diffeomorphism classes named, in view o f 16.3. (Recall that the PL and DIFF classifications coincide since Tf = ^ (PL/O) = 0 , / < 5). Hence, by Remark 16.4, there are exactly four - two in each homeomorphism class.

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Correction to p ro o f o f 16.5 : Glen Bredon has informed me that [101] is incorrect, and that in fact (X5 ,a r) can be identified to (M2r+ j , j8r + 1 ) . Thus, a different argument is required to show that the DIFF manifolds , d = 1, 3, 5, 7 mod 8 , respectively, occupy the four distinct diffeomorphism classes of DIFF 5-manifolds homotopy equivalent to P5 . The only proof of this available in 1975 is the one provided by M. F. Atiyah in the note reproduced overleaf. So many mistakes, small and large, have been committed with these involutions that it would perhaps be wise to seek several proofs.

338

Note on involutions

by M. F. Atiyah This is an addendum to Atiyah-Bott “A Lefschetz fixed-point formula for elliptic complexes: II” . Let X be a closed spin-manifold of dimension 4k + 2 and let T : X -»• X be an involution preserving orientation and spin-structure. /\ Then T can be lifted to a map T of the principal spin bundle of X . The Lefschetz number Spin(T , X) €E Z[i] is then defined. Assume now that T is of odd type, so that T = - 1 (as explained in Atiyah-Bott p. 488, this happens if T has an isolated fixed-point) . In this case the eigenvalues of T on the harmonic spinors are ±i (the eigenvalues ± 1 aj do not occur since T = —1 ) . Let a,b denote the multiplicity of +i, —i respectively in H+- H (the spaces of harmonic spinors). Then Spin(T , X ) = ( a - b ) i , while the index of the Dirac operator = a+b . Since dimX = 4k + 2 this index = A ( X ) = 0 , so a = ~b , hence Spin(T ,X) = 2ai . Applying this to the exotic Brieskorn involutions we pick up an extra factor of 2 in Theorem 9.8 [in fact this factor of 2 is already incorporated (by mistake) in the statement of 9.8 —the proof gives 22m_l (see the last line of paper on p. 490) ] . A

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[V]

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353

IN D E X * Many definitions of very standard concepts (manifolds, imbeddings, etc.) are given in Essay I, § 2 . They are not all indicated here. Concordance: —of CAT structures on a manifold 3 ,1 2 , 24; — of CAT structures on a microbundle 162 (stable 163); — of microbundles 203; bc a t 19 1 sliced — 1 2 ; sliced A-parametered - 73 ®CAT 236 Concordance Extension Theorem 3 ,2 6 Block transverse, 98, 102 Concordance Implies Isotopy Theorem Blocked equivalence (o f 3 ,2 5 ,3 1 6 s-decompositions) 130 Concordance Implies Sliced Concordance Bundle theorem 7 0 ,2 1 7 Lemma 110 T to p 233 Conditioned (hom otopy, concordance, CAT 10 etc.) 1 1 ,7 6 CAT(£) , CAT(£ ) 221 Corners 7, 8, 29, 119 Corner set 8 C A T r(M ) 2 2 1 Critical point 108 CATm (A) 224 css 220 C A T (£n earA ) 234 dem 146 CATc1s(£) 2 3 4 ,2 3 8 DIFF 6 Cairns-Hirsch Theorem 37 DIFF transversality theorem 85 Cardinal fact (about STABLE structures) DIFF triangulation 9 139 Differential 221 Casson-Sullivan periodicity 327, 278 A-set 278 Chart 7, product - 59, submersion Elementary Engulfing Lemma 257 59 Elementary expansion 117 Classical smoothing theory 156, 209, Embed = imbed 6 cf. 169-171 Expansion 118 Classification theorems for CAT structures 155, 176, 194, 313 Farrell’s Fibration Theorem 211 Clean collaring 40 Finiteness of compact types 301, 123, 117 Clean submanifold 12 Cobordism: oriented — 322, unoriented Fully normal space 46 - 324, spin — 325, relative - 19 General position 147-149 Cofibration 127 Generalized Klein bottle 284 Collapsing Lemma 256 Geometric realisation (o f a css set) 222 Collaring theorems 40-42 Gradient like field 112 Composed microbundle 203 G/CAT 1 0 3 ,2 6 5 -2 6 7 (see G/TOP) Compressible (neighborhood) 66 G/TOP 102, 103, 273ff, 318, 326, 327ff Annulus Conjecture 291 Approximation (for majorant topology) 46-54

7 J.-C. Hausmann, J. Polevoi, and L. S., 1976.

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Handle 1 5 ,1 0 4 ; — Lemma 74; — Problem 15 Handlebody 104; — decomposition 104; — Theorem 104 Hauptvermutung 213, 292, 299, 308 Hole 257 Hom otopy 11; support of — 11; conditioned — 11 Homotopy-CAT structure 265 Hom otopy theorem for microbundles 161 Hom otopy tori 208, 264ff, 275, 287, 291 Hom otopy real projective space 331, 338 Imbed = embed 6 Immersion theoretic method 220-230, 256-263 Involutions 3 3 1 -3 3 3 ,3 3 7 ,3 3 8 Immersion of (Tn - point) in Rn 43 Isotopy 11; — approximation theorem 78; — extension lemma 206; microbundle — 203 Kan fibration 223 Kister-Mazur Theorem 15 9 k-LC imbedded ( = k-LCC ) 146 k-smooth homeomorphism 198 LCk imbedded 146 Lm(7T) 266, Z.(X) , L(7T) 2 6 9 ,2 7 8 Local collaring 40 (uniqueness of — 40) Local finiteness theorem 123 Locally CAT transverse 90 Locally flat submanifold 13 Locally tame 143 Main diagram 16, 297, 302 Majorant topology: target — 46; source - 52 Microbundle 10, 159; composed — 203; concordance of — 203; —hom otopy 162; — isotopy 203; — map 161; reduction of — 162; rooted — 182 Microflexible 260 Micro-identical 159 Micro-isomorphism 159

Morlet’s Theorem 241 Morse function 108 A/(X) 265, N(X) 279 Normal (cobordism, invariant, map) 265 Normal microbundle 160, 203-208 Periodicity theorem: - o f Wall 266-7; —for G/TOP 277, structure — 283 Pinching Lemma 86 PL 6 Poly-Z of rank n 284 Product along P near a point (or a subset) 1 0 , 1 1 Product Structure Theorem 3 ,3 1 ,3 1 5 ; local version 36 Proper map 47 Pseudo-group on a space 7 Pull back rule 166 Punctured torus 43 Reduction of micro bundle structure 162, 220; - of bundle structure 120, 187 Regular hom otopy (o f open immersions) 147 rel = fixing a neighborhood of 11 Relative cobordism 19 Rohlin’s signature theorem 210 Rooted microbundle 182 §CA T ^)

anc^ ^ C A T ^ re^ ^ > 2 o )

164 S ( X ) 264, S Q(X) 264, S(X ) 282 s-cobordism Theorem 4; non-compact — 19 s-decomposition 128; block equivalence of — 130; complete — 130; TOP — 131 Sharply relative 161 Sheaf of quasi-spaces 259; — o f sets 48 Shortcut 65 Simple space 118 Simple type 118 (recent theorems on 117) Simple type of a TOP manifold 117, 119, 126ff, 301, 318

Index

Sliced 12 , 57 Sliced concordance 12 Sliced Concordance Theorem 73; Alternate - 74; Variant of - 77 Sliced Concordance Extension Theorem 77 Sliced ( A-parametered) concordance 73 Smoothed (handle which can be) 15 Smoothing Lemma (for open immersions) 149 Smoothing rule 167

355

Sullivan-Wall exact sequence 269 Surgery 113 ( § 3 .4 ) , 139, 145, 151, 152, 264-289 T2 ( = Hausdorff) 69 Tangent microbundle 163 Technical Bundle Theorem 60 TOP 6 TOP/CAT 192, 196, 200ff, 246ff, 317ff TOP/CAT(£) , TO P/CA T(£relC ,7?) 163

Torsion: topological invariance o f — 117, 119, 126ff, 301, 318 Spaced simple block decomposition: (see Transverse (to a micro bundle) 84, s-decomposition) 128 imbedded — 91 Spine 107 Transversality Lemma for surgery 151 STABLE homeomorphism 3 3 , 1 3 9 , 2 9 1 Transversality theorem: First - 85; STABLE manifold 5 , 1 3 9 , 1 4 0 D IF F — 85; TOP imbedded micro­ bundle — 91; PL imbedded micro­ Stable normal microbundles (stable bundle — 94; PL imbedded block existence and stable uniqueness) 204 ff bundle - 98; General - 100; Stability (of solvability of handle problem) General —(for map) 102; — for problem) 23 surgery 145ff Stability theorem (for 7Tjc(TOPn/CATn) Triangulation Conjecture 213, 292, 299, 192, 247, 3 0 5 , 3 1 7 3 0 8 ff; unrestricted — 312; Stably microbundle transverse ( $ ) unrestricted - for manifolds 10 100, 102 V Unbending corners 119 Stanko hom otopy dimension of imbedding U -near 46 146 Wandering point 2 31 Straightening handles 15 Weak stability lemma 192 Straightening polyhedra in codimension Whitehead compatible 10 > 3 143 Whitehead triangulation 9 Straightening corners 119 Windowblind Lemma 35 Strongly flexible 232

Structure: CAT manifold - 7; CAT microbundle — 220 Submersion 10, — chart 59, — of the punctured torus 43

ANNALS OF MATHEMATICS STUDIES Edited by Wu-chung Hsiang, John Milnor, and Elias M. Stein 88. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, by R o bion C. K irby and L a u r e n c e C. S ie b e n m a n n 87. Scattering Theory for Automorphic Functions, by P e ter D. L ax and R al ph S. P h il l ips 86. 3 -Manifolds, by J o h n H e m p e l 85. Entire Holomorphic Mappings in One and Several Complex Variables, by P h il l ip A . G r iff it h s 84. Knots, Groups, and 3-Manifolds, edited by L. P. N e u w ir t h 83. Automorphic Forms on Adele Groups, by S t e p h e n S. G elbar t 82. Braids, Links, and Mapping Class Groups, by J oan S. B ir m a n 81. The Discrete Series of GL„ over a Finite Field, by G eorge L usztig 80. Smoothings of Piecewise Linear Manifolds, by M. W. H irsch and B. M a zu r 79. Discontinuous Groups and Riemann Surfaces, edited by L eo n G reen ber g 78. Strong Rigidity of Locally Symmetric Spaces, by G. D. M o sto w 77. Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics, by J u r g e n M oser 76. Characteristic Classes, by J o h n W. M iln o r and J a m e s D. S t a s h e ff 75. The Neumann Problem for the Cauchy-Riemann Complex, by G. B. F o lland and J. J. K o h n 74. Lectures on p-Adic L-Functions by K e n k ic h i I w asaw a 73. Lie Equations by A n t o n io K u m p e r a and D onald S p e n c e r 72. Introduction to Algebraic K-Theory, by J o h n M iln o r 71. Normal Two-Dimensional Singularities, by H e n r y B. L a u fe r 70. Prospects in Mathematics, by F. H ir z e b r u c h , L ars H o r m a n d e r , Jo h n M il n o r , Je a n P ier re S e r r e , and I. M. S in ger 69. Symposium on Infinite Dimensional Topology, edited by R. D. A n d e r so n 68. On Group-Theoretic Decision Problems and Their Classification, by C harles F. M il l e r , III 67. Profinite Groups, Arithmetic, and Geometry, by S t e p h e n S. S h a t z 66. Advances in the Theory of Riemann Surfaces, edited by L. V . A h l f o r s , L. B e r s , H. M. F arkas , R. C. G u n n in g , I. K ra , and H. E. R a u ch 65. Lectures on Boundary Theory for Markov Chains, by K ai L ai C h u n g 64. The Equidistribution Theory of Holomorphic Curves, by H u n g -H si W u 63. Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, by E lias M. S t e in 62. Generalized Feynman Amplitudes, by E. R. S pe e r 61. Singular Points of Complex Hypersurfaces, by J o h n M iln o r 60. Topology Seminar, Wisconsin, 1965, edited by R. H. B ing and R. J. B ean 59. Lectures on Curves on an Algebraic Surface, by D avid M u m f o r d 58. Continuous Model Theory, by C. C. C h ang and H. J. K eisler 56. Knot Groups, by L. P. N e u w ir t h 55. Degrees of Unsolvability, by G. E. S acks 54. Elementary Differential Topology (Rev. edn., 1966), by J. R. M u n k r e s 52. Advances in Game Theory, edited by M. D r e sh e r , L. S h a p l e y , and A. W. T uc ker 51. Morse Theory, by J. W. M il n o r 50. Cohomology Operations, lectures by N. E. S t e e n r o d , written and revised by D.B.A. E p s t e in

48. Lectures on Modular Forms, by R. C. G u n n in g 47. Theory of Formal Systems, by R. S m u l l y a n A complete catalogue of Princeton mathematics and science books, with prices, is available upon request. P R IN C E T O N U N IV E R S IT Y PR E SS P r in c e t o n , N e w J e r se y 08540

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Kirby, Robion C 1938— Foundational essays on topological manifolds, smoothings, and triangulations. (Annals of mathematics studies; no. 88) Bibliography: p. Includes index. 1. Manifolds (Mathematics)—Addresses, essays, lectures. 2. Piecewise linear topology—Addresses, essays, lectures. 3. Triangulating manifolds—Addresses, essays, lectures. I. Siebenmann, L., joint author. II. Title. III. Series. QA613.K57 51477 76-45918 ISBN 0-691-08190-5 ISBN 0-691-08191-3 pbk.

9780691081915 '78069