In this book, which may be used as a self-contained text for a beginning course, Professor Lefschetz aims to give the re

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*Table of contents : PrefaceContentsIntroduction, a Survey of Some Topological Concepts 1. Theory of Sets. Topological Spaces 2. Questions Related to Curves 3. Polyhedra 4. Coincidences and Fixed Points 5. Vector Fields 6. Integration and TopologyChapter I. Basic Information about Sets, Spaces, Vectors, Groups 1. Questions of Notation and Terminology 2. Euclidean Spaces, Metric Spaces, Topological Spaces 3. Compact Spaces 4. Vector Spaces 5. Products of Sets, Spaces and Groups. Homotopy ProblemsChapter II. Two-dimensional Polyhedral Topology 1. Elements of the Theory of Complexes. Geometric Consideration 2. Elements of the Theory of Complexes. Modulo Two Theory 3. The Jordan Curve Theorem 4. Proof of the Jordan Curve Theorem 5. Some Additional Properties of Complexes 6. Closed Surfaces. Generalities 7. Closed Surfaces. Reduction to a Normal Form ProblemsChapter III. Theory of Complexes 1. Intuitive Approach 2. Simplexes and Simplicial Complexes 3. Chains, Cycles, Homology Groups 4. Geometric Complexes 5. Calculation of the Betti Numbers. The Euler-Poincaré Characteristic 6. Relation between Connectedness and Homology 7. Circuits ProblemsChapter IV. Transformations of Complexes. Simplicial Approximations and Related Questions 1. Set-transformations. Chain-mappings 2. Derivation 3. The Brouwer Fixed Point Theorem 4. Simplicial Approximation 5. The Brouwer Degree 6. Hopf's Classification of Mappings of ?-spheres on n-spheres 7. Some Theorems on the Sphere ProblemsChapter V. Further Properties of Homotopy. Fixed Points. Fundamental Group. Homotopy Groups 1. Homotopy of Chain-mappings 2. Homology in Polyhedra. Relation to Homotopy 3. The Lefschetz Fixed Point Theorem for Polyhedra 4. The Fundamental Group 5. The Homotopy Groups ProblemsChapter VI. Introduction to Manifolds. Duality Theorems 1. Differentiable and Other Manifolds 2. The Poincar6 Duality Theorem 3. Relative Homology Theory 4. Relative Manifolds and Related Duality Theory (Elementary Theory). Alexander's Duality Theorem ProblemsBibliographyList of SymbolsIndex*

INTRODUCTION TO TOPOLOGY

PRINCETON MATHEMATICAL SERIES

Editors: MABSTON MOKSB1 H. P. ROBERTSON, A. W. TUCKER 1. The Classical Groups, Their Invariants and Representations. By HERMANN WEYL. 2. Topological Groups. By L. PONTRJAGIN. Translated by EMMA LEHMER.

3. An Introduction to Differential Geometry with Use of the Tensor Calculus. By LUTHER PFAHLER EISENHART. 4. Dimension Theory. By WITOLD HUREWICZ and HENRY WALLMAN. 5. The Analytical Foundations of Celestial Mechanics. By ATJREL WINTNER.

6. The Laplace Transform. By DAVID VERNON WIDDER. 7. Integration. By EDWARD JAMES MCSHANE. 8. Theory of Lie Groups: I. By CLAUDE CHEVALLEY. 9. Mathematical Methods of Statistics. By HARALD CRAMER. 10. Several Complex Variables. By S. BOCHNER and W. T. MARTIN. 11. Introduction to Topology. By SOLOMON LEFSCHETZ.

INTRODUCTION TO TOPOLOGY

By SOLOMON LEFSGHETZ

PRINCETON, NEW JERSEY · 1949

PRINCETON UNIVERSITY PRESS

Copyright, 1949, by Princeton University Press PRINTED IN THE UNITED STATES OF AMERICA

LONDON: GEOFFREY CUMBERLEGE OXFORD UNIVERSITY PRESS

Preface The present work originated from a short course delivered in 1944 before the Institute of Mathematics of the National University of Mexico. Lectures on a number of the topics dealt with in the book were also given from time to time at Princeton University. Generally the auditors had ample training in algebra and general topology but little in algebraic topology. While this topic was to some extent emphasized in the lectures, it was found advisable to provide considerable back ground material. This has also been the author's purpose throughout the present volume. The main topic presented here is the topology of polyhedra, and its treatment begins properly with the third chapter. The indispensable preliminary survey is meant to give a bird's eye view of topology with out much regard to the specific subjects presented later. The first chap ter, on foundations, contains practically no proofs. Chapter II stands apart from the main text. It is selfcontained and deals only with dimen sions at most 2 and chains mod 2. A surprising number of the basic concepts shows up already at that early stage so that the chapter is an excellent introduction to the rest of the book. Here and there throughout the volume, but above all in the first chapter, proofs of certain propositions are omitted, a fact indicated by an asterisk before the statements. A sparse bibliography will be found at the end of this book. The references are given by the author's name followed by a specific indica tion in square brackets (the caps refer to books and the lower case letters to papers). From Chapter II on, and except for a few problems, the only spaces considered are polyhedra. Armed with this knowledge the reader will be well prepared for the more advanced and more complete treatises. The author wishes to express here his grateful appreciation for many valuable suggestions to Dr. H. F. Tuan, of the National Tsing Hua University, to Mr. Jaime Lifshitz, of the National University of Mexico, who read the manuscript, and to Dr. Ε. E. Floyd and Mr. H. W. Kuhn who read the galleys. S. LEFSCHETZ

Contents Preface

ν

Introduction, a Survey of Some Topological Concepts 1. 2. 3. 4. 5. 6.

...

Theory of Sets. Topological Spaces Questions Related to Curves Polyhedra Coincidences and Fixed Points Vector Fields Integration and Topology

3 3 5 8 14 17 19

Chapter I. Basic Information about Sets, Spaces, Vectors, Groups 1. 2. 3. 4. 5.

Questions of Notation and Terminology Euclidean Spaces, Metric Spaces, Topological Spaces. Compact Spaces Vector Spaces Products of Sets, Spaces and Groups. Homotopy. Problems

.

26 28 34 38 40 43

Chapter II. Two-dimensional Polyhedral Topology 1. Elements of the Theory of Complexes. Geometric Consider ation 2. Elements of the Theory of Complexes. Modulo Two Theory. 3. The Jordan Curve Theorem 4. Proof of the Jordan Curve Theorem 5. Some Additional Properties of Complexes 6. Closed Surfaces. Generalities 7. Closed Surfaces. Reduction to a Normal Form. Problems

45 50 61 65 68 72 73 84

Chapter III. Theory of Complexes 1. 2. 3. 4.

Intuitive Approach Simplexes and Simplicial Complexes Chains, Cycles, Homology Groups Geometric Complexes vii

86 87 89 95

viii

CONTENTS

5. Calculation of the Betti Numbers. The Euler-Poincar^ Char acteristic 99 6. Relation between Connectedness and Homology. . . . 103 7. Circuits 105 Problems 107 Chapter IV. Transformations of Complexes. Simplicial Approxi mations and Related Questions 1. 2. 3. 4. 5. 6. 7.

Set-transformations. Chain-mappings 110 Derivation 112 The Brouwer Fixed Point Theorem 117 Simplicial Approximation 119 The Brouwer Degree 124 Hopf's Classification of Mappings of η-spheres on n-spheres. 132 Some Theorems on the Sphere 134 Problems 140

Chapter V. Further Properties of Homotopy. Fixed Points. Fun damental Group. Homotopy Groups 1. 2. 3. 4. 5.

Homotopy of Chain-mappings Homology in Polyhedra. Relation to Homotopy. . The Lefschetz Fixed Point Theorem for Polyhedra. . The Fundamental Group The Homotopy Groups Problems

. .

. .

142 148 153 157 170 180

Chapter VI. Introduction to Manifolds. Duality Theorems 1. 2. 3. 4.

Differentiable and Other Manifolds 183 The Poincar6 Duality Theorem 188 Relative Homology Theory 195 Relative Manifolds and Related Duality Theory (Elemen tary Theory). Alexander's Duality Theorem 202 Problems 206

Bibliography

208

List of Symbols

211

Index

213

INTRODUCTION TO TOPOLOGY

Introduction: A Survey of Some Topological Concepts The concepts which it is proposed to examine in this survey are above all those related to algebraic topology. Little regard will be paid to historical matters as the existing and very accessible bibliographies make it easy to trace most questions under discussion to their sources.

1. THEORY OF SETS. TOPOLOGICAL SPACES. Topology begins where sets are implemented with some cohesive properties enabling one to define continuity. The sets are then called topological spaces. Thus topology is a branch of general set theory, the creation of Georg Cantor (around 1880). That his ideas have had a profound influence in all mathematics is well known. In topology this influence has been decisive. The cohesion giving rise to continuity may be obtained in various ways, notably by means of limits, closure relations, distances, or finally by the assignment of a collection of subsets: the open sets, or the closed sets, satisfying certain structural axioms. The choice of axioms is dic tated by the behavior of the open or closed subsets of Euclidean spaces. For the choice made must be such that among the selected spaces are included the classical spaces. The topological spaces of various kinds began to come into their own through Hausdorff's classical treatise: Grundzuge der Mengenlehre (1912). The most important topological spaces are the metric spaces first introduced in their full generality by Fr^chet (Paris thesis 1906), together with a number of other important notions, notably the notion of compactness. A discussion of these con cepts will be found in Chapter I. Once continuity is solidly established one may define continuous transformations or functions, henceforth called mappings. One may also define therefore bicontinuous transformations, also called topolog ical transformations or homeomorphisms. Whatever such transforma tions maintain invariant constitutes the subject matter of topology proper. In many questions, notably in algebra and in analysis, one has occaS

4

TOPOLOGICAL CONCEPTS

sion to topologize sets of some sort and sometimes even in different ways for the same set. InparticularagroupG= { c(g) from some Fi. *(11.2) Corresponding to any % there is a positive number ¢2(¾), called the Lebesgue number of 5, such that if A C SR and diam A < d(%) and A meets a collection of sets of 5 then the sets of the collection intersect. *(11.3) If U is a covering there is a corresponding positive number dx(U), called the Lebesgue number of U, such that: (a) every point χ of St is in some Ui and at a distance > di(U) from its complement 8Ϊ — £/,·; (b) if A C 9ί and diam A < di(U), then A is contained in some set Ui-

88

SETS, SPACES, VECTORS, GROUPS

[CHAP. I

§4. VECTOR SPACES. 12. Due to their simplicity these spaces will be extensively utilized in connection with homology. However, before taking them up we shall say a few words regarding groups and fields. Groups. We shall merely mention that unless otherwise stated all groups are assumed additive, i.e. commutative, with addition as the group operation, and unity written 0. A subgroup G' of a group G is a subset of G such that if g\, g2 are in G' so is their difference g\ — g'2. If G, H are two groups, then a homomorphism τ : G —»H is a transforma tion such that rg — rg' = r(g — g'). The inverse G' = τ-1O is a sub group of G called the kernel of τ. If τ is one-one it is known as an iso morphism, of G with H. Let G be a group and H a subgroup. By a coset of G relative to H (also called a coset of G mod H) we understand a subset II' of Gy such that there exists a fixed g GC? such that H' = {g + h\h £1 H}. Two cosets are clearly either disjoint or coincident. The cosets of G relative to H from a group called the factor-group of G mod H, written G/H. The transformation χ :G—> G/H sending each element of G into its coset is a homomorphism, called the projection of G into G/H, and its kernel is H. We shall have repeated occasion to deal with the following question: Given two groups G,H and a homomorphism τ : G —»Η, to prove that τ is an isomorphism. This will always be done by reference to one of the following two criteria: *(12.1) If τ is univalent and onto then it is an isomorphism. ""(12.2) If there exist homomorphisms 0,0' : H —> G such that θτ = 1, τ0' = 1 then τ is an isomorphism and θ = θ' = τ-1. It may be observed that both criteria are also valid for multiplicative (i.e. perhaps noncommutative) groups. Fields. We recall that a field is merely a collection of elements Ω = {α} subject to the same rules of rational operations as ordinary (real or complex) numbers. Examples of fields are: (a) the set of all complex numbers; (b) the set of all real numbers; (c) the set of all ra tional numbers; (d) the set of all (integral) residues modulo a prime number v. This last field is finite and consists of τ elements. The two fields (c) and (d) will be particularly important in dealing with the homology groups. 13. Vector spaces. A vector space V over a field Ω is an additive group {ζ} such that there is defined a product ax for every pair (α,χ), α G Ω, iCF, with values in V, and with the properties:

§ 4]

VECTOR SPACES

39

When 0 is the real or the rational field or the set of residues mod x, V is said to be real or rational or mod w. Example. Let be n arbitrary symbols and define as a vector any linear form with addition in the obvious way and the conventions

Then is a vector space over When is the field of the residues mod x the vector space just defined is finite and consists of elements. Let be a vector space over The elements of V are said to be linearly dependent if there exists a relation with the not all zero; in the contrary case the Xi are said to be linearly independent. If there is a finite maximum n of linearly independent elements we say that n is the dimension of V, written dim V. When no finite maximum exists we merely say that the dimension of V is infinite. The field 12 itself is a one-dimensional vector space. Suppose n = dim V is finite and let be independent elements. If x is any other element whatsoever we have a relation

with the not all zero. We cannot have a = 0 since then the £» would be linearly dependent. Thus and so if we set we have (13.1) Moreover, this representation of x is unique. For suppose

where the are not all zero. Since the would be linearly dependent. Thus every element x is represented in one and only one way in the form (13.1) and all expressions (13.1) represent elements of V. For this reason is called a base for V. Example. In the first example of the present number the {&} form a base and so the dimension is n. Linear transformations. Isomorphism. If V,W are vector spaces over Q and if

O is a two-cell.

§5. THE HOMOTOPY GROUPS. 20. A central problem of topology is the determination of the homotopy classes of mappings. An important step toward its solution would be the solution for mappings of polyhedra into one another. It is not difficult to show that this problem depends upon the possibility of extending a mapping Φ from the bases of a prism I X \K\ to the cells of successive dimensions, and hence from the boundary sphere of a cell I X σρ to the cell. In short, it would seem desirable from this, and other considerations, to solve first the homotopy problem for spheres. The preceding general remarks suggest the extension of the funda mental group to mappings of spheres of all dimensions. The required generalization has been carried out and extensively investigated by W. Hurewicz [a] whose work we follow in the present section. For ref erences on the most recent investigations on this topic, see Eilenberg [a], Pontrjagin [a], Steenrod [a], J. H. C. Whitehead [b], George Whitehead [a]. We shall be dealing primarily with the mappings of a "punctured" oriented η-sphere, that is to say, with an oriented η-sphere Sn together with an assigned point Q of Snt written (S",Q). Unless otherwise stated, it is assumed that η > 1. The case η = O offers no interest and η = 1 corresponds to the fundamental group already treated. The departures from the latter are such that the case η > 1 is altogether dissimilar from the case η = 1. The space into which the mappings are made is assumed arc-wise connected and a pre-assigned point P, called base point, is chosen in 3i. We shall consider mappings a of the punctured sphere (Sn,Q) —> 9t, such that aQ = P. If (S'n,Q') is another such sphere and / is a homeomorphism S'n—>Sn of degree +1 (sense-preserving homeomorphism) such that fQ' = Q, then af is a mapping S'n —> 9ΐ sending Q' into P. We agree to identify a with af and this will enable us to deal only with a fixed punctured Euclidean η-sphere in a fixed orientation. Let a,a' be of the same type as above, I the segment O < u < 1, and suppose that there exists a mapping Φ : I X Sn —> 9ί such that Φ (I X Q) = P and that Φ agrees with a on O X Sn and with a' on 1 X S". Under the circumstances we shall say that a is homotopic to a' relative to P, written a a' rel P. Thus the homotopies here consid ered are subjected to the restriction that the path of Q is P. It is shown

§5]

THE HOMOTOPY GROUPS

171

as for ordinary homotopy that homotopy rel P is an equivalence rela tion and thus gives rise to homotopy classes rel P. If / is as above and a ^ a' rel P, then likewise af ^ a'f rel P and conversely. Hence if we agree to identify a with a' whenever they are homotopic rel P, this will be consistent with the identifications already introduced. We shall utilize the preceding remark to reduce α to a certain simple normal form. To that effect take any point R of Sn — Q and a closed spherical region Ω of center R which does not contain Q. Define now a mapping/ of S n into itself in the following way:/OS n — Ω) = Q,fR = R. If SB_1 is the boundary sphere of Ω and M £ Ω — R, there is a unique plane containing Q,M,R; it intersects Sn in a circumference C with a Sn"

Figure 68

unique arc RMQ. This arc intersects