Algebraic Topology [Reprint 2020 ed.] 9783112318522, 9783112307250

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ALGEBRAIC TOPOLOGY WOLFGANG FRANZ Professor of Mathematics University of Frankfurt-on-Main

Translated from the German by LEO F. BORON University

of Idaho

With the collaboration of SAMUEL

D.

SHORE

University of New JAMES J . HARRY F .

JOINER

ROBERT C . KYOSHI

FREDERICK

University

ISEKI

University

W UNGAR NEW

II

MOORE

The Florida State Kobe

Hampshire

ANDREWS

PUBLISHING

YORK

CO.

Translated from the German original Topologie II: Algebraische

Topologie

By arrangement with Walter de Gruyter & Co., Berlin

Copyright© 1968 by Frederick Ungar Publishing Co., Inc.

Printed in the United States of America

Library of Congress Catalog Card No. 65-28047

CONTENTS INTRODUCTION

1.

Analytic and Algebraic Topology

1

2.

Problems and Examples

5

Part I SIMPLICIAL COMPLEXES Chaper 1.

GEOMETRY OF SIMPLICIAL COMPLEXES

3.

Hulls and Stars

15

4.

Barycentric Stars

18

5.

Simplicial Mappings

19

6.

Neighboring Mappings

25

Chapter 2.

HOMOLOGY GROUPS AND COHOMOLOGY GROUPS

7.

Orientation. Incidence Numbers

29

8.

Homology Groups

31

9.

Examples and Applications

34

10.

Cohomology Groups

42

11.

Homotopic Mappings

48 iii

iv

CONTENTS

Part II CHAIN COMPLEXES AND THEIR APPLICATIONS Chapter 3.

GENERAL THEORY

12.

Homology Groups of Chain Complexes

55

13.

Subcomplexes and Factor Complexes

57

14.

The Boundary Operator

63

Chapter 4.

F R E E CHAIN COMPLEXES

15.

Modules and Dual Modules

65

16.

Mappings and Dual Mappings

67

17.

Free Chain Complexes. Canonical Bases

73

Part

III

CELL COMPLEXES. INVARIANCE Chapter 5.

CELL COMPLEXES

18.

Cell Decompositions

85

19.

The Homology Groups of Cell Decompositions

90

20.

Normal Subdivisions

93

Chapter 6.

INVARIANCE OF T H E HOMOLOGY GROUPS

21.

Proof of Invariance

98

22.

Supplements. Generalizations

101

23.

Results and Applications

105

24.

Local Homology Groups

115

V

CONTENTS

Part IV D E V E L O P M E N T OF T H E

Chapter 7.

THEORY

PRODUCTS IN POLYHEDRA

25.

The Cohomology Ring

123

26.

The Cap Product

132

Chapter 8.

MANIFOLDS

27.

Definitions

137

28.

Complementary Cell Decompositions

142

29.

The Poincaré Duality Theorem

144

Chapter 9.

T H E COHOMOLOGY RING OF A MANIFOLD

30.

Products in Manifolds

154

31.

Product Matrices

159

BIBLIOGRAPHY

165

INDEX

167

INTRODUCTION § 1. Analytic and Algebraic Topology In Volume I, General Topology, we discussed the most general possible classes of spaces and derived their properties deductively from a small number of axioms. One may also therefore designate this branch of topology as "Analytic Topology." Accordingly, the assertions of general topology are very extensive and general, and for that reason relatively non-concrete. In the present Volume I I we treat the other branch of contemporary topology, Algebraic Topology. Here we will deal with more special problems and shall place particular emphasis on concepts and theorems which are closely related to direct geometric intuition. In this connection, we must essentially limit the domain of the basic spaces within our framework: We consider polyhedra almost exclusively. The real difference between the two branches of topology consists in the method. The nomenclature "Algebraic Topology" indicates that algebraic tools are drawn upon for the treatment of topological questions. We can explain even at this point in rough outline the procedure to be used for this. We consider a so-called category 9? which consists of spaces and continuous mappings or, more specifically, of the spaces B, 8, . . . of & fixed class (for instance, of all Hausdorff spaces or of all compacta or as in our case of all polyhedra B, 8,. . .) and of all continuous, not necessarily monomorphic or epimorphic mappings / : B S among them. By means of a procedure which will be developed later, we shall assign to each space B of 91, for each natural number q = 0, 1,. . ., an abelian group Hq(B), the so-

2

INTRODUCTION

called q-dimensional homology group of R. Furthermore, for each q, we shall assign to each continuous mapping f : R - ^ - S a homomorphism : Hq(R) —Hq(S), the /-induced homomorphism /„ of the corresponding homology groups. This double correspondence— groups to spaces and homomorphisms to mappings—has the following properties: (I) If i: R -> R is the identity mapping, then, for each q, it is the identity homomorphism of Hq{R). (II) I f / : R 8 and g: 8 ->T are continuous mappings of spaces in 5R and gof: R is the composition mapping, then (gof)* = R, which maps each point of A into itself, is called the inclusion mapping of A into R; it must of course be distinguished from iA: A —> A.—We first point out the two most important problems for topological spaces, the homeomorphism problem and the homotopy problem.

6

INTRODUCTION

A. The homeomorphism problem. Two spaces R and S are called homeomorphic, in symbols R « 8 (cf. Volume I, Def. 5.1), if there exists a one-to-one bicontinuous function from R onto S which carries over the topological structure of R (for instance, the open sets or the neighborhoods in R) into the topological structure of S. The theorem (Volume I, Theorem 5.6) that R and S are homeomorphic if and only if there exists a one-to-one, that is, monomorphic and epimorphic, mapping / : JR —> $ which is continuous as well as its inverse / - 1 : S R is valid. I f if is a compactum, in particular a polyhedron, then one can omit the last-named requirement on since in this case it follows from the other requirements (Volume I, Theorem 18.2). The homeomorphism problem asks whether or not two given topological spaces R and S are homeomorphic. If we assign the same space type to two homeomorphic spaces, it is therefore to be decided whether R and S belong to the same space type or not. I t is further required to produce, for each space type, a special space R as representative, also called the normal form. For the time being, the homeomorphism problem is unsolved in this form. In the sequel, we shall be able to produce with the aid of homology theory only relatively coarse necessary conditions for the homeomorphism of two spaces. We shall give a solution of the homeomorphism problem including the construction of normal forms only for the very special class of two-dimensional surfaces. B. The homotopy problem 2.1 Definition: Two continuous mappings/, g: R-+S are called homotopic, denoted / ~ g or more precisely / ~ g: R S, if either one of the following two equivalent conditions is satisfied: (1) There exists a family of mappings F(p, t): R S, which are continuous in the two variables pe R and the real number t in the interval 0 f i t 5^1, such that F(p, 0) =f(p) and F(p, 1) = g(p). (2) There exists a continuous mapping F: R x I S with g : R S, then F'(p, t) = F(p, 1 — t) is a homotopy connecting g with / ; it therefore follows from f ~ g that g ~ f.—Now let F(p, t) be as just defined and let 0(p, t) be a homotopy connecting g with h: R -> S. Then, as one can easily verify, the homotopy H(p, t) =

'F(p, 21) for 0 ^ t