Algebraic Topology: Homology and Cohomology

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S Title

Algebraic Topology Homology and Cohomology

Copyright
     © 1970 by W. A. Benjamin, Inc
     ISBN 0-8053-9482-6
     LCCN 79-108005

Copyright Dover
     1970, 1993 by Andrew H. Wallac
     ISBN-I 3: 978-0-486-46239-4
     ISBN-10: 0-486-46239-0
     QA612.W33 2007 514'.2--dc22
     LCCN 2007010621

Preface

Contents

1  Singular Homology Theory

     1-1. EUCLIDEAN SIMPLEXES

     1-2. LINEAR MAPS

     1-3. SINGULAR SIMPLEXES AND CHAINS

     1-4. THE BOUNDARY OPERATOR

     1-5. CYCLES AND HOMOLOGY

     1-6. INDUCED HOMOMORPHISMS

     1-7. THE MAIN THEOREMS

     1-8. THE DIMENSION THEOREM

     1-9. THE EXACTNESS THEOREM

     1-10. THE HOMOTOPY THEOREM

     1-11. THE EXCISION THEOREM

2  Singular and Simplicial Homology

     2-1. THE AXIOMATIC APPROAC

     2-2. SIMPLICIAL COMPLEXES

     2-3. THE DIRECT SUM THEOREM

     2-4. THE DIRECT SUM THEOREM FOR COMPLEXES

     2-5. HOMOLOGY GROUPS OF CELLS AND SPHERES

     2-6. ORIENTATION

     2-7. HOMOLOGY GROUPS OF A SIMPLICIAL PAIR

     2-8. FORMAL DESCRIPTION OF SIMPLICIAL HOMOLOGY

     2-9. CELL COMPLEXES

     2-10. CANONICAL BASES

3  Chain Complexes-Homologyand Cohomology

     3-1. A PAUSE FOR MOTIVATION

     3-2. CHAIN COMPLEXES

     3-3. CHAIN HOMORPHISMS

     3-4. INDUCED HOMOMORPHISMS ON HOMOLOGY AND COHOMOLOGY GROUPS

     3-5. CHAIN HOMOTOPY

     3-6. THE ALGEBRAIC HOMOTOPY THEOREM

     3-7. SOME APPLICATIONS OF ALGEBRAIC HOMOTOPY

     3-8. SUBCOMPLEXES AND QUOTIENT COMPLEXES

     3-9. COMPUTATION OF COHOMOLOGY GROUPS

     3-10. ATTACHING CONES AND CELLS

4  The Cohomology Ring

     4-1. MOTIVATION

     4-2. U PRODUCT FOR SINGULAR COHOMOLOGY

     4-3. DEFINITION OF THE SINGULAR COHOMOLOGY RING

     4-4. SOME EXAMPLES OF COMPUTATIONS

     4-5. TENSOR PRODUCTS AND CUP PRODUCTS

     4-6. TENSOR PRODUCTS OF CHAIN AND COCHAIN COMPLEXES

     4-7. COHOMOLOGY OF A TENSOR PRODUCT

     4-8. TENSOR PRODUCTS AND TOPOLOGICAL PRODUCTS

     4-9. SINGULAR CHAINS ON A PRODUCT

     4-10. SOME ACYCLICITY ARGUMENTS

     4-11. THE DIAGONAL MAP

     4-12. SUMMING UP

     4-13. THE x PRODUCT

     4-14. ANTICOMMUTATIVITY OF THE U PRODUCT

     4-15. THE COHOMOLOGY RING OF A PRODUCT SPACE

     4-16. SOME COMPUTATIONS

5  Cech Homology Theory -- The Construction

     5-1. INTRODUCTION

     5-2. THE NERVE OF A COVERING

     5-3. HOMOLOGY GROUPS OF A COVERING

     5-4. INVERSE LIMITS

     5-5. LIMITS OF HOMOMORPHISMS

     5-6. CONSTRUCTION OF THE CECH HOMOLOGY GROUPS

     5-7. CECH HOMOLOGY FOR SIMPLICIAL PAIRS

6  Further Properties of Cech Homology

     6-1. INTRODUCTION

     6-2. THE HOMOTOPY THEOREM FOR CECH HOMOLOGY

     6-3. THE EXCISION THEOREM IN CECH HOMOLOGY

     6-4. THE PARTIAL EXACTNESS THEOREM

     6-5. THE CONTINUITY THEOREM

     6-6. COMPARISON OF SINGULAR AND CECH HOMOLOGY THEORIES

     6-7. RELATIVE HOMEOMORPHISMS

     6-8. THE UNIQUENESS THEOREM FOR CECH HOMOLOGY

7  Cech Cohomology Theory

     7-1. INTRODUCTION

     7-2. DIRECT LIMITS

     7-3. THE DEFINITION OF CECH COHOMOLOGY GRO UPS

     7-4. INDUCED HOMOMORPHISMS

     7-5. THE EXCISION AND HOMOTOPY THEOREMS

     7-6. THE EXACTNESS THEOREM

     7-7. THE CONTINUITY THEOREM

Appendix A  The Fundamental Group

     A-1. PATHS, THEIR COMPOSITIONS AND HOMOTOPIES

     A-2. GROUP SPACES

Appendix B  General Topology

     B-1. SET THEORY

     B-2. GEOMETRIC NOTIONS

     B-3. TOPOLOGICAL SPACES

     B-4. CONTINUOUS MAPS

     B-5. PRODUCT AND QUOTIENT SPACES

     B-6. COMPACT SPACES

     B-7. CONNECTED SPACES

Bibliography

Index

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