دانلود کتاب Algebraic Topology: Homology and Cohomology

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Cover 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 Back Cover