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代数拓扑学（英文）电子书
Introduction 1
Chapter 1. The fundamental group and some of its applications 5
1. What is algebraic topology? 5
2. The fundamental group 6
3. Dependence on the basepoint 7
4. Homotopy invariance 7
5. Calculations: 1(R) = 0 and 1(S1) = Z 8
6. The Brouwer fixed point theorem 10
7. The fundamental theorem of algebra 10
Chapter 2. Categorical language and the van Kampen theorem 13
1. Categories 13
2. Functors 13
3. Natural transformations 14
4. Homotopy categories and homotopy equivalences 14
5. The fundamental groupoid 15
6. Limits and colimits 16
7. The van Kampen theorem 17
8. Examples of the van Kampen theorem 19
Chapter 3. Covering spaces 21
1. The definition of covering spaces 21
2. The unique path lifting property 22
3. Coverings of groupoids 22
4. Group actions and orbit categories 24
5. The classification of coverings of groupoids 25
6. The construction of coverings of groupoids 27
7. The classification of coverings of spaces 28
8. The construction of coverings of spaces 30
Chapter 4. Graphs 35
1. The definition of graphs 35
2. Edge paths and trees 35
3. The homotopy types of graphs 36
4. Covers of graphs and Euler characteristics 37
5. Applications to groups 37
Chapter 5. Compactly generated spaces 39
1. The definition of compactly generated spaces 39
2. The category of compactly generated spaces 40
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Chapter 6. Cofibrations 43
1. The definition of cofibrations 43
2. Mapping cylinders and cofibrations 44
3. Replacing maps by cofibrations 45
4. A criterion for a map to be a cofibration 45
5. Cofiber homotopy equivalence 46
Chapter 7. Fibrations 49
1. The definition of fibrations 49
2. Path lifting functions and fibrations 49
3. Replacing maps by fibrations 50
4. A criterion for a map to be a fibration 51
5. Fiber homotopy equivalence 52
6. Change of fiber 53
Chapter 8. Based cofiber and fiber sequences 57
1. Based homotopy classes of maps 57
2. Cones, suspensions, paths, loops 57
3. Based cofibrations 58
4. Cofiber sequences 59
5. Based fibrations 61
6. Fiber sequences 61
7. Connections between cofiber and fiber sequences 63
Chapter 9. Higher homotopy groups 65
1. The definition of homotopy groups 65
2. Long exact sequences associated to pairs 65
3. Long exact sequences associated to fibrations 66
4. A few calculations 66
5. Change of basepoint 68
6. n-Equivalences, weak equivalences, and a technical lemma 69
Chapter 10. CW complexes 73
1. The definition and some examples of CW complexes 73
2. Some constructions on CW complexes 74
3. HELP and the Whitehead theorem 75
4. The cellular approximation theorem 76
5. Approximation of spaces by CW complexes 77
6. Approximation of pairs by CW pairs 78
7. Approximation of excisive triads by CW triads 79
Chapter 11. The homotopy excision and suspension theorems 83
1. Statement of the homotopy excision theorem 83
2. The Freudenthal suspension theorem 85
3. Proof of the homotopy excision theorem 86
Chapter 12. A little homological algebra 91
1. Chain complexes 91
2. Maps and homotopies of maps of chain complexes 91
3. Tensor products of chain complexes 92
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4. Short and long exact sequences 93
Chapter 13. Axiomatic and cellular homology theory 95
1. Axioms for homology 95
2. Cellular homology 97
3. Verification of the axioms 100
4. The cellular chains of products 101
5. Some examples: T, K, and RPn 103
Chapter 14. Derivations of properties from the axioms 107
1. Reduced homology; based versus unbased spaces 107
2. Cofibrations and the homology of pairs 108
3. Suspension and the long exact sequence of pairs 109
4. Axioms for reduced homology 110
5. Mayer-Vietoris sequences 112
6. The homology of colimits 114
Chapter 15. The Hurewicz and uniqueness theorems 117
1. The Hurewicz theorem 117
2. The uniqueness of the homology of CW complexes 119
Chapter 16. Singular homology theory 123
1. The singular chain complex 123
2. Geometric realization 124
3. Proofs of the theorems 125
4. Simplicial objects in algebraic topology 126
5. Classifying spaces and K(, n)s 128
Chapter 17. Some more homological algebra 131
1. Universal coefficients in homology 131
2. The K¨unneth theorem 132
3. Hom functors and universal coefficients in cohomology 133
4. Proof of the universal coefficient theorem 135
5. Relations between
and Hom 136
Chapter 18. Axiomatic and cellular cohomology theory 137
1. Axioms for cohomology 137
2. Cellular and singular cohomology 138
3. Cup products in cohomology 139
4. An example: RPn and the Borsuk-Ulam theorem 140
5. Obstruction theory 142
Chapter 19. Derivations of properties from the axioms 145
1. Reduced cohomology groups and their properties 145
2. Axioms for reduced cohomology 146
3. Mayer-Vietoris sequences in cohomology 147
4. Lim1 and the cohomology of colimits 148
5. The uniqueness of the cohomology of CW complexes 149
Chapter 20. The Poincar´e duality theorem 151
1. Statement of the theorem 151
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2. The definition of the cap product 153
3. Orientations and fundamental classes 155
4. The proof of the vanishing theorem 158
5. The proof of the Poincar´e duality theorem 160
6. The orientation cover 163
Chapter 21. The index of manifolds; manifolds with boundary 165
1. The Euler characteristic of compact manifolds 165
2. The index of compact oriented manifolds 166
3. Manifolds with boundary 168
4. Poincar´e duality for manifolds with boundary 169
5. The index of manifolds that are boundaries 171
Chapter 22. Homology, cohomology, and K(, n)s 175
1. K(, n)s and homology 175
2. K(, n)s and cohomology 177
3. Cup and cap products 179
4. Postnikov systems 182
5. Cohomology operations 184
Chapter 23. Characteristic classes of vector bundles 187
1. The classification of vector bundles 187
2. Characteristic classes for vector bundles 189
3. Stiefel-Whitney classes of manifolds 191
4. Characteristic numbers of manifolds 193
5. Thom spaces and the Thom isomorphism theorem 194
6. The construction of the Stiefel-Whitney classes 196
7. Chern, Pontryagin, and Euler classes 197
8. A glimpse at the general theory 200
Chapter 24. An introduction to K-theory 203
1. The definition of K-theory 203
2. The Bott periodicity theorem 206
3. The splitting principle and the Thom isomorphism 208
4. The Chern character; almost complex structures on spheres 211
5. The Adams operations 213
6. The Hopf invariant one problem and its applications 215
Chapter 25. An introduction to cobordism 219
1. The cobordism groups of smooth closed manifolds 219
2. Sketch proof that N is isomorphic to (TO) 220
3. Prespectra and the algebra H(TO;Z2) 223
4. The Steenrod algebra and its coaction on H(TO) 226
5. The relationship t