A concise course in algebraic topology:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chicago [u.a.]
Univ. of Chicago Press
2006
|
| Edition: | [Nachdr.] |
| Series: | Chicago lectures in mathematics series
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| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015822231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | IX, 243 S. graph. Darst. |
| ISBN: | 0226511839 9780226511832 |
Staff View
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Record in the Search Index
| _version_ | 1819258132784742400 |
|---|---|
| adam_text | Contents
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: tt^R) = 0 and Tr^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 23
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 29
Chapter 4. Graphs 33
1. The definition of graphs 33
2. Edge paths and trees 33
3. The homotopy types of graphs 34
4. Covers of graphs and Euler characteristics 35
5. Applications to groups 35
Chapter 5. Compactly generated spaces 37
1. The definition of compactly generated spaces 37
2. The category of compactly generated spaces 38
V
vi CONTENTS
Chapter 6. Cofibrations 41
1. The definition of cofibrations 41
2. Mapping cylinders and cofibrations 42
3. Replacing maps by cofibrations 43
4. A criterion for a map to be a cofibration 43
5. Cofiber homotopy equivalence 44
Chapter 7. Fibrations 47
1. The definition of fibrations 47
2. Path lifting functions and fibrations 47
3. Replacing maps by fibrations 48
4. A criterion for a map to be a fibration 49
5. Fiber homotopy equivalence 50
6. Change of fiber 51
Chapter 8. Based cofiber and fiber sequences 55
1. Based homotopy classes of maps 55
2. Cones, suspensions, paths, loops 55
3. Based cofibrations 56
4. Cofiber sequences 57
5. Based fibrations 59
6. Fiber sequences 59
7. Connections between cofiber and fiber sequences 61
Chapter 9. Higher homotopy groups 63
1. The definition of homotopy groups 63
2. Long exact sequences associated to pairs 63
3. Long exact sequences associated to fibrations 64
4. A few calculations 64
5. Change of basepoint 66
6. n Equivalences, weak equivalences, and a technical lemma 67
Chapter 10. CW complexes 71
1. The definition and some examples of CW complexes 71
2. Some constructions on CW complexes 72
3. HELP and the Whitehead theorem 73
4. The cellular approximation theorem 74
5. Approximation of spaces by CW complexes 75
6. Approximation of pairs by CW pairs 76
7. Approximation of excisive triads by CW triads 77
Chapter 11. The homotopy excision and suspension theorems 81
1. Statement of the homotopy excision theorem 81
2. The Freudenthal suspension theorem 83
3. Proof of the homotopy excision theorem 84
Chapter 12. A little homological algebra 89
1. Chain complexes 89
2. Maps and homotopies of maps of chain complexes 89
3. Tensor products of chain complexes 90
4. Short and long exact sequences 91
CONTENTS vii
Chapter 13. Axiomatic and cellular homology theory 93
1. Axioms for homology 93
2. Cellular homology 94
3. Verification of the axioms 98
4. The cellular chains of products 99
5. Some examples: T, K, and RPn 101
Chapter 14. Derivations of properties from the axioms 105
1. Reduced homology; based versus unbased spaces 105
2. Cofibrations and the homology of pairs 106
3. Suspension and the long exact sequence of pairs 107
4. Axioms for reduced homology 108
5. Mayer Vietoris sequences 110
6. The homology of colimits 112
Chapter 15. The Hurewicz and uniqueness theorems 115
1. The Hurewicz theorem 115
2. The uniqueness of the homology of CW complexes 117
Chapter 16. Singular homology theory 121
1. The singular chain complex 121
2. Geometric realization 122
3. Proofs of the theorems 123
4. Simplicial objects in algebraic topology 124
5. Classifying spaces and K(n,n)s 126
Chapter 17. Some more homological algebra 129
1. Universal coefficients in homology 129
2. The Kiinneth theorem 130
3. Horn functors and universal coefficients in cohomology 131
4. Proof of the universal coefficient theorem 133
5. Relations between (g and Horn 133
Chapter 18. Axiomatic and cellular cohomology theory 135
1. Axioms for cohomology 135
2. Cellular and singular cohomology 136
3. Cup products in cohomology 137
4. An example: RP and the Borsuk Ulam theorem 138
5. Obstruction theory 140
Chapter 19. Derivations of properties from the axioms 143
1. Reduced cohomology groups and their properties 143
2. Axioms for reduced cohomology 144
3. Mayer Vietoris sequences in cohomology 145
4. Lim1 and the cohomology of colimits 146
5. The uniqueness of the cohomology of CW complexes 147
Chapter 20. The Poincare duality theorem 149
1. Statement of the theorem 149
2. The definition of the cap product 151
3. Orientations and fundamental classes 153
viLi CONTENTS
4. The proof of the vanishing theorem 155
5. The proof of the Poincare duality theorem 158
6. The orientation cover l61
Chapter 21. The index of manifolds; manifolds with boundary 163
1. The Euler characteristic of compact manifolds 163
2. The index of compact oriented manifolds 164
3. Manifolds with boundary 166
4. Poincare duality for manifolds with boundary 167
5. The index of manifolds that are boundaries I69
Chapter 22. Homology, cohomology, and K(n, n)s 171
1. K(n,n)s and homology 1^1
2. K(n, n)s and cohomology I73
3. Cup and cap products 1^5
4. Postnikov systems 1^8
5. Cohomology operations 180
Chapter 23. Characteristic classes of vector bundles 183
1. The classification of vector bundles 183
2. Characteristic classes for vector bundles 185
3. Stiefel Whitney classes of manifolds 187
4. Characteristic numbers of manifolds 189
5. Thom spaces and the Thom isomorphism theorem 190
6. The construction of the Stiefel Whitney classes 192
7. Chern, Pontryagin, and Euler classes 193
8. A glimpse at the general theory 196
Chapter 24. An introduction to /( theory 199
1. The definition of ^ theory 1
2. The Bott periodicity theorem 202
3. The splitting principle and the Thom isomorphism 204
4. The Chern character; almost complex structures on spheres 207
5. The Adams operations 209
6. The Hopf invariant one problem and its applications 211
Chapter 25. An introduction to cobordism 215
1. The cobordism groups of smooth closed manifolds 215
2. Sketch proof that Jf, is isomorphic to n,(TO) 216
3. Prespectra and the algebra H,{TO; Z2) 219
4. The Steenrod algebra and its coaction on H* (TO) 222
5. The relationship to Stiefel Whitney numbers 224
6. Spectra and the computation of 7r»(TO) = w,(MO) 226
7. An introduction to the stable category 228
Suggestions for further reading 231
1. A classic book and historical references 231
2. Textbooks in algebraic topology and homotopy theory 231
3. Books on CW complexes 232
4. Differential forms and Morse theory 232
5. Equivariant algebraic topology 233
CONTENTS ix
6. Category theory and homological algebra 233
7. Simplicial sets in algebraic topology 233
8. The Serre spectral sequence and Serre class theory 233
9. The Eilenberg Moore spectral sequence 233
10. Cohomology operations 234
11. Vector bundles 234
12. Characteristic classes 234
13. if theory 235
14. Hopf algebras; the Steenrod algebra, Adams spectral sequence 235
15. Cobordism 236
16. Generalized homology theory and stable homotopy theory 236
17. Quillen model categories 236
18. Localization and completion; rational homotopy theory 237
19. Infinite loop space theory 237
20. Complex cobordism and stable homotopy theory 238
21. Follow ups to this book 238
Index 239
|
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| author | May, Jon Peter 1939- |
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| discipline | Mathematik |
| edition | [Nachdr.] |
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| id | DE-604.BV022616095 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:59:35Z |
| institution | BVB |
| isbn | 0226511839 9780226511832 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015822231 |
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| physical | IX, 243 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Univ. of Chicago Press |
| record_format | marc |
| series2 | Chicago lectures in mathematics series |
| spellingShingle | May, Jon Peter 1939- A concise course in algebraic topology Algebraische Topologie (DE-588)4120861-4 gnd |
| subject_GND | (DE-588)4120861-4 |
| title | A concise course in algebraic topology |
| title_auth | A concise course in algebraic topology |
| title_exact_search | A concise course in algebraic topology |
| title_full | A concise course in algebraic topology J. P. May |
| title_fullStr | A concise course in algebraic topology J. P. May |
| title_full_unstemmed | A concise course in algebraic topology J. P. May |
| title_short | A concise course in algebraic topology |
| title_sort | a concise course in algebraic topology |
| topic | Algebraische Topologie (DE-588)4120861-4 gnd |
| topic_facet | Algebraische Topologie |
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