Lecture notes in algebraic topology:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
AMS
2001
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| Series: | Graduate studies in mathematics
volume 35 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 367 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821821602 0821821601 |
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MARC
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| 245 | 1 | 0 | |a Lecture notes in algebraic topology |c James F. Davis, Paul Kirk |
| 264 | 1 | |a Providence, Rhode Island |b AMS |c 2001 | |
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| 490 | 1 | |a Graduate studies in mathematics |v volume 35 | |
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Record in the Search Index
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| adam_text | Contents
Preface xi
Projects xiv
Chapter 1. Chain Complexes, Homology, and Cohomology 1
§1.1. Chain complexes associated to a space 1
§1.2. Tensor products, adjoint functors, and Horn 8
§1.3. Tensor and Horn functors on chain complexes 12
§1.4. Singular cohomology 14
§1.5. The Eilenberg Steenrod axioms 19
§1.6. Projects for Chapter 1 22
Chapter 2. Homological Algebra 23
§2.1. Axioms for Tor and Ext; projective resolutions 23
§2.2. Projective and injective modules 29
§2.3. Resolutions 33
§2.4. Definition of Tor and Ext existence 35
§2.5. The fundamental lemma of homological algebra 36
§2.6. Universal coefficient theorems 43
§2.7. Projects for Chapter 2 49
Chapter 3. Products 51
v
vi Contents
§3.1. Tensor products of chain complexes and the algebraic Kiinneth
theorem 51
§3.2. The Eilenberg Zilber maps 54
§3.3. Cross and cup products 56
§3.4. The Alexander Whitney diagonal approximation 64
§3.5. Relative cup and cap products 67
§3.6. Projects for Chapter 3 70
Chapter 4. Fiber Bundles 77
§4.1. Group actions 77
§4.2. Fiber bundles 78
§4.3. Examples of fiber bundles 81
§4.4. Principal bundles and associated bundles 84
§4.5. Reducing the structure group 89
§4.6. Maps of bundles and pullbacks 90
§4.7. Projects for Chapter 4 92
Chapter 5. Homology with Local Coefficients 95
§5.1. Definition of homology with twisted coefficients 96
§5.2. Examples and basic properties 98
§5.3. Definition of homology with a local coefficient system 103
§5.4. Functoriality 105
§5.5. Projects for Chapter 5 108
Chapter 6. Fibrations, Cofibrations and Homotopy Groups 111
§6.1. Compactly generated spaces 111
§6.2. Fibrations 114
§6.3. The fiber of a fibration 116
§6.4. Path space fibrations 120
§6.5. Fiber homotopy 123
§6.6. Replacing a map by a fibration 123
§6.7. Cofibrations 127
Contents vii
§6.8. Replacing a map by a cofibration 131
§6.9. Sets of homotopy classes of maps 134
§6.10. Adjoint of loops and suspension; smash products 136
§6.11. Fibration and cofibration sequences 138
§6.12. Puppe sequences 141
§6.13. Homotopy groups 143
§6.14. Examples of fibrations 145
§6.15. Relative homotopy groups 152
§6.16. The action of the fundamental group on homotopy sets 155
§6.17. The Hurewicz and Whitehead theorems 160
§6.18. Projects for Chapter 6 163
Chapter 7. Obstruction Theory and Eilenberg MacLane Spaces 165
§7.1. Basic problems of obstruction theory 165
§7.2. The obstruction cocycle 168
§7.3. Construction of the obstruction cocycle 169
§7.4. Proof of the extension theorem 172
§7.5. Obstructions to finding a homotopy 175
§7.6. Primary obstructions 176
§7.7. Eilenberg MacLane spaces 177
§7.8. Aspherical spaces 183
§7.9. CW approximations and Whitehead s theorem 185
§7.10. Obstruction theory in fibrations 189
§7.11. Characteristic classes 191
§7.12. Projects for Chapter 7 192
Chapter 8. Bordism, Spectra, and Generalized Homology 195
§8.1. Framed bordism and homotopy groups of spheres 196
§8.2. Suspension and the Freudenthal theorem 202
§8.3. Stable tangential framings 204
§8.4. Spectra 210
viii Contents
§8.5. More general bordism theories 213
§8.6. Classifying spaces 217
§8.7. Construction of the Thorn spectra 219
§8.8. Generalized homology theories 227
§8.9. Projects for Chapter 8 234
Chapter 9. Spectral Sequences 237
§9.1. Definition of a spectral sequence 237
§9.2. The Leray Serre Atiyah Hirzebruch spectral sequence 241
§9.3. The edge homomorphisms and the transgression 245
§9.4. Applications of the homology spectral sequence 249
§9.5. The cohomology spectral sequence 254
§9.6. Homology of groups 261
§9.7. Homology of covering spaces 264
§9.8. Relative spectral sequences 266
§9.9. Projects for Chapter 9 266
Chapter 10. Further Applications of Spectral Sequences 267
§10.1. Serre classes of abelian groups 267
§10.2. Homotopy groups of spheres 276
§10.3. Suspension, looping, and the transgression 279
§10.4. Cohomology operations 283
§10.5. The mod 2 Steenrod algebra 288
§10.6. The Thom isomorphism theorem 295
§10.7. Intersection theory 299
§10.8. Stiefel Whitney classes 306
§10.9. Localization 312
§10.10. Construction of bordism invariants 317
§10.11. Projects for Chapter 10 319
Chapter 11. Simple Homotopy Theory 323
§11.1. Introduction 323
Contents ix
§11.2. Invertible matrices and Ki(R) 326
§11.3. Torsion for chain complexes 334
§11.4. Whitehead torsion for CW complexes 343
§11.5. Reidemeister torsion 346
§11.6. Torsion and lens spaces 348
§11.7. The s cobordism theorem 357
§11.8. Projects for Chapter 11 357
Bibliography 359
Index 363
|
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| isbn | 9780821821602 0821821601 |
| language | English |
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| spelling | Davis, James F. 1955- Verfasser (DE-588)142627968 aut Lecture notes in algebraic topology James F. Davis, Paul Kirk Providence, Rhode Island AMS 2001 xv, 367 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics volume 35 Algebraische Topologie Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Algebraische Topologie (DE-588)4120861-4 s DE-604 Kirk, Paul Verfasser aut Erscheint auch als Online-Ausgabe 978-1-4704-2088-8 Graduate studies in mathematics volume 35 (DE-604)BV009739289 35 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009543467&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Davis, James F. 1955- Kirk, Paul Lecture notes in algebraic topology Graduate studies in mathematics Algebraische Topologie Algebraische Topologie (DE-588)4120861-4 gnd |
| subject_GND | (DE-588)4120861-4 |
| title | Lecture notes in algebraic topology |
| title_auth | Lecture notes in algebraic topology |
| title_exact_search | Lecture notes in algebraic topology |
| title_full | Lecture notes in algebraic topology James F. Davis, Paul Kirk |
| title_fullStr | Lecture notes in algebraic topology James F. Davis, Paul Kirk |
| title_full_unstemmed | Lecture notes in algebraic topology James F. Davis, Paul Kirk |
| title_short | Lecture notes in algebraic topology |
| title_sort | lecture notes in algebraic topology |
| topic | Algebraische Topologie Algebraische Topologie (DE-588)4120861-4 gnd |
| topic_facet | Algebraische Topologie |
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