A basic course in algebraic topology:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1997
|
| Ausgabe: | Corr. 3. print. |
| Schriftenreihe: | Graduate texts in mathematics
127 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 428 S. graph. Darst. |
| ISBN: | 9780387974309 038797430X 354097430X |
Internformat
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| 245 | 1 | 0 | |a A basic course in algebraic topology |c William S. Massey |
| 250 | |a Corr. 3. print. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1997 | |
| 300 | |a XVI, 428 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 127 | |
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Datensatz im Suchindex
| _version_ | 1816443392038535168 |
|---|---|
| adam_text |
CONTENTS
PREFACE
V
NOTATION AND TERMINOLOGY
XV
CHAPTER I
TWO-DIMENSIONAL MANIFOLDS J
§1. INTRODUCTION
1
§2. DEFINITION AND EXAMPLES OF N-MANIFOLDS 2
§3. ORIENTABLE VS. NONORIENTABLE MANIFOLDS
3
§4. EXAMPLES OF COMPACT, CONNECTED 2-MANIFOLDS 5
§5. STATEMENT OF THE CLASSIFICATION THEOREM FOR COMPACT SURFACES 9
§6. TRIANGULATIONS OF COMPACT SURFACES
14
§7. PROOF OF THEOREM 5.1
16
§8. THE EULER CHARACTERISTIC OF A SURFACE 26
REFERENCES
34
CHAPTER II
THE FUNDAMENTAL GROUP 35
§1. INTRODUCTION 35
§2. BASIC NOTATION AND TERMINOLOGY 36
§3. DEFINITION OF THE FUNDAMENTAL GROUP OF A SPACE 38
§4. THE EFFECT OF A CONTINUOUS MAPPING ON THE FUNDAMENTAL GROUP 42
§5. THE FUNDAMENTAL GROUP OF A CIRCLE IS INFINITE CYCLIC 47
BIBLIOGRAFISCHE INFORMATIONEN
HTTP://D-NB.INFO/931597242
X
CONTENTS
§6. APPLICATION: THE BROUWER FIXED-POINT THEOREM IN DIMENSION 2 50
§7. THE FUNDAMENTAL GROUP OF A PRODUCT SPACE 52
§8. HOMOTOPY TYPE AND HOMOTOPY EQUIVALENCE OF SPACES 54
REFERENCES 59
CHAPTER III
FREE GROUPS AND FREE PRODUCTS OF GROUPS 60
§1. INTRODUCTION 60
§2. THE WEAK PRODUCT OF ABELIAN GROUPS 60
§3. FREE ABELIAN GROUPS 63
§4. FREE PRODUCTS OF GROUPS 71
§5. FREE GROUPS 75
§6. THE PRESENTATION OF GROUPS BY GENERATORS AND RELATIONS 78
§7. UNIVERSAL MAPPING PROBLEMS 81
REFERENCES 85
CHAPTER IV
SEIFERT AND VAN KAMPEN THEOREM ON THE FUNDAMENTAL GROUP
OF THE UNION OF TWO SPACES. APPLICATIONS 86
§1. INTRODUCTION 86
§2. STATEMENT AND PROOF OF THE THEOREM OF SEIFERT AND VAN KAMPEN 87
§3. FIRST APPLICATION OF THEOREM 2.1 91
§4. SECOND APPLICATION OF THEOREM 2.1 95
§5. STRUCTURE OF THE FUNDAMENTAL GROUP OF A COMPACT SURFACE 96
§6. APPLICATION TO KNOT THEORY 103
§7. PROOF OF LEMMA 2.4 108
REFERENCES 116
CHAPTER V
COVERING SPACES
117
§1. INTRODUCTION 117
§2. DEFINITION AND SOME EXAMPLES OF COVERING SPACES 117
§3. LIFTING OF PATHS TO A COVERING SPACE 123
§4. THE FUNDAMENTAL GROUP OF A COVERING SPACE 126
§5. LIFTING OF ARBITRARY MAPS TO A COVERING SPACE 127
§6. HOMOMORPHISMS AND AUTOMORPHISMS OF COVERING SPACES 130
CONTENTS XI
§7. THE ACTION OF THE GROUP 7T(A',X) ON THE SET P"'(X) 133
§8. REGULAR COVERING SPACES AND QUOTIENT SPACES 135
§9. APPLICATION: THE BORSUK-ULAM THEOREM FOR THE 2-SPHERE 138
§10. THE EXISTENCE THEOREM FOR COVERING SPACES 140
REFERENCES 146
CHAPTER VI
BACKGROUND AND MOTIVATION FOR HOMOLOGY THEORY 147
§1. INTRODUCTION 147
§2. SUMMARY OF SOME OF THE BASIC PROPERTIES OF HOMOLOGY THEORY 147
§3. SOME EXAMPLES OF PROBLEMS WHICH MOTIVATED THE DEVELOPMENT
OF HOMOLOGY THEORY IN THE NINETEENTH CENTURY 149
REFERENCES 157
CHAPTER VII
DEFINITIONS AND BASIC PROPERTIES OF HOMOLOGY THEORY 158
§1. INTRODUCTION 158
§2. DEFINITION OF CUBICAL SINGULAR HOMOLOGY GROUPS 158
§3. THE HOMOMORPHISM INDUCED BY A CONTINUOUS MAP 163
§4. THE HOMOTOPY PROPERTY OF THE INDUCED HOMOMORPHISMS 166
§5. THE EXACT HOMOLOGY SEQUENCE OF A PAIR 169
§6. THE MAIN PROPERTIES OF RELATIVE HOMOLOGY GROUPS 173
§7. THE SUBDIVISION OF SINGULAR CUBES AND THE PROOF OF THEOREM 6.4 178
CHAPTER VIII
DETERMINATION OF THE HOMOLOGY GROUPS OF CERTAIN SPACES;
APPLICATIONS AND FURTHER PROPERTIES OF HOMOLOGY THEORY 186
§1. INTRODUCTION 186
§2. HOMOLOGY GROUPS OF CELLS AND SPHERES-APPLICATIONS 192
§3. HOMOLOGY OF FINITE GRAPHS 201
§4. HOMOLOGY OF COMPACT SURFACES 206
§5. THE MAYER-VIETORIS EXACT SEQUENCE 207
§6. THE JORDAN-BROUWER SEPARATION THEOREM AND
INVARIANCE OF DOMAIN 211
§7. THE RELATION BETWEEN THE FUNDAMENTAL GROUP AND
THE FIRST HOMOLOGY GROUP 217
REFERENCES 224
CONTENTS
XII
CHAPTER IX
HOMOLOGY OF CW-COMPLEXES 225
§1. INTRODUCTION 225
§2. ADJOINING CELLS TO A SPACE 225
§3. CW-COMPLEXES 228
§4. THE HOMOLOGY GROUPS OF A CW-COMPLEX 232
§5. INCIDENCE NUMBERS AND ORIENTATIONS OF CELLS 238
§6. REGULAR CW-COMPLEXES 243
§7. DETERMINATION OF INCIDENCE NUMBERS FOR
A REGULAR CELL COMPLEX 244
§8. HOMOLOGY GROUPS OF A PSEUDOMANIFOLD 249
REFERENCES 253
CHAPTER
X
HOMOLOGY WITH ARBITRARY COEFFICIENT GROUPS 254
§1. INTRODUCTION 254
§2. CHAIN COMPLEXES 254
§3. DEFINITION AND BASIC PROPERTIES OF HOMOLOGY WITH
ARBITRARY COEFFICIENTS 262
§4. INTUITIVE GEOMETRIC PICTURE OF A CYCLE WITH COEFFICIENTS IN G 266
§5. COEFFICIENT HOMOMORPHISMS AND COEFFICIENT EXACT SEQUENCES 267
§6. THE UNIVERSAL COEFFICIENT THEOREM 269
§7. FURTHER PROPERTIES OF HOMOLOGY WITH ARBITRARY COEFFICIENTS 274
REFERENCES 278
CHAPTER XI
THE HOMOLOGY OF PRODUCT SPACES 279
§1. INTRODUCTION 279
§2. THE PRODUCT OF CW-COMPLEXES AND THE TENSOR PRODUCT OF
CHAIN COMPLEXES 280
§3. THE SINGULAR CHAIN COMPLEX OF A PRODUCT SPACE 282
§4. THE HOMOLOGY OF THE TENSOR PRODUCT OF CHAIN COMPLEXES
(THE KIINNETH THEOREM) 284
§5. PROOF OF THE EILENBERG-ZILBER THEOREM 286
§6. FORMULAS FOR THE HOMOLOGY GROUPS OF PRODUCT SPACES 300
REFERENCES 303
CONTENTS
XIII
CHAPTER XII
COHOMOLOGY THEORY
305
§1. INTRODUCTION 305
§2. DEFINITION OF COHOMOLOGY GROUPS-PROOFS OF
THE BASIC PROPERTIES 306
§3. COEFFICIENT HOMOMORPHISMS AND THE BOCKSTEIN OPERATOR
IN COHOMOLOGY 309
§4. THE UNIVERSAL COEFFICIENT THEOREM FOR COHOMOLOGY GROUPS 310
§5. GEOMETRIC INTERPRETATION OF COCHAINS, COCYCLES, ETC. 316
§6. PROOF OF THE EXCISION PROPERTY; THE MAYER-VIETORIS SEQUENCE 319
REFERENCES 322
CHAPTER XIII
PRODUCTS IN HOMOLOGY AND COHOMOLOGY 323
§1. INTRODUCTION 323
§2. THE INNER PRODUCT 324
§3. AN OVERALL VIEW OF THE VARIOUS PRODUCTS 324
§4. EXTENSION OF THE DEFINITION OF THE VARIOUS PRODUCTS TO
RELATIVE HOMOLOGY AND COHOMOLOGY GROUPS 329
§5. ASSOCIATIVITY, COMMUTATIVITY, AND EXISTENCE OF A UNIT
OF THE VARIOUS PRODUCTS 333
§6. DIGRESSION: THE EXACT SEQUENCE OF A TRIPLE OR A TRIAD 336
§7. BEHAVIOR OF PRODUCTS WITH RESPECT TO THE BOUNDARY AND
COBOUNDARY OPERATOR OF A PAIR 338
§8. RELATIONS INVOLVING THE INNER PRODUCT 341
§9. CUP AND CAP PRODUCTS IN A PRODUCT SPACE 342
§10. REMARKS ON THE COEFFICIENTS FOR THE VARIOUS PRODUCTS-
THE COHOMOLOGY RING 343
§11. THE COHOMOLOGY OF PRODUCT SPACES (THE KIINNETH THEOREM
FOR COHOMOLOGY) 344
REFERENCES 349
CHAPTER XIV
DUALITY THEOREMS FOR THE HOMOLOGY OF MANIFOLDS 350
§1. INTRODUCTION 350
§2. ORIENTABILITY AND THE EXISTENCE OF ORIENTATIONS FOR MANIFOLDS 351
§3. COHOMOLOGY WITH COMPACT SUPPORTS 358
§4. STATEMENT AND PROOF OF THE POINCARE DUALITY THEOREM 360
XIV CONTENTS
§5. APPLICATIONS OF THE POINCARE DUALITY THEOREM TO
COMPACT MANIFOLDS 365
§6. THE ALEXANDER DUALITY THEOREM 370
§7. DUALITY THEOREMS FOR MANIFOLDS WITH BOUNDARY 375
§8. APPENDIX: PROOF OF TWO LEMMAS ABOUT CAP PRODUCTS 380
REFERENCES 393
CHAPTER XV
CUP PRODUCTS IN PROJECTIVE SPACES AND APPLICATIONS OF CUP PRODUCTS 394
§1. INTRODUCTION 394
§2. THE PROJECTIVE SPACES 394
§3. THE MAPPING CYLINDER AND MAPPING CONE 399
§4. THE HOPF INVARIANT 402
REFERENCES 406
APPENDIX A
A PROOF OF DE RHAM'S THEOREM 407
§1. INTRODUCTION 407
§2. DIFFERENTIABLE SINGULAR CHAINS 408
§3. STATEMENT AND PROOF OF DE RHAM'S THEOREM 411
REFERENCES 417
APPENDIX B
PERMUTATION GROUPS OR TRANSFORMATION GROUPS 419
§1. BASIC DEFINITIONS 419
§2. HOMOGENEOUS G-SPACES 421
INDEX 424 |
| any_adam_object | 1 |
| author | Massey, William S. 1920- |
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| dewey-ones | 514 - Topology |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Corr. 3. print. |
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| indexdate | 2024-11-22T17:10:08Z |
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| language | English |
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| spelling | Massey, William S. 1920- Verfasser (DE-588)1104850397 aut A basic course in algebraic topology William S. Massey Corr. 3. print. New York [u.a.] Springer 1997 XVI, 428 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 127 Algebraic topology Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Algebraische Topologie (DE-588)4120861-4 s DE-604 Graduate texts in mathematics 127 (DE-604)BV000000067 127 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007800530&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Massey, William S. 1920- A basic course in algebraic topology Graduate texts in mathematics Algebraic topology Algebraische Topologie (DE-588)4120861-4 gnd |
| subject_GND | (DE-588)4120861-4 |
| title | A basic course in algebraic topology |
| title_auth | A basic course in algebraic topology |
| title_exact_search | A basic course in algebraic topology |
| title_full | A basic course in algebraic topology William S. Massey |
| title_fullStr | A basic course in algebraic topology William S. Massey |
| title_full_unstemmed | A basic course in algebraic topology William S. Massey |
| title_short | A basic course in algebraic topology |
| title_sort | a basic course in algebraic topology |
| topic | Algebraic topology Algebraische Topologie (DE-588)4120861-4 gnd |
| topic_facet | Algebraic topology Algebraische Topologie |
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