Verfügbarkeit: Algebraic topology

Algebraic topology:

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1. Verfasser:	Tom Dieck, Tammo 1938- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Zürich Europ. Math. Soc. 2008
Schriftenreihe:	EMS textbooks in mathematics
Schlagworte:	Algebraïsche topologie Topologia algébrica Algebraic topology Homology theory Homotopy theory Algebraische Topologie
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XI, 567 S. graph. Darst.
ISBN:	9783037190487

Internformat

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Datensatz im Suchindex

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adam_text	Contents Preface v 1 Topological Spaces 1 1.1 Basic Notions ............................ 1 1.2 Subspaces. Quotient Spaces .................... 5 1.3 Products and Sums ......................... 8 1.4 Compact Spaces .......................... 11 1.5 Proper Maps ............................ 14 1.6 Paracompact Spaces ........................ 15 1.7 Topological Groups ........................ 15 1.8 Transformation Groups ....................... 17 1.9 Projective Spaces. Grassmann Manifolds ............. 21 2 The Fundamental Group 24 2.1 The Notion of Homotopy ..................... 25 2.2 Further Homotopy Notions ..................... 30 2.3 Standard Spaces .......................... 34 2.4 Mapping Spaces and Homotopy .................. 37 2.5 The Fundamental Groupoid .................... 41 2.6 The Theorem of Seifert and van Kampen............. 45 2.7 The Fundamental Group of the Circle ............... 47 2.8 Examples .............................. 52 2.9 Homotopy Groupoids ....................... 58 3 Covering Spaces 62 3.1 Locally Trivial Maps. Covering Spaces .............. 62 3.2 Fibre Transport. Exact Sequence ................. 66 3.3 Classification of Coverings ..................... 70 3.4 Connected Groupoids ....................... 72 3.5 Existence of Liftings ........................ 76 3.6 The Universal Covering ...................... 78 4 Elementary Homotopy Theory 81 4.1 The Mapping Cylinder ....................... 81 4.2 The Double Mapping Cylinder ................... 84 4.3 Suspension. Homotopy Groups .................. 86 4.4 Loop Space ............................. 89 viii Contents 4.5 Groups and Cogroups ....................... 90 4.6 The Cofibre Sequence ....................... 92 4.7 The Fibre Sequence ........................ 97 5 Cofibrations and Fibrations 101 5.1 The Homotopy Extension Property ................ 101 5.2 Transport .............................. 107 5.3 Replacing a Map by a Cofibration ................. 110 5.4 Characterization of Cofibrations .................. 113 5.5 The Homotopy Lifting Property .................. 115 5.6 Transport .............................. 119 5.7 Replacing a Map by a Fibration .................. 120 6 Homotopy Groups 121 6.1 The Exact Sequence of Homotopy Groups ............ 122 6.2 The Role of the Base Point ..................... 126 6.3 Serre Fibrations .......................... 129 6.4 The Excision Theorem ....................... 133 6.5 The Degree ............................. 135 6.6 The Brouwer Fixed Point Theorem ................ 137 6.7 Higher Connectivity ........................ 141 6.8 Classical Groups .......................... 146 6.9 Proof of the Excision Theorem ................... 148 6.10 Further Applications of Excision .................. 152 7 Stable Homotopy. Duality 159 7.1 A Stable Category ......................... 159 7.2 Mapping Cones ........................... 164 7.3 Euclidean Complements ...................... 168 7.4 The Complement Duality Functor ................. 169 7.5 Duality ............................... 175 7.6 Homology and Cohomology for Pointed Spaces ......... 179 7.7 Spectral Homology and Cohomology ............... 181 7.8 Alexander Duality ......................... 185 7.9 Compactly Generated Spaces ................... 186 8 Cell Complexes 196 8.1 Simplicial Complexes ....................... 197 8.2 Whitehead Complexes ....................... 199 8.3 CW-Complexes ........................... 203 8.4 Weak Homotopy Equivalences ................... 207 8.5 Cellular Approximation ...................... 210 8.6 CW-Approximation ........................ 211 Contents ix 8.7 Homotopy Classification ...................... 216 8.8 Eilenberg-Mac Lane Spaces.................... 217 9 Singular Homology 223 9.1 Singular Homology Groups .................... 224 9.2 The Fundamental Group...................... 227 9.3 Homotopy............................. 228 9.4 Barycentric Subdivision. Excision................. 231 9.5 Weak Equivalences and Homology ................ 235 9.6 Homology with Coefficients .................... 237 9.7 The Theorem of Eilenberg and Zilber ............... 238 9.8 The Homology Product ...................... 241 10 Homology 244 10.1 The Axioms of Eilenberg and Steenrod .............. 244 10.2 Elementary Consequences of the Axioms ............. 246 10.3 Jordan Curves. Invariance of Domain ............... 249 10.4 Reduced Homology Groups .................... 252 10.5 The Degree ............................. 256 10.6 The Theorem of Borsuk and Ulam ................. 261 10.7 Mayer-Vietoris Sequences ..................... 265 10.8 Colimits .............................. 270 10.9 Suspension ............................. 273 11 Homological Algebra 275 11.1 Diagrams .............................. 275 11.2 Exact Sequences .......................... 279 11.3 Chain Complexes .......................... 283 11.4 Cochain complexes ......................... 285 11.5 Natural Chain Maps and Homotopies ............... 286 11.6 Chain Equivalences ........................ 287 11.7 Linear Algebra of Chain Complexes ................ 289 11.8 The Functors Tor and Ext ..................... 292 11.9 Universal Coefficients ....................... 295 11.10 The Kiinneth Formula ........................ 298 12 Cellular Homology 300 12.1 Cellular Chain Complexes ..................... 300 12.2 Cellular Homology equals Homology ............... 304 12.3 Simplicial Complexes ....................... 306 12.4 The Euler Characteristic ...................... 308 12.5 Euler Characteristic of Surfaces .................. 311 χ Contents 13 Partitions of Unity in Homotopy Theory 318 13.1 Partitions of Unity ......................... 318 13.2 The Homotopy Colimit of a Covering ............... 321 13.3 Homotopy Equivalences ...................... 324 13.4 Fibrations .............................. 325 14 Bundles 328 14.1 Principal Bundles .......................... 328 14.2 VectorBundles ........................... 335 14.3 The Homotopy Theorem ...................... 342 14.4 Universal Bundles. Classifying Spaces .............. 344 14.5 Algebra of Vector Bundles ...................... 351 14.6 Grothendieck Rings of Vector Bundles .............. 355 15 Manifolds 358 15.1 Differentiable Manifolds ...................... 358 15.2 Tangent Spaces and Differentials ................. 362 15.3 Smooth Transformation Groups .................. 366 15.4 Manifolds with Boundary ..................... 369 15.5 Orientation ............................. 372 15.6 Tangent Bundle. Normal Bundle .................. 374 15.7 Embeddings ............................ 379 15.8 Approximation ........................... 383 15.9 Transversality ........................... 384 15.10 Gluing along Boundaries ...................... 388 16 Homology of Manifolds 392 16.1 Local Homology Groups ...................... 392 16.2 Homological Orientations ..................... 394 16.3 Homology in the Dimension of the Manifold ........... 396 16.4 Fundamental Class and Degree .................. 399 16.5 Manifolds with Boundary ..................... 402 16.6 Winding and Linking Numbers .................. 403 17 Cohomology 405 17.1 Axiomatic Cohomoiogy ...................... 405 17.2 Multiplicative Cohomology Theories ............... 409 17.3 External Products .......................... 413 17.4 Singular Cohomology ....................... 416 17.5 Eiienberg-Mac Lane Spaces and Cohomology .......... 419 17.6 The Cup Product in Singular Cohomology ............ 422 17.7 Fibration over Spheres ....................... 425 17.8 The Theorem of Leray and Hirsch................. 427 Contents xi 17.9 The Thom Isomorphism ...................... 431 18 Duality 438 18.1 The Cap Product .......................... 438 18.2 Duality Pairings .......................... 441 18.3 The Duality Theorem ........................ 444 18.4 Euclidean Neighbourhood Retracts ................ 447 18.5 Proof of the Duality Theorem ................... 451 18.6 Manifolds with Boundary ..................... 455 18.7 The Intersection Form. Signature ................. 457 18.8 The Euler Number ......................... 461 18.9 Euler Class and Euler Characteristic ................ 464 19 Characteristic Classes 467 19.1 Projective Spaces .......................... 468 19.2 Projective Bundles ......................... 471 19.3 Chern Classes ........................... 472 19.4 Stiefel-Whitney Classes ...................... 478 19.5 Pontrjagin Classes ......................... 479 19.6 Hopf Algebras ........................... 482 19.7 Hopf Algebras and Classifying Spaces ............... 486 19.8 Characteristic Numbers ...................... 491 20 Homology and Homotopy 495 20.1 The Theorem of Hurewicz ..................... 495 20.2 Realization of Chain Complexes .................. 501 20.3 Serre Classes ............................ 504 20.4 Qualitative Homology of Fibrations ................ 505 20.5 Consequences of the Fibration Theorem .............. 508 20.6 Hurewicz and Whitehead Theorems modulo Serre classes ..... 510 20.7 Cohomology of Eilenberg-Mac Lane Spaces ........... 513 20.8 Homotopy Groups of Spheres ................... 514 20.9 Rational Homology Theories .................... 518 21 Bordism 521 21.1 Bordism Homology ........................ 521 21.2 The Theorem of Pontrjagin and Thom ............... 529 21.3 Bordism and Thom Spectra .................... 535 21.4 Oriented Bordism ......................... 537 Bibliography 541 Symbols 551 Index 557
adam_txt	Contents Preface v 1 Topological Spaces 1 1.1 Basic Notions . 1 1.2 Subspaces. Quotient Spaces . 5 1.3 Products and Sums . 8 1.4 Compact Spaces . 11 1.5 Proper Maps . 14 1.6 Paracompact Spaces . 15 1.7 Topological Groups . 15 1.8 Transformation Groups . 17 1.9 Projective Spaces. Grassmann Manifolds . 21 2 The Fundamental Group 24 2.1 The Notion of Homotopy . 25 2.2 Further Homotopy Notions . 30 2.3 Standard Spaces . 34 2.4 Mapping Spaces and Homotopy . 37 2.5 The Fundamental Groupoid . 41 2.6 The Theorem of Seifert and van Kampen. 45 2.7 The Fundamental Group of the Circle . 47 2.8 Examples . 52 2.9 Homotopy Groupoids . 58 3 Covering Spaces 62 3.1 Locally Trivial Maps. Covering Spaces . 62 3.2 Fibre Transport. Exact Sequence . 66 3.3 Classification of Coverings . 70 3.4 Connected Groupoids . 72 3.5 Existence of Liftings . 76 3.6 The Universal Covering . 78 4 Elementary Homotopy Theory 81 4.1 The Mapping Cylinder . 81 4.2 The Double Mapping Cylinder . 84 4.3 Suspension. Homotopy Groups . 86 4.4 Loop Space . 89 viii Contents 4.5 Groups and Cogroups . 90 4.6 The Cofibre Sequence . 92 4.7 The Fibre Sequence . 97 5 Cofibrations and Fibrations 101 5.1 The Homotopy Extension Property . 101 5.2 Transport . 107 5.3 Replacing a Map by a Cofibration . 110 5.4 Characterization of Cofibrations . 113 5.5 The Homotopy Lifting Property . 115 5.6 Transport . 119 5.7 Replacing a Map by a Fibration . 120 6 Homotopy Groups 121 6.1 The Exact Sequence of Homotopy Groups . 122 6.2 The Role of the Base Point . 126 6.3 Serre Fibrations . 129 6.4 The Excision Theorem . 133 6.5 The Degree . 135 6.6 The Brouwer Fixed Point Theorem . 137 6.7 Higher Connectivity . 141 6.8 Classical Groups . 146 6.9 Proof of the Excision Theorem . 148 6.10 Further Applications of Excision . 152 7 Stable Homotopy. Duality 159 7.1 A Stable Category . 159 7.2 Mapping Cones . 164 7.3 Euclidean Complements . 168 7.4 The Complement Duality Functor . 169 7.5 Duality . 175 7.6 Homology and Cohomology for Pointed Spaces . 179 7.7 Spectral Homology and Cohomology . 181 7.8 Alexander Duality . 185 7.9 Compactly Generated Spaces . 186 8 Cell Complexes 196 8.1 Simplicial Complexes . 197 8.2 Whitehead Complexes . 199 8.3 CW-Complexes . 203 8.4 Weak Homotopy Equivalences . 207 8.5 Cellular Approximation . 210 8.6 CW-Approximation . 211 Contents ix 8.7 Homotopy Classification . 216 8.8 Eilenberg-Mac Lane Spaces. 217 9 Singular Homology 223 9.1 Singular Homology Groups . 224 9.2 The Fundamental Group. 227 9.3 Homotopy. 228 9.4 Barycentric Subdivision. Excision. 231 9.5 Weak Equivalences and Homology . 235 9.6 Homology with Coefficients . 237 9.7 The Theorem of Eilenberg and Zilber . 238 9.8 The Homology Product . 241 10 Homology 244 10.1 The Axioms of Eilenberg and Steenrod . 244 10.2 Elementary Consequences of the Axioms . 246 10.3 Jordan Curves. Invariance of Domain . 249 10.4 Reduced Homology Groups . 252 10.5 The Degree . 256 10.6 The Theorem of Borsuk and Ulam . 261 10.7 Mayer-Vietoris Sequences . 265 10.8 Colimits . 270 10.9 Suspension . 273 11 Homological Algebra 275 11.1 Diagrams . 275 11.2 Exact Sequences . 279 11.3 Chain Complexes . 283 11.4 Cochain complexes . 285 11.5 Natural Chain Maps and Homotopies . 286 11.6 Chain Equivalences . 287 11.7 Linear Algebra of Chain Complexes . 289 11.8 The Functors Tor and Ext . 292 11.9 Universal Coefficients . 295 11.10 The Kiinneth Formula . 298 12 Cellular Homology 300 12.1 Cellular Chain Complexes . 300 12.2 Cellular Homology equals Homology . 304 12.3 Simplicial Complexes . 306 12.4 The Euler Characteristic . 308 12.5 Euler Characteristic of Surfaces . 311 χ Contents 13 Partitions of Unity in Homotopy Theory 318 13.1 Partitions of Unity . 318 13.2 The Homotopy Colimit of a Covering . 321 13.3 Homotopy Equivalences . 324 13.4 Fibrations . 325 14 Bundles 328 14.1 Principal Bundles . 328 14.2 VectorBundles . 335 14.3 The Homotopy Theorem . 342 14.4 Universal Bundles. Classifying Spaces . 344 14.5 Algebra of Vector Bundles . 351 14.6 Grothendieck Rings of Vector Bundles . 355 15 Manifolds 358 15.1 Differentiable Manifolds . 358 15.2 Tangent Spaces and Differentials . 362 15.3 Smooth Transformation Groups . 366 15.4 Manifolds with Boundary . 369 15.5 Orientation . 372 15.6 Tangent Bundle. Normal Bundle . 374 15.7 Embeddings . 379 15.8 Approximation . 383 15.9 Transversality . 384 15.10 Gluing along Boundaries . 388 16 Homology of Manifolds 392 16.1 Local Homology Groups . 392 16.2 Homological Orientations . 394 16.3 Homology in the Dimension of the Manifold . 396 16.4 Fundamental Class and Degree . 399 16.5 Manifolds with Boundary . 402 16.6 Winding and Linking Numbers . 403 17 Cohomology 405 17.1 Axiomatic Cohomoiogy . 405 17.2 Multiplicative Cohomology Theories . 409 17.3 External Products . 413 17.4 Singular Cohomology . 416 17.5 Eiienberg-Mac Lane Spaces and Cohomology . 419 17.6 The Cup Product in Singular Cohomology . 422 17.7 Fibration over Spheres . 425 17.8 The Theorem of Leray and Hirsch. 427 Contents xi 17.9 The Thom Isomorphism . 431 18 Duality 438 18.1 The Cap Product . 438 18.2 Duality Pairings . 441 18.3 The Duality Theorem . 444 18.4 Euclidean Neighbourhood Retracts . 447 18.5 Proof of the Duality Theorem . 451 18.6 Manifolds with Boundary . 455 18.7 The Intersection Form. Signature . 457 18.8 The Euler Number . 461 18.9 Euler Class and Euler Characteristic . 464 19 Characteristic Classes 467 19.1 Projective Spaces . 468 19.2 Projective Bundles . 471 19.3 Chern Classes . 472 19.4 Stiefel-Whitney Classes . 478 19.5 Pontrjagin Classes . 479 19.6 Hopf Algebras . 482 19.7 Hopf Algebras and Classifying Spaces . 486 19.8 Characteristic Numbers . 491 20 Homology and Homotopy 495 20.1 The Theorem of Hurewicz . 495 20.2 Realization of Chain Complexes . 501 20.3 Serre Classes . 504 20.4 Qualitative Homology of Fibrations . 505 20.5 Consequences of the Fibration Theorem . 508 20.6 Hurewicz and Whitehead Theorems modulo Serre classes . 510 20.7 Cohomology of Eilenberg-Mac Lane Spaces . 513 20.8 Homotopy Groups of Spheres . 514 20.9 Rational Homology Theories . 518 21 Bordism 521 21.1 Bordism Homology . 521 21.2 The Theorem of Pontrjagin and Thom . 529 21.3 Bordism and Thom Spectra . 535 21.4 Oriented Bordism . 537 Bibliography 541 Symbols 551 Index 557
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spelling	Tom Dieck, Tammo 1938- Verfasser (DE-588)124473091 aut Algebraic topology Tammo tom Dieck Zürich Europ. Math. Soc. 2008 XI, 567 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier EMS textbooks in mathematics Algebraïsche topologie gtt Topologia algébrica larpcal Algebraic topology Homology theory Homotopy theory Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Algebraische Topologie (DE-588)4120861-4 s DE-604 Erscheint auch als Tom Dieck, Tammo, 1938- Algebraic topology Online-Ausgabe 978-3-03719-548-2 (DE-604)BV036713255 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016700582&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Tom Dieck, Tammo 1938- Algebraic topology Algebraïsche topologie gtt Topologia algébrica larpcal Algebraic topology Homology theory Homotopy theory Algebraische Topologie (DE-588)4120861-4 gnd
subject_GND	(DE-588)4120861-4
title	Algebraic topology
title_auth	Algebraic topology
title_exact_search	Algebraic topology
title_exact_search_txtP	Algebraic topology
title_full	Algebraic topology Tammo tom Dieck
title_fullStr	Algebraic topology Tammo tom Dieck
title_full_unstemmed	Algebraic topology Tammo tom Dieck
title_short	Algebraic topology
title_sort	algebraic topology
topic	Algebraïsche topologie gtt Topologia algébrica larpcal Algebraic topology Homology theory Homotopy theory Algebraische Topologie (DE-588)4120861-4 gnd
topic_facet	Algebraïsche topologie Topologia algébrica Algebraic topology Homology theory Homotopy theory Algebraische Topologie
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