This book is written as a textbook on algebraic topology. The first part covers the material for two introductory courses about homotopy and homology. The second part presents more advanced applications and concepts (duality, characteristic classes, homotopy groups of spheres, bordism). The author recommends starting an introductory course with homotopy theory. For this purpose, classical results are presented with new elementary proofs. Alternatively, one could start more traditionally with singular and axiomatic homology. Additional chapters are devoted to the geometry of manifolds, cell complexes and fibre bundles. A special feature is the rich supply of nearly 500 exercises and problems. Several sections include topics which have not appeared before in textbooks as well as simplified proofs for some important results. Prerequisites are standard point set topology (as recalled in the first chapter), elementary algebraic notions (modules, tensor product), and some terminology from category theory. The aim of the book is to introduce advanced undergraduate and graduate (master's) students to basic tools, concepts and results of algebraic topology. Sufficient background material from geometry and algebra is included. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Tammo tom Dieck
Publisher: European Mathematical Society
Year: 2008
Language: English
Pages: 578
Cover......Page 1
EMS Textbooks in Mathematics......Page 3
Algebraic Topology......Page 4
9783037190487......Page 5
Preface......Page 6
Contents......Page 8
1.1 Basic Notions ......Page 14
1.2 Subspaces. Quotient Spaces ......Page 18
1.3 Products and Sums ......Page 21
1.4 Compact Spaces ......Page 24
1.5 Proper Maps ......Page 27
1.7 Topological Groups ......Page 28
1.8 Transformation Groups ......Page 30
1.9 Projective Spaces. Grassmann Manifolds ......Page 34
2 The Fundamental Group ......Page 37
2.1 The Notion of Homotopy ......Page 38
2.2 Further Homotopy Notions ......Page 43
2.3 Standard Spaces ......Page 47
2.4 Mapping Spaces and Homotopy ......Page 50
2.5 The Fundamental Groupoid ......Page 54
2.6 The Theorem of Seifert and van Kampen ......Page 58
2.7 The Fundamental Group of the Circle ......Page 60
2.8 Examples ......Page 65
2.9 Homotopy Groupoids ......Page 71
3.1 Locally Trivial Maps. Covering Spaces ......Page 75
3.2 Fibre Transport. Exact Sequence ......Page 79
3.3 Classification of Coverings ......Page 83
3.4 Connected Groupoids ......Page 85
3.5 Existence of Liftings ......Page 89
3.6 The Universal Covering ......Page 91
4.1 The Mapping Cylinder ......Page 94
4.2 The Double Mapping Cylinder ......Page 97
4.3 Suspension. Homotopy Groups ......Page 99
4.4 Loop Space ......Page 102
4.5 Groups and Cogroups ......Page 103
4.6 The Cofibre Sequence ......Page 105
4.7 The Fibre Sequence ......Page 110
5.1 The Homotopy Extension Property ......Page 114
5.2 Transport ......Page 120
5.3 Replacing a Map by a Cofibration ......Page 123
5.4 Characterization of Cofibrations ......Page 126
5.5 The Homotopy Lifting Property ......Page 128
5.6 Transport ......Page 132
5.7 Replacing a Map by a Fibration ......Page 133
6 Homotopy Groups ......Page 134
6.1 The Exact Sequence of Homotopy Groups ......Page 135
6.2 The Role of the Base Point ......Page 139
6.3 Serre Fibrations ......Page 142
6.4 The Excision Theorem ......Page 146
6.5 The Degree ......Page 148
6.6 The Brouwer Fixed Point Theorem ......Page 150
6.7 Higher Connectivity ......Page 154
6.8 Classical Groups ......Page 159
6.9 Proof of the Excision Theorem ......Page 161
6.10 Further Applications of Excision ......Page 165
7.1 A Stable Category ......Page 172
7.2 Mapping Cones ......Page 177
7.3 Euclidean Complements ......Page 181
7.4 The Complement Duality Functor ......Page 182
7.5 Duality ......Page 188
7.6 Homology and Cohomology for Pointed Spaces ......Page 192
7.7 Spectral Homology and Cohomology ......Page 194
7.8 Alexander Duality ......Page 198
7.9 Compactly Generated Spaces ......Page 199
8 Cell Complexes ......Page 209
8.1 Simplicial Complexes ......Page 210
8.2 Whitehead Complexes ......Page 212
8.3 CW-Complexes ......Page 216
8.4 Weak Homotopy Equivalences ......Page 220
8.5 Cellular Approximation ......Page 223
8.6 CW-Approximation ......Page 224
8.7 Homotopy Classification ......Page 229
8.8 Eilenberg–Mac Lane Spaces ......Page 230
9 Singular Homology ......Page 236
9.1 Singular Homology Groups ......Page 237
9.2 The Fundamental Group ......Page 240
9.3 Homotopy ......Page 241
9.4 Barycentric Subdivision. Excision ......Page 244
9.5 Weak Equivalences and Homology ......Page 248
9.6 Homology with Coefficients ......Page 250
9.7 The Theorem of Eilenberg and Zilber ......Page 251
9.8 The Homology Product ......Page 254
10.1 The Axioms of Eilenberg and Steenrod ......Page 257
10.2 Elementary Consequences of the Axioms ......Page 259
10.3 Jordan Curves. Invariance of Domain ......Page 262
10.4 Reduced Homology Groups ......Page 265
10.5 The Degree ......Page 269
10.6 The Theorem of Borsuk and Ulam ......Page 274
10.7 Mayer–Vietoris Sequences ......Page 278
10.8 Colimits ......Page 283
10.9 Suspension ......Page 286
11.1 Diagrams ......Page 288
11.2 Exact Sequences ......Page 292
11.3 Chain Complexes ......Page 296
11.4 Cochain complexes ......Page 298
11.5 Natural Chain Maps and Homotopies ......Page 299
11.6 Chain Equivalences ......Page 300
11.7 Linear Algebra of Chain Complexes ......Page 302
11.8 The Functors Tor and Ext ......Page 305
11.9 Universal Coefficients ......Page 308
11.10 The Künneth Formula ......Page 311
12.1 Cellular Chain Complexes ......Page 313
12.2 Cellular Homology equals Homology ......Page 317
12.3 Simplicial Complexes ......Page 319
12.4 The Euler Characteristic ......Page 321
12.5 Euler Characteristic of Surfaces ......Page 324
13.1 Partitions of Unity ......Page 331
13.2 The Homotopy Colimit of a Covering ......Page 334
13.3 Homotopy Equivalences ......Page 337
13.4 Fibrations ......Page 338
14.1 Principal Bundles ......Page 341
14.2 Vector Bundles ......Page 348
14.3 The Homotopy Theorem ......Page 355
14.4 Universal Bundles. Classifying Spaces ......Page 357
14.5 Algebra of Vector Bundles ......Page 364
14.6 Grothendieck Rings of Vector Bundles ......Page 368
15.1 Differentiable Manifolds ......Page 371
15.2 Tangent Spaces and Differentials ......Page 375
15.3 Smooth Transformation Groups ......Page 379
15.4 Manifolds with Boundary ......Page 382
15.5 Orientation ......Page 385
15.6 Tangent Bundle. Normal Bundle ......Page 387
15.7 Embeddings ......Page 392
15.8 Approximation ......Page 396
15.9 Transversality ......Page 397
15.10 Gluing along Boundaries ......Page 401
16.1 Local Homology Groups ......Page 405
16.2 Homological Orientations ......Page 407
16.3 Homology in the Dimension of the Manifold ......Page 409
16.4 Fundamental Class and Degree ......Page 412
16.5 Manifolds with Boundary ......Page 415
16.6 Winding and Linking Numbers ......Page 416
17.1 Axiomatic Cohomology ......Page 418
17.2 Multiplicative Cohomology Theories ......Page 422
17.3 External Products ......Page 426
17.4 Singular Cohomology ......Page 429
17.5 Eilenberg–Mac Lane Spaces and Cohomology ......Page 432
17.6 The Cup Product in Singular Cohomology ......Page 435
17.7 Fibration over Spheres ......Page 438
17.8 The Theorem of Leray and Hirsch ......Page 440
17.9 The Thom Isomorphism ......Page 444
18.1 The Cap Product ......Page 451
18.2 Duality Pairings ......Page 454
18.3 The Duality Theorem ......Page 457
18.4 Euclidean Neighbourhood Retracts ......Page 460
18.5 Proof of the Duality Theorem ......Page 464
18.6 Manifolds with Boundary ......Page 468
18.7 The Intersection Form. Signature ......Page 470
18.8 The Euler Number ......Page 474
18.9 Euler Class and Euler Characteristic ......Page 477
19 Characteristic Classes ......Page 480
19.1 Projective Spaces ......Page 481
19.2 Projective Bundles ......Page 484
19.3 Chern Classes ......Page 485
19.4 Stiefel–Whitney Classes ......Page 491
19.5 Pontrjagin Classes ......Page 492
19.6 Hopf Algebras ......Page 495
19.7 Hopf Algebras and Classifying Spaces ......Page 499
19.8 Characteristic Numbers ......Page 504
20.1 The Theorem of Hurewicz ......Page 508
20.2 Realization of Chain Complexes ......Page 514
20.3 Serre Classes ......Page 517
20.4 Qualitative Homology of Fibrations ......Page 518
20.5 Consequences of the Fibration Theorem ......Page 521
20.6 Hurewicz and Whitehead Theorems modulo Serre classes ......Page 523
20.7 Cohomology of Eilenberg–Mac Lane Spaces ......Page 526
20.8 Homotopy Groups of Spheres ......Page 527
20.9 Rational Homology Theories ......Page 531
21.1 Bordism Homology ......Page 534
21.2 The Theorem of Pontrjagin and Thom ......Page 542
21.3 Bordism and Thom Spectra ......Page 548
21.4 Oriented Bordism ......Page 550
Bibliography ......Page 554
Symbols ......Page 564
Index ......Page 570
