This book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field. To keep the length reasonable and the focus clear, the author assumes the reader knows or can easily learn the necessary algebra, but wants to see the topology done in detail. The central subject of the book is the theory of ends. Here the author adopts a new algebraic approach which is geometric in spirit.
Author(s): Ross Geoghegan
Series: Graduate Texts in Mathematics
Edition: 1
Publisher: Springer
Year: 2007
Language: English
Pages: 490
Preface......Page 7
Contents......Page 11
PART I: ALGEBRAIC TOPOLOGY FOR GROUP THEORY......Page 15
1.1 Review of general topology......Page 17
1.2 CW complexes......Page 24
1.3 Homotopy......Page 37
1.4 Maps between CW complexes......Page 42
1.5 Neighborhoods and complements......Page 45
2.1 Review of chain complexes......Page 49
2.2 Review of singular homology......Page 51
2.3 Cellular homology: the abstract theory......Page 54
2.4 The degree of a map from a sphere to itself......Page 57
2.5 Orientation and incidence number......Page 66
2.6 The geometric cellular chain complex......Page 74
2.7 Some properties of cellular homology......Page 76
2.8 Further properties of cellular homology......Page 79
2.9 Reduced homology......Page 84
3.1 Combinatorial fundamental group, Tietze transformations, Van Kampen Theorem......Page 87
Appendix: Presentations......Page 97
3.2 Combinatorial description of covering spaces......Page 98
Appendix: Cayley graphs......Page 106
3.3 Review of the topologically de.ned fundamental group......Page 108
3.4 Equivalence of the two de.nitions of the fundamental group of a CW complex......Page 110
4.1 Altering a CW complex within its homotopy type......Page 115
Appendix: the equivariant case......Page 123
4.2 Cell trading......Page 124
4.3 Domination, mapping tori, and mapping telescopes......Page 126
4.4 Review of homotopy groups......Page 130
4.5 GeometricproofoftheHurewiczTheorem......Page 133
5.1 Review of topological manifolds......Page 139
5.2 Simplicial complexes and combinatorial manifolds......Page 143
5.3 Regular CW complexes......Page 149
5.4 Incidence numbers in simplicial complexes......Page 153
PART II: FINITENESS PROPERTIES OF GROUPS......Page 155
6.1 The Borel construction, stacks, and rebuilding......Page 157
6.2 Decomposing groups which act on trees (Bass-Serre Theory)......Page 162
Appendix: Generalized graphs of groups......Page 171
7.1 K(G, 1) complexes......Page 175
7.2 Finiteness properties and dimensions of groups......Page 183
7.3 Recognizing the .niteness properties and dimension of a group......Page 190
7.4 Brown’s Criterion for .niteness......Page 191
8.1 Homology of groups......Page 195
8.2 Homological .niteness properties......Page 199
8.3 Synthetic Morse theory and the Bestvina-Brady Theorem......Page 201
9.1 Finiteness properties of Coxeter groups......Page 211
9.2 Thompson’s group F and homotopy idempotents......Page 215
9.3 Finiteness properties of Thompson’s Group......Page 220
9.4 Thompson’s simple group......Page 226
9.5 The outer automorphism group of a free group......Page 228
PART III: LOCALLY FINITE ALGEBRAIC TOPOLOGY FOR GROUP THEORY......Page 231
10.1 Proper maps and proper homotopy theory......Page 233
10.2 CW-proper maps......Page 241
11.1 In.nite cellular homology......Page 243
11.2 Review of inverse and direct systems......Page 249
11.3 The derived limit......Page 255
11.4 Homology of ends......Page 262
12.1 Cohomology based on in.nite and .nite (co)chains......Page 273
12.2 Cohomology of ends......Page 279
12.3 A special case: Orientation of pseudomanifolds and manifolds......Page 281
12.4 Review of more homological algebra......Page 287
12.5 Comparison of the various homology and cohomology theories......Page 291
12.6 Homology and cohomology of products......Page 295
PART IV: TOPICS IN THE COHOMOLOGY OF INFINITE GROUPS......Page 297
13.1 Cohomology of groups......Page 299
13.2 Homology and cohomology of highly connected covering spaces......Page 300
13.3 Topological interpretation of H∗(G,RG)......Page 307
13.4 Ends of spaces......Page 309
Appendix: Topology of the space of ends......Page 312
13.5 Ends of groups and the structure of H^1(G,RG)......Page 314
13.6 Proof of Stallings’ Theorem......Page 322
13.7 The structure of H^2(G,RG)......Page 328
13.8 Asphericalization and an example of H^3(G, ZG)......Page 335
13.9 Coxeter group examples of H^n(G, ZG)......Page 338
Appendix: Homology of connected sums......Page 341
13.10 The case H∗(G,RG) = 0......Page 344
13.11 An example of H∗(G,RG) = 0......Page 345
14.1 Filtered homotopy theory......Page 347
14.2 Filtered chains......Page 352
14.3 Filtered ends of spaces......Page 355
14.4 Filtered cohomology of pairs of groups......Page 358
14.5 Filtered ends of pairs of groups......Page 360
15.1 CW manifolds and dual cells......Page 367
15.2 Poincar´e and Lefschetz Duality......Page 370
15.3 Poincar´e Duality groups and duality groups......Page 376
PART V: HOMOTOPICAL GROUP THEORY......Page 381
16.1 Connectedness at in.nity......Page 383
Appendix. Semistability and trees of modules......Page 390
16.2 Analogs of the fundamental group......Page 393
16.3 Necessary conditions for a free Z-action......Page 397
16.4 Example: Whitehead’s contractible 3-manifold......Page 401
16.5 Group invariants: simple connectivity, stability, and semistability......Page 407
16.6 Example: Coxeter groups and Davis manifolds......Page 410
16.7 Free topological groups......Page 411
16.8 Products and group extensions......Page 413
16.9 Sample theorems on simple connectivity and semistability......Page 415
17.1 Higher proper homotopy......Page 425
17.2 Higher connectivity invariants of groups......Page 427
17.3 Higher invariants of group extensions......Page 429
17.4 The space of proper rays......Page 432
17.5 Z-set compactifications......Page 435
17.6 Compacti.ability at in.nity as a group invariant......Page 439
17.7 Strong shape theory......Page 440
PART VI: THREE ESSAYS......Page 445
18.1 l2-Poincar´e duality......Page 447
18.2 Quasi-isometry invariants......Page 449
Appendix: Quasi-isometry and geometry......Page 454
18.3 The Bieri-Neumann-Strebel invariant......Page 455
References......Page 467
Index......Page 477
