Topological Methods in Group Theory

by Geoghegan, Ross

• ISBN13: 9780387746111
• ISBN10: 0387746110
• Format: Hardcover
• Copyright: 2007-11-30
• Publisher: Springer Nature

Summary

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 Biography

Ross Geoghegan is Professor of Mathematics at the State University of New York at Binghamton (Binghamton University).

Table of Contents

Algebraic Topology for Group Theory	p. 1
CW Complexes and Homotopy	p. 3
Review of general topology	p. 3
CW complexes	p. 10
Homotopy	p. 23
Maps between CW complexes	p. 28
Neighborhoods and complements	p. 31
Cellular Homology	p. 35
Review of chain complexes	p. 35
Review of singular homology	p. 37
Cellular homology: the abstract theory	p. 40
The degree of a map from a sphere to itself	p. 43
Orientation and incidence number	p. 52
The geometric cellular chain complex	p. 60
Some properties of cellular homology	p. 62
Further properties of cellular homology	p. 65
Reduced homology	p. 70
Fundamental Group and Tietze Transformations	p. 73
Fundamental group, Tietze transformations, Van Kampen Theorem	p. 73
Combinatorial description of covering spaces	p. 84
Review of the topologically defined fundamental group	p. 94
Equivalence of the two definitions	p. 96
Some Techniques in Homotopy Theory	p. 101
Altering a CW complex within its homotopy type	p. 101
Cell trading	p. 110
Domination, mapping tori, and mapping telescopes	p. 112
Review of homotopy groups	p. 116
Geometric proof of the Hurewicz Theorem	p. 119
Elementary Geometric Topology	p. 125
Review of topological manifolds	p. 125
Simplicial complexes and combinatorial manifolds	p. 129
Regular CW complexes	p. 135
Incidence numbers in simplicial complexes	p. 139
Finiteness Properties of Groups	p. 141
The Borel Construction and Bass-Serre Theory	p. 143
The Borel construction, stacks, and rebuilding	p. 143
Decomposing groups which act on trees (Bass-Serre Theory)	p. 148
Topological Finiteness Properties and Dimension of Groups	p. 161
K(G, 1) complexes	p. 161
Finiteness properties and dimensions of groups	p. 169
Recognizing the finiteness properties and dimension of a group	p. 176
Brown's Criterion for finiteness	p. 177
Homological Finiteness Properties of Groups	p. 181
Homology of groups	p. 181
Homological finiteness properties	p. 185
Synthetic Morse theory and the Bestvina-Brady Theorem	p. 187
Finiteness Properties of Some Important Groups	p. 197
Finiteness properties of Coxeter groups	p. 197
Thompson's group F and homotopy idempotents	p. 201
Finiteness properties of Thompson's Group F	p. 206
Thompson's simple group T	p. 212
The outer automorphism group of a free group	p. 214
Locally Finite Algebraic Topology for Group Theory	p. 217
Locally Finite CW Complexes and Proper Homotopy	p. 219
Proper maps and proper homotopy theory	p. 219
CW-proper maps	p. 227
Locally Finite Homology	p. 229
Infinite cellular homology	p. 229
Review of inverse and direct systems	p. 235
The derived limit	p. 241
Homology of ends	p. 248
Cohomology of CW Complexes	p. 259
Cohomology based on infinite and finite (co)chains	p. 259
Cohomology of ends	p. 265
A special case: Orientation of pseudomanifolds and manifolds	p. 267
Review of more homological algebra	p. 273
Comparison of the various homology and cohomology theories	p. 277
Homology and cohomology of products	p. 281
Topics in the Cohomology of Infinite Groups	p. 283
Cohomology of Groups and Ends Of Covering Spaces	p. 285
Cohomology of groups	p. 285
Homology and cohomology of highly connected covering spaces	p. 286
Topological interpretation of H*(G, RG)	p. 293
Ends of spaces	p. 295
Ends of groups and the structure of H[superscript 1](G, RG)	p. 300
Proof of Stallings' Theorem	p. 308
The structure of H[superscript 2](G, RG)	p. 314
Asphericalization and an example of H[superscript 3](G, ZG)	p. 321
Coxeter group examples of H[superscript n](G, ZG)	p. 324
The case H*(G, RG) = 0	p. 330
An example of H*(G, RG) = 0	p. 331
Filtered Ends of Pairs of Groups	p. 333
Filtered homotopy theory	p. 333
Filtered chains	p. 333
Filtered ends of spaces	p. 341
Filtered cohomology of pairs of groups	p. 344
Filtered ends of pairs of groups	p. 346
Poincare Duality in Manifolds and Groups	p. 353
CW manifolds and dual cells	p. 353
Poincare and Lefschetz Duality	p. 356
Poincare Duality groups and duality groups	p. 362
Homotopical Group Theory	p. 367
The Fundamental Group At Infinity	p. 369
Connectedness at infinity	p. 369
Analogs of the fundamental group	p. 379
Necessary conditions for a free Z-action	p. 383
Example: Whitehead's contractible 3-manifold	p. 387
Group invariants: simple connectivity, stability, and semistability	p. 393
Example: Coxeter groups and Davis manifolds	p. 396
Free topological groups	p. 397
Products and group extensions	p. 399
Sample theorems on simple connectivity and semistability	p. 401
Higher homotopy theory of groups	p. 411
Higher proper homotopy	p. 411
Higher connectivity invariants of groups	p. 413
Higher invariants of group extensions	p. 415
The space of proper rays	p. 418
Z-set compactifications	p. 421
Compactifiability at infinity as a group invariant	p. 425
Strong shape theory	p. 426
Three Essays	p. 431
Three Essays	p. 433
l[subscript 2]-Poincare duality	p. 433
Quasi-isometry invariants	p. 435
The Bieri-Neumann-Strebel invariant	p. 441
References	p. 453
Index	p. 463
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