The geometric and algebraic aspects of two-dimensional homotopy theory are both important areas of current research. Basic work on two-dimensional homotopy theory dates back to Reidemeister and Whitehead. The contributors to this book consider the current state of research beginning with introductory chapters on low-dimensional topology and covering crossmodules, Peiffer-Reid identities, and concretely discussing P2 theory. The chapters have been skillfully woven together to form a coherent picture, and the geometric nature of the subject is illustrated by over 100 diagrams. The final chapters round off neatly with a look at the present status of the conjectures of Zeeman, Whitehead and Andrews-Curtis.
Author(s): Cynthia Hog-Angeloni, Wolfgang Metzler, Allan J. Sieradski
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 1994
Language: English
Pages: 425
Front Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Editors' Preface......Page 10
Addresses of Authors......Page 12
1 Complexes of Low Dimensions and Group Presentations......Page 14
2 Simple-Homotopy and Low Dimensions......Page 24
3 P.L. Embeddings of 2-Complexes into Manifolds......Page 42
4 Three Conjectures and Further Problems......Page 57
1 Techniques in Homotopy......Page 64
2 Homotopy Groups for 2-Complexes......Page 75
3 Equivariant World for 2-Complexes......Page 88
4 Mac Lane-Whitehead Algebraic Types......Page 101
1 Bias Invariant & Homology Classification......Page 110
2 Classifications for Finite Abelian irl........Page 124
3 Classifications for Non-Finite 7r1 (with Cynthia Hog-Angeloni)......Page 130
1 Introduction......Page 138
2 Crossed and Precrossed Modules......Page 139
3 On the Second Homotopy Module of a 2-Complex......Page 153
4 Identity Properties......Page 161
1 The Theory of Pictures......Page 170
2 Generation of H2......Page 180
3 Applications and Results......Page 189
1 Introduction......Page 202
2 Decidability and Dehn's Algorithm......Page 203
3 Cayley Graph and van Kampen Diagrams......Page 205
4 Word Hyperbolic Groups and Combings......Page 210
5 Curvature Tests......Page 216
VII Fox Ideals, A(-Torsion and Applications to Groups and 3-Manifolds......Page 232
1 Fox ideals......Page 233
and the homological dimension of a group......Page 238
3 JI-torsion: Basic theory......Page 243
4 JV1(G), Nielsen equivalence of generating systems and Heegaard splittings......Page 246
5 N-torsion as generalization of the bias and (simple)-homotopy of (G, m)-complexes......Page 258
1 3-Manifolds......Page 264
2 Singular 3-Manifolds......Page 287
1 A Cancellation Theorem for 2-Complexes......Page 294
2 Stable Classification of 4-Manifolds......Page 299
3 A Cancellation Theorem for Topological 4-Manifolds......Page 303
4-Manifolds......Page 309
5 A Non-Cancellation Example for Simple-Homotopy Equivalent Topological 4-Manifolds......Page 312
6 Application of Cancellation to Exotic Structures on 4-Manifolds......Page 315
4-Manifolds and Pseudo-free Group Actions......Page 318
1 Introduction......Page 322
2 The Context of Whitehead's Question......Page 323
3 Structural Results......Page 325
4 Reductions, Evidence and Test Cases......Page 327
5 On the 7rl-Kernel......Page 333
6 Acyclic Coverings......Page 335
7 Finitely Generated Perfect Subgroups......Page 339
8 Kaplansky's Theorem......Page 341
9 Framed Links......Page 343
10 Open Questions (with J. Howie)......Page 346
1 Introduction......Page 348
2 Collapsing......Page 351
3 Some Special Ways of Collapsing P2 x I......Page 352
4 1-Collapsibility Modulo 2-Expansions......Page 361
5 Zeeman Conjecture for Special Polyhedra......Page 362
6 Generalizing (Z) to Higher Dimensions......Page 374
7 Open Problems......Page 376
1 Introduction......Page 378
2 Strategies and Characterizations......Page 379
3 Q**-Transformations and Presentations of Free Products......Page 386
4 Some Further Results......Page 393
Bibliography......Page 394
Index......Page 421
