Lectures on Polyhedral Topology

Table of contents......Page 2
Introduction......Page 5
1 Definition of polyhedra......Page 8
2 Convexity......Page 9
3 Open convex sets......Page 16
4 Calculus of boundaries......Page 20
5 Convex cells......Page 26
6 Presentations of polyhedra......Page 31
7 Refinement by bisection......Page 34
1 Triangulation of polyhedra......Page 39
2 Triangulation of maps......Page 41
III Topology and approximation......Page 47
1 Neighborhoods that retract......Page 48
2 Approximation theorem......Page 50
3 Mazur's criterion......Page 55
1 Abstract theory I......Page 58
2 Abstract theory II......Page 62
3 Geometric theory......Page 65
4 Polyhedral cells, spheres and manifolds......Page 76
5 Recalling homotopy facts......Page 83
V General position......Page 86
1 Nondegeneracy......Page 87
2 ND(n)-spaces. Definition and elementary properties......Page 90
3 Characterizations of ND(n)-spaces......Page 93
4 Singularity dimension......Page 102
VI Regular neighborhoods......Page 119
1 Isotopy......Page 120
2 Centerings, isotopies and neighborhoods of subpolyhedra......Page 122
3 Definition of 'regular neighborhood'......Page 128
4 Collaring......Page 131
5 Absolute regular neighborhhods and some Newmanish theorems......Page 140
6 Collapsing......Page 145
7 Homogeneous collapsing......Page 149
8 The regular neighborhood theorem......Page 154
9 Some applications and remarks......Page 166
10 Conclusion......Page 170
1 Regular collapsing......Page 173
2 Applications......Page 184
Appendix to Chapter VII......Page 189
1 Handles......Page 196
2 Relative n-manifolds and their handle presentations......Page 198
3 Statement of the theorems, applications, comments......Page 203
4 Modification of handle presentations......Page 215
5 Cancellation of handles......Page 220
6 Insertion of cancelling pairs of handles......Page 227
7 Elimination of 0- and 1-handles......Page 236
8 Dualisation......Page 240
9 Algebraic description......Page 243
10 Proofs of the main theorems (A and B)......Page 251
11 Proof of the Isotopy Lemma......Page 258