Combinatorial Algebraic Topology

Dmitry Kozlov (auth.)354071961X, 978-3-540-71961-8

Combinatorial algebraic topology is a fascinating and dynamic field at the crossroads of algebraic topology and discrete mathematics. This volume is the first comprehensive treatment of the subject in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology, including Stiefel-Whitney characteristic classes, which are needed for the later parts. Readers – graduate students and working mathematicians alike – will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Our presentation of standard topics is quite different from that of existing texts. In addition, several new themes, such as spectral sequences, are included. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms. The main benefit for the reader will be the prospect of fairly quickly getting to the forefront of modern research in this active field.

Table of contents :
Front Matter….Pages I-XIX
Overture….Pages 1-4
Cell Complexes….Pages 7-35
Homology Groups….Pages 37-58
Concepts of Category Theory….Pages 59-75
Exact Sequences….Pages 77-87
Homotopy….Pages 89-100
Cofibrations….Pages 101-110
Principal Γ-Bundles and Stiefel—Whitney Characteristic Classes….Pages 111-125
Combinatorial Complexes Melange….Pages 129-149
Acyclic Categories….Pages 151-178
Discrete Morse Theory….Pages 179-209
Lexicographic Shellability….Pages 211-224
Evasiveness and Closure Operators….Pages 225-243
Colimits and Quotients….Pages 245-257
Homotopy Colimits….Pages 259-273
Spectral Sequences….Pages 275-289
Chromatic Numbers and the Kneser Conjecture….Pages 293-308
Structural Theory of Morphism Complexes….Pages 309-326
Using Characteristic Classes to Design Tests for Chromatic Numbers of Graphs….Pages 327-347
Applications of Spectral Sequences to Hom Complexes….Pages 349-376
Back Matter….Pages 377-389