Cellular structures in topology

Rudolf Fritsch, Renzo Piccinini0521327849, 9780521327848

This is the only up-to-date book describing the construction and the properties of CW-complexes. These spaces are important because firstly they are the correct framework for homotopy theory, and secondly most spaces that arise in pure mathematics are of this type. The authors discuss the foundations and also recent developments, for example, the theory of finite CW-complexes, CW-complexes in relation to the theory of fibrations, and Milnor’s work on spaces of the type of CW-complexes. They establish very clearly the relationship between CW-complexes and the theory of simplicial complexes, which is developed in great detail. Exercises are provided throughout the book; some are straightforward, others extend the text in a non-trivial way. For the latter; further reference is given for their solution. Each chapter ends with a section sketching the historical development. An appendix gives basic results from topology, homology and homotopy theory. These features will aid graduate students, who can use the work as a course text. As a contemporary reference work it will be essential reading for the more specialised workers in algebraic topology and homotopy theory.

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Table of contents :
Contents……Page 4
Preface……Page 6
1.0 Balls, spheres and projective spaces……Page 9
1.1 Adjunction of n-cells……Page 19
1.2 CW-complexes……Page 30
1.3 Some topological properties……Page 35
1.4 Subcomplexes……Page 41
1.5 Finiteness and countability……Page 48
1.6 Whitehead complexes……Page 59
Notes……Page 62
2.1 Morphisms……Page 64
2.2 Coproducts and products……Page 65
2.3 Some special constructions in the category CW^c……Page 70
2.4 The cellular approximation theorem and some related topics……Page 76
2.5 Whitehead’s realizability theorem……Page 84
2.6 Computation of the fundamental group……Page 86
2.7 Increasing the connectivity of maps……Page 91
Notes……Page 95
3.1 Geometric simplices and cubes……Page 97
3.2 Euclidean complexes……Page 105
3.3 Simplicial complexes……Page 117
3.4 Triangulations……Page 136
Notes……Page 139
4.1 The category Delta of finite ordinals……Page 140
4.2 Simplicial and cosimplicial sets……Page 147
4.3 Properties of the geometric realization functor……Page 160
4.4 Presimplicial sets……Page 173
4.5 Kan fibrations and Kan sets……Page 178
4.6 Subdivision and triangulation of simplicial sets……Page 206
Notes……Page 228
5.1 Preliminaries……Page 231
5.2 CW-complexes and absolute neighbourhood retracts……Page 234
5.3 n-ads and function spaces……Page 237
5.4 Spaces of the type of CW-complexes and fibrations……Page 244
Notes……Page 247
A.1 Weak hausdorff k-spaces……Page 249
A.2 Topologies determined by families of subspaces……Page 254
A.3 Coverings……Page 256
A.4 Cofibrations and fibrations; pushouts and pullbacks; adjunction spaces……Page 258
A.5 Union spaces of expanding sequences……Page 281
A.6 Absolute neighbourhood retracts in the category of metric spaces……Page 289
A.7 Simplicial homology……Page 291
A.8 Homotopy groups, n-connectivity, fundamental groupoid……Page 294
A.9 Dimension and embedding……Page 307
A.10 The adjoint functor generating principle……Page 311
Bibliography……Page 314
List of symbols……Page 320
Index……Page 327