Topology and groupoids

Ronald Brown9781419627224, 1419627228

The book is intended as a text for a two-semester course in topology and algebraic topology at the advanced undergraduate or beginning graduate level. There are over 500 exercises, 114 figures, numerous diagrams. The general direction of the book is toward homotopy theory with a geometric point of view. This book would provide a more than adequate background for a standard algebraic topology course that begins with homology theory. For more information see www.bangor.ac.uk/r.brown/topgpds.html This version dated April 19, 2006, has a number of corrections made.

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Table of contents :
Contents……Page 3
Preface to the first edition……Page 7
Preface to the second edition……Page 10
Preface to the third edition……Page 18
1.1 Neighbourhoods in R……Page 27
1.2 Continuity……Page 30
1.3 Open sets, closed sets, closure……Page 36
1.4 Some generalisations……Page 41
2 Topological spaces……Page 45
2.1 Axioms for neighbourhoods……Page 46
2.2 Open sets……Page 48
2.3 Product spaces……Page 53
2.4 Relative topologies and subspaces……Page 56
2.5 Continuity……Page 59
2.6 Other conditions for continuity……Page 64
2.7 Comparison of topologies, homeomorphism……Page 68
2.8 Metric spaces and normed vector spaces……Page 71
2.9 Distance from a subset……Page 83
2.10 Hausdorff spaces……Page 85
3.1 The sum of topological spaces……Page 91
3.2 Connected spaces……Page 94
3.3 Components and locally connected spaces……Page 98
3.4 Path-connectedness……Page 103
3.5 Compactness……Page 108
3.6 Further properties of compactness……Page 114
4.1 Introduction……Page 123
4.2 Final topologies, identification topologies……Page 127
4.3 Subspaces, products, and identification maps……Page 131
4.4 Cells and spheres……Page 139
4.5 Adjunction spaces……Page 144
4.6 Properties of adjunction spaces……Page 152
4.7 Cell complexes……Page 157
5.1 Quaternions……Page 167
5.2 Normed vector spaces again……Page 172
5.3 Projective spaces……Page 173
5.4 Isometries of inner product spaces……Page 178
5.5 Simplicial complexes……Page 184
5.6 Bases and sub-bases for open sets; initial topologies……Page 188
5.7 Joins……Page 194
5.8 The smash product……Page 203
5.9 Spaces of functions, and the compact-open topology……Page 207
6 The fundamental groupoid……Page 227
6.1 Categories……Page 228
6.2 Construction of the fundamental groupoid……Page 233
6.3 Properties of groupoids……Page 241
6.4 Functors and morphisms of groupoids……Page 245
6.5 Homotopies……Page 251
6.6 Coproducts and pushouts……Page 260
6.7 The fundamental groupoid of a union of spaces……Page 266
7 Cofibrations……Page 279
7.1 The track groupoid……Page 280
7.2 Fibrations of groupoids……Page 288
7.3 Examples……Page 303
7.4 The gluing theorem for homotopy equivalences of closed unions……Page 309
7.5 The homotopy type of adjunction spaces……Page 316
7.6 The cellular approximation theorem……Page 329
8 Some combinatorial groupoid theory……Page 337
8.1 Universal morphisms……Page 338
8.2 Free groupoids……Page 349
8.3 Quotient groupoids……Page 355
8.4 Some computations……Page 358
9.1 The Van Kampen theorem for adjunction spaces……Page 365
9.2 The Jordan Curve Theorem……Page 378
10 Covering spaces, covering groupoids……Page 385
10.1 Covering maps and covering homotopies……Page 386
10.2 Covering groupoids……Page 391
10.3 On lifting sums and morphisms……Page 395
10.4 Existence of covering groupoids……Page 399
10.5 Lifted topologies……Page 405
10.6 The equivalence of categories……Page 413
10.7 Induced coverings and pullbacks……Page 419
10.8 Applications to subgroup theorems in group theory……Page 425
11.1 Groups acting on spaces……Page 435
11.2 Groups acting on groupoids……Page 441
11.3 General normal subgroupoids and quotient groupoids……Page 445
11.4 The semidirect product groupoid……Page 450
11.5 Semidirect product and orbit groupoids……Page 452
11.6 Full subgroupoids of orbit groupoids……Page 457
12 Conclusion……Page 463
A.1 Functions……Page 469
A.2 Finite, countable and uncountable sets……Page 474
A.3 Products and the axiom of choice……Page 480
A.4 Universal properties……Page 483
Glossary of terms from set theory……Page 491
Bibliography……Page 496
Glossary of symbols……Page 519
Index……Page 525