Complex Topological K-Theory

Efton Park0521856345, 9780521856348, 9780511388699

Topological K-theory is a key tool in topology, differential geometry and index theory, yet this is the first contemporary introduction for graduate students new to the subject. No background in algebraic topology is assumed; the reader need only have taken the standard first courses in real analysis, abstract algebra, and point-set topology. The book begins with a detailed discussion of vector bundles and related algebraic notions, followed by the definition of K-theory and proofs of the most important theorems in the subject, such as the Bott periodicity theorem and the Thom isomorphism theorem. The multiplicative structure of K-theory and the Adams operations are also discussed and the final chapter details the construction and computation of characteristic classes. With every important aspect of the topic covered, and exercises at the end of each chapter, this is the definitive book for a first course in topological K-theory.

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Table of contents :
Cover……Page 1
Series-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
Preface……Page 11
1.1 Complex inner product spaces……Page 13
1.2 Matrices of continuous functions……Page 17
1.3 Invertibles……Page 22
1.4 Idempotents……Page 29
1.5 Vector bundles……Page 33
1.6 Abelian monoids and the Grothendieck completion……Page 41
1.7 Vect(X) vs. Idem(C(X))……Page 43
1.8 Some homological algebra……Page 51
1.9 A very brief introduction to category theory……Page 55
Exercises……Page 59
2.1 Definition of K(X)……Page 63
2.2 Relative K-theory……Page 66
2.3 Invertibles and K……Page 74
2.4 Connecting K and K……Page 81
2.5 Reduced K-theory……Page 88
2.6 K-theory of locally compact topological spaces……Page 90
2.7 Bott periodicity……Page 95
2.8 Computation of some K groups……Page 115
2.9 Cohomology theories and K-theory……Page 119
2.10 Notes……Page 120
Exercises……Page 121
3.1 Mayer–Vietoris……Page 123
3.2 Tensor products……Page 126
3.3 Multiplicative structures……Page 131
3.4 An alternate picture of relative K……Page 142
3.5 The exterior algebra……Page 153
3.6 Thom isomorphism theorem……Page 159
3.7 The splitting principle……Page 169
3.8 Operations……Page 178
3.9 The Hopf invariant……Page 182
Exercises……Page 185
4.1 De Rham cohomology……Page 188
4.2 Invariant polynomials……Page 193
4.3 Characteristic classes……Page 199
4.4 The Chern character……Page 208
4.5 Notes……Page 212
Exercises……Page 213
References……Page 215
Symbol index……Page 216
Subject index……Page 218