Complex topological K-theory / Efton Park.

Format:

Book

Author/Creator:

Park, Efton.

Series:

Cambridge studies in advanced mathematics ; 111.

Cambridge studies in advanced mathematics ; 111

Language:

English

Subjects (All):

Algebraic topology.

K-theory.

Physical Description:

x, 208 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2008.

Summary:

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.

Contents:

1 Preliminaries 1

1.1 Complex inner product spaces 1

1.2 Matrices of continuous functions 5

1.3 Invertibles 10

1.4 Idempotents 17

1.5 Vector bundles 21

1.6 Abelian monoids and the Grothendieck completion 29

1.7 Vect(X) vs. Idem(C(X)) 31

1.8 Some homological algebra 39

1.9 A very brief introduction to category theory 43

1.10 Notes 47

Exercises 47

2 K-theory 51

2.1 Definition of K0(X) 51

2.2 Relative K-theory 54

2.3 Invertibles and K-1 62

2.4 Connecting K0 and K-1 69

2.5 Reduced K-theory 76

2.6 K-theory of locally compact topological spaces 78

2.7 Bott periodicity 83

2.8 Computation of some K groups 103

2.9 Cohomology theories and K-theory 107

2.10 Notes 108

Exercises 109

3 Additional structure 111

3.1 Mayer-Vietoris 111

3.2 Tensor products 114

3.3 Multiplicative structures 119

3.4 An alternate picture of relative K0 130

3.5 The exterior algebra 141

3.6 Thom isomorphism theorem 147

3.7 The splitting principle 157

3.8 Operations 166

3.9 The Hopf invariant 170

3.10 Notes 173

Exercises 173

4 Characteristic classes 176

4.1 De Rham cohomology 176

4.2 Invariant polynomials 181

4.3 Characteristic classes 187

4.4 The Chern character 196

4.5 Notes 200

Exercises 201.

Notes:

Includes bibliographical references (page 203) and index.

ISBN:

9780521856348

0521856345

OCLC:

183264824