K-Theory

Michael Atiyah0201407922, 9780201407921

These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.

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Table of contents :
Table of Contents……Page 3
Introduction……Page 4
§1.1 Basic definitions……Page 5
§1.2 Operations on vector bundles……Page 10
§1.3 Sub-bundles and quotient bundles……Page 14
§1.4 Vector bundles on compact spaces……Page 19
§1.5 Additional structures……Page 36
§1.6 G-bundles over G-spaces……Page 39
§2.1 Definitions……Page 46
§2.2 The periodicity theorem……Page 48
§2.3 K_G(X)……Page 69
§2,4 Cohomology properties of K……Page 70
§2.5 Computations of K^*(X) for some……Page 84
§2.6 Multiplication in K^*(X,Y)……Page 89
§2.7 The Thom isomorphism……Page 107
§3.1 Exterior powers……Page 122
§3.2 The Adams operations……Page 140
§3.3 The group J(X)……Page 151
Appendix The space of Fredholm operators……Page 158
Power operations in K-theory……Page 172
K-theory and reality……Page 201