1. Survey of Topological K-Theory

This expository section is intended only as motivation and historical perspective for the theory to be developed in these notes. See [Atiyah 1967; Karoubi 1978] for a complete development of the topological theory.

K-theory is the branch of algebraic topology concerned with the study of vector bundles by algebraic means. Vector bundles have long been important in geometry and topology. The first notions of K-theory were developed by Grothendieck in his work on the Riemann-Roch theorem in algebraic geometry, K-theory as a part of algebraic topology was begun by Atiyah and Hirzebruch [1961].

1.1. Vector Bundles

Informally, a vector bundle over a base space X (which we will usually take to be a compact Hausdorff space) is formed by attaching a finite-dimensional vector space to each point of X and tying them together in an appropriate manner so that the bundle itself is a topological space. More specifically:

DEFINITION 1.1.1. A vector bundle over X is a topological space E, a continuous map p : E → X, and a finite-dimensional vector space structure on each Ex = p-1(x) compatible with the induced topology, such that E is locally trivial: for each x ∈ X there is a neighborhood U of x such that E| u = p-1 (U) is isomorphic to a trivial bundle over U.

An isomorphism of vector bundles E and F over X is a homeomorphism from E to F which takes Ex to Fx for each x ∈ X and which is linear on each fiber.