Equivariant K-Theory

Abstract

In this chapter we discuss fairly straightforward generalization of vector bundles and K-theory to the category of G-spaces where G is a topological group. A G-vector bundle over a G-space is a vector bundle which is compatible with the group action. The set of isomorphism classes of these bundles over a G-space X form a ring K _G (X) just as in the case when there is no group action, that is, when G is a trivial group. All the elementary theory developed so far for the non-equivariant case goes over without essential change to K _G . One of the main topics of this chapter is the equivariant Bott periodicity theorem which can be obtained using equivariant Toeplitz operators just as in the non-equivariant case. Our previous method also leads to equivariant Thorn isomorphism theorem when G is a commutative Lie group. Lastly we prove the localization theorem.