Equivariant K-theory functor KG: Periodicity, thom isomorphism, localization, and completion

Abstract

Using the Grothendieck construction in the preceding chapter, we defined the functors KG for real and complex G-bundles and the Atiyah real KRG by mapping the semiring of G-vector bundles into its ring envelope. We saw that the basic properties of the equivariant versions of vector bundle theory have close parallels with the usual vector bundle theory, and the same is true for the related relative K-theories. This we carry further in this chapter for the version of topological K-theory that has close relations to index theory. The simple form of Bott periodicity for complex K-theory is the isomorphism K(X) ⊗ K(S2) → K(X × S2). The 2-sphere S2 which also can be identified as the one-dimensional complex projective space P1 (ℂ) plays a basic role.We can say that this is the periodicity theorem for a trivial line bundle X × ℂ → X over X. For the case of KRG periodicity and Thom isomorphism, it is convenient to have a version of the Bott isomorphism for any line bundle L→X over a space X or space X with involution τ. We survey two basic results of Atiyah and Segal: the localization theory and the completion theorem for the calculation of K-theory of classifying spaces. © Springer-Verlag Berlin Heidelberg 2008.