G-spaces, G-bundles, and G-vector bundles

Abstract

In this chapter, we introduce the basic notions related to objects being acted on by a group G. Since the objects like spaces and bundles have topologies, we will assume that the group is a topological group. Let G be a topological group which is eventually a compact Lie group.We consider G-spaces X and G-vector bundles with a base G-space. The aim is to develop the theory in a parallel fashion to ordinary bundle theory taking into account the G-action on both the base space and the total space together with actions between fibres. There are some generalities which apply to bundles in general including principal bundles which we outline in the first sections. A G-vector bundle over a point is just a representation of G on a vector space. In particular, we analyze part of the properties of G-vector bundles in terms of representation theory of G. Since the representation theory of compact groups is well understood, we will restrict ourselves to this case for the more precise theory, but compact groups will play a role for topological reasons too. For example, we show that G-homotopic maps always induce isomorphic G-vector bundles when G is compact. Of special interest is the group G of two elements. The symbol (τ) is a symbol for the group on two elements with generator τ which is always an involution under any action. The first example is that of complex conjugation τ(a+ib) = a-ib, where τ: ℂ → ℂ.We incorporate the concept real structures in K-theory using (τ)-bundles and mix this with the G-equivariant theory, for it plays a basic role in the relation between complex and real structures on vector bundles. © Springer-Verlag Berlin Heidelberg 2008.