Bott periodicity maps and Clifford algebras

Abstract

Before there was K-theory, there was Bott periodicity. It was a result of the analysis of homogeneous spaces, their loop spaces, and classifying spaces of limits of Lie groups. In a parallel study of Clifford algebras, we see a periodicity which has much to do with Bott periodicity. This is pointed out in this chapter. When topological K-theory was introduced by analogy with Grothendieck's coherence sheaf K-theory, Bott periodicity was there and made possible an entire cohomology theory. Immediately, the vector bundle class approach to K-theory was coupled with the homotopy theory as in the discussion of the universal bundles and classifying spaces. The universal bundles of a group became the classifying space of K-theory by a simple limiting process. These limit spaces is the domain of the first forms of K-theory. For complex vector bundles, the group GL(n,ℂ) and its compact subgroup U(n) both play a role. The fact that the use of principal bundles over U(n) or over GL(n,ℂ) follows from the existence of a Hermitian metric on vector bundles. There is another approach which we take up in the next chapter by the classical Gram--Schmidt process.We use the compact forms, that is, U(n), for the homotopy classification of K-theory. After the homotopy analysis of Bott periodicity, we consider the Clifford algebra description. This we do in the context of KR-theory as in the previous chapter. © Springer-Verlag Berlin Heidelberg 2008.