Bott periodicity via simplicial spaces

Abstract

We give a proof of complex Bott periodicity that uses some of the general ideas of Segal and Quillen on classifying spaces [I, 2, 41. We attempt to understand which essential properties of the complex numbers and the unitary group are responsible for the periodicity, as contrasted with general facts on classifying spaces and linear groups over general topological rings (i.e., we compare complex K-theory and general algebraic K-theory). From this point of view the only essential properties of the complex numbers are: the “spectral theorem” (n x n unitary matrices can be diagonalized) and the fact that the Stiefel manifold of k-frames in Cd” is highly connected for large IV.