Rational algebraic $K$–theory of topological $K$–theory

Abstract

We show that after rationalization there is a homotopy fiber sequence

$${BBU}_{\otimes }\to K\left(ku\right)\to K\left(\mathbb{Z}\right).$$

We interpret this as a correspondence between the virtual $2$–vector bundles over a space $X$ and their associated anomaly bundles over the free loop space $\mathcal{ℒ}X$. We also rationally compute $K\left(KU\right)$ by using the localization sequence, and $K\left(MU\right)$ by a method that applies to all connective $S$–algebras.