Volume 16, issue 4 (2012)

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

Christian Ausoni and John Rognes

Geometry & Topology 16 (2012) 2037–2065
Abstract

We show that after rationalization there is a homotopy fiber sequence

${BBU}_{\otimes }\to K\left(ku\right)\to K\left(ℤ\right)\phantom{\rule{0.3em}{0ex}}.$

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 $\mathsc{ℒ}X$. 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.

Keywords
algebraic $K$–theory, rational homotopy, topological $K$–theory, bordism spectra, determinants
Mathematical Subject Classification 2000
Primary: 55N15
Secondary: 18F25, 19Lxx