Rational algebraic K-theory of topological K-theory

TL;DR

This paper explores the rational algebraic K-theory of topological K-theory, establishing homotopy fiber sequences and computing K-theory for various spectra, revealing deep connections between algebraic and topological invariants.

Contribution

It introduces a homotopy fiber sequence relating K(ku) and K(Z) after rationalization and provides methods to compute K-theory for spectra like KU and MU.

Findings

Homotopy fiber sequence BBU -> K(ku) -> K(Z) after rationalization

Rational computation of K(KU) using localization sequence

Rational computation of K(MU) applicable to all connective S-algebras

Abstract

We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). 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 LX. We also rationally compute K(KU) by using the localization sequence, and K(MU) by a method that applies to all connective S-algebras.