Homotopy automorphisms of R-module bundles, and the K-theory of string topology

TL;DR

This paper determines the homotopy type of automorphism groups of R-module bundles, generalizing previous results, and applies this to compute the K-theory of string topology spectra via mapping spaces.

Contribution

It identifies the homotopy type of homotopy automorphism groups of R-module bundles, extending prior work on R-line bundles, and connects this to string topology K-theory calculations.

Findings

Homotopy type of automorphism groups of R-module bundles is characterized.

Generalization of results from R-line bundles to higher rank bundles.

K-theory of string topology spectrum expressed via mapping spaces from manifolds.

Abstract

Let $R$ be a ring spectrum and $ E\to X$ an $R$-module bundle of rank $n$. Our main result is to identify the homotopy type of the group-like monoid of homotopy automorphisms of this bundle, $hAut^R(E)$. This will generalize the result regarding $R$-line bundles previously proven by the authors. The main application is the calculation of the homotopy type of $BGL_n(End ((L))$ where $L \to X$ is any $R$-line bundle, and $End (L)$ is the ring spectrum of endomorphisms. In the case when such a bundle is the fiberwise suspension spectrum of a principal bundle over a manifold, $G \to P \to M$, this leads to a description of the $K$-theory of the string topology spectrum in terms of the mapping space from $M$ to $BGL (\Sigma^\infty (G_+))$.