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

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 $BGL (\Sigma^\infty (G_+))$.