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

Abstract

Let R be a ring spectrum and $\mathcal {E}\rightarrow X$ 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, $hAut^R(\mathcal {E})$. This will generalize the result regarding R-line bundles proven by Cohen and Jones (Mex Bull Math, 2016). The main application is the calculation of the homotopy type of $BGL_n(End ((\mathcal {L}))$ where $\mathcal {L}\rightarrow X$ is any R-line bundle, and $End^R(\mathcal {L})$ 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, $G \rightarrow P \rightarrow M$, this leads to a description of the K-theory of the string topology spectrum in terms of the mapping space from M to $BGL \left( \Sigma ^\infty (G_+)\right)$.