Homotopy automorphisms of \(R\)-module bundles, and the \(K\)-theory of string topology (Q2398329)

Let \(R\) be a ring spectrum and let \({\mathcal E} \to X\) be an \(R\)-module bundle of rank \(n\), i.e. a parametrized \(R\)-module spectrum over \(X\) such that there is an \(R\)-module equivalence \({\mathcal E}_x\simeq\vee_nR\) for each \(x \in X\). Denote by \(End^R_X({\mathcal E}) \to X\) the parametrized ring spectrum consisting of endomorphisms of \({\mathcal E}\) and write \(End^R({\mathcal E})=\Gamma_X(End^R_X({\mathcal E}))\) for the module of its sections; it naturally inherits the structure of a ring spectrum. Then if we define the group-like monoid \(hAut^R({\mathcal E})\) to be the units of \(End^R({\mathcal E})\), the main result of this paper is that there is an equivalence of group-like monoids \[ hAut^R({\mathcal E}) \simeq\Omega Map_{\mathcal E}(X, BGL_n(R)) \] where the subscript \({\mathcal E}\) indicates the path-component of maps which classify \({\mathcal E}\). This is a generalization of the result in the case when \(n=1\) proved by the authors in [Bol. Soc. Mat. Mex., III. Ser. 23, No. 1, 233--255 (2017; Zbl 1378.55006)]. The proof follows from applying the theorem of \textit{M. F. Atiyah} and \textit{R. Bott} [Philos. Trans. R. Soc. Lond., Ser. A 308, 523--615 (1983; Zbl 0509.14014)] to the principal \(GL(n,R)\)-bundle \(P_{\mathcal E} \to X\) associated to \({\mathcal E}\), which tells us that setting \({\mathcal G}(P_{\mathcal E})=\Gamma_X(P_{\mathcal E}^{Ad})\), \(P_{\mathcal E}^{Ad}\) being the associated adjoint bundle, we have \[ B{\mathcal G}(P_{\mathcal E})\simeq Map_{\mathcal E}(X, BGL_n(R)). \] So we find that in essence the proof here relies on the discovery of the fact that there is an equivalence between \({\mathcal G}(P_{\mathcal E})\) and \(hAut^R({\mathcal E})\). All the other results of the paper are applications of the one stated above, where the authors discuss mainly subjects related to the string topology spectrum \({\mathcal S}(P)\) of a principal bundle \(P\) over a manifold \(M\).