hcal{L}}, \oplus_n {\mathcal{L}})$, where $Mor^R$ refers to the $R$-module morphisms in the category of parameterized spectra over $X$. Thus when $S = End^R({\mathcal{L}})$, we have a natural equivalence $$End^S(\vee_n S) \simeq \prod_n(Mor^R({\mathcal{L}}, \oplus_n {\mathcal{L}}) = End^R (\oplus_n {\mathcal{L}}).$$ Furthermore this equivalence clearly preserves the ring structure. It therefore induces an equivalence of their group-like monoids of units, $$GL_1(End^S(\vee_n S)) \simeq GL_1(End^R(\oplus_n {\mathcal{L}}).$$ The left side is by definition $GL_n (End^R({\mathcal{L}}))$, and the right side is by definition ${hAut^R}(\oplus_n {\mathcal{L}})$.

We now notice that Theorem \[main\] and Lemma \[GLN\] together imply the following.

\[bgln\] There is a homotopy equivalence $$BGL_n({End^R_X {\mathcal{L}}}) \simeq Map_{\oplus_n{\mathcal{L}}}(X, BGL_nR).$$

When $G \to P \to M$ is a principal bundle over a manifold and ${\mathcal{L}}= \Sigma^\infty_M (P_+)$, we obtain the following result about the general linear groups of the string topology spectrum,

\[string\] There is a homotopy equivalence $$\beta : BGL_n ({\mathcal{S}}(P)) {\xrightarrow}{\simeq} Map_{\oplus_n {\mathcal{L}}}(M, BGL_n(\Sigma^\infty(G_+))$$ where ${\mathcal{L}}$ is the line bundle $\Sigma^\infty (G_+) \to \Sigma^\infty_M(P_+) \to M$. In particular there is an equivalence $$BGL_n(LM^{-TM}) \simeq Map_{\iota_n}(M, BGL_n(\Sigma^\infty(\Omega M_+)).$$

$K$-theoretic implications
==========================

The goal of this section is to use Corollary \[string\] to prove Theorem \[ktheory\] as stated in the introduction. We will then use it to derive descriptions of the $K$-theory of the connective covers of the Spanier-Whitehead dual $D(M) = Map (\Sigma^\infty (M_+), {\mathbb{S}})$ and of the string topology ring spectrum ${LM^{-TM}}$.

To do this we need to understand the equivalence given in Corollary \[bgln\] more carefully, so that we can deduce $K$-theoretic consequences. As in that corollary, let ${\mathcal{L}}\to M$ be an $R$-line bundle