---
abstract: 'Let $R$ be a ring spectrum and $ {\mathcal{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, $hAut^R({\mathcal{E}})$. This will generalize the result regarding $R$-line bundles proven by the authors in [@cjgauge]. The main application is the calculation of the homotopy type of $BGL_n(End (({\mathcal{L}}))$ where ${\mathcal{L}}\to X$ is any $R$-line bundle, and $End ({\mathcal{L}})$ 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, $G \to P \to M$, this leads to a description of the $K$-theory of the string topology spectrum in terms of the mapping space from $M$ to $BGL (\Sigma^\infty (G_+))$.'
author:
- |
    Ralph L. Cohen [^1]\
    Department of Mathematics\
    Stanford University\
    Bldg. 380\
    Stanford, CA 94305, USA
- |
    John D.S Jones\
    Mathematics Institute\
    Zeeman Building\
    Warwick University\
    Coventry, CV4 7AL, UK
title: 'Homotopy automorphisms of $R$-module bundles, and the $K$-theory of string topology'
---

Introduction {#introduction .unnumbered}
============

Let $R$ be a ring spectrum. In several places in the recent literature, the notion of an $R$-module bundle ${\mathcal{E}}\to X$ of rank $n$ has been defined and described [@units], [@5author], [@lind]. This is a parameterized $R$-module spectrum ${\mathcal{E}}$ over $X$, where each fiber $E_x$ admits an $R$-module equivalence $E_x {\xrightarrow}{\simeq} \vee_{n} R$. In analogy to vector bundles, it was proved in [@lind] that equivalence classes of rank $n$ $R$-module bundles over $X$ are in bijective correspondence with the set of homotopy classes, $[X, BGL_n(R)] = \pi_0(Map(X, BGL_n(R))$.

The main theorem in this paper is the identification of the homotopy type of the group-like monoid of homotopy automorphisms, ${hAut^R}({\mathcal{E}})$. This is the space of self equivalences of ${\mathcal{E}}$ living over the identity of $X$ that preserve the $R$-module structure. A precise definition will be given in the text of the paper.

\[main\] Let $R$ be a ring spectrum and $X$ a connected space of the homotopy type of a $CW$-complex. Let ${\mathcal{E}}\to X$ be an $R$-module bundle of rank $n$. Then there is an equivalence of group-like monoids, $$hAut^R({\mathcal{E}}) \simeq \Omega Map_{\mathcal{E}}(X, BGL_n(R))$$ where the subscript in this mapping space refers to the path component of maps that classify ${\mathcal{E}}$.

The special case of this theorem when $n=1$ was proved by the authors in [@cjgauge]. As discussed there, this result is important in string topology. Namely, given a principal bundle over a manifold $G \to P \to M$, if we let $${\mathcal{L}}= \Sigma^\infty_M (P_+)$$ be the fiberwise suspension spectrum of $P$ with a fiberwise disjoint basepoint, then the string topology spectrum of $P$, ${\mathcal{S}}(P) =  P^{-TM}$ is equivalent, as ring spectra, to the endomorphism ring $End^{\Sigma^\infty (G_+)}(\Sigma^\infty_M (P_+))$. Thus if $R = \Sigma^\infty (G_+)$, and ${\mathcal{E}}= \Sigma^\infty_M (P_+)$, then the above theorem, in the case $n= 1$, describes the homotopy type of the group-like monoid of units, $GL_1({\mathcal{S}}(P))$.

Theorem \[main\] in its general setting will have following implication to string topology.

Let ${\mathcal{L}}\to X$ be an $R$-line bundle, and let $\oplus_n {\mathcal{L}}\to X$ be the Whitney-sum of $n$-copies of ${\mathcal{L}}$. This is an $R$-module bundle of rank $n$.

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

As a special case we obtain the following result about the general linear groups of the string topology spectrum.

\[string\] If $G \to P \to M$ is a principal bundle over a manifold and ${\mathcal{L}}= \Sigma^\infty_M (P_+)$, there is a homotopy equivalence $$BGL_n ({\mathcal{S}}(P)) \simeq Map_{\oplus_n{\mathcal{L}}}(M, BGL_n(\Sigma^\infty(G_+)).$$ In particular there is an equivalence, $$BGL_n(LM^{-TM}) \simeq Map_{\iota_n}(M, BGL_n(\Sigma^\infty(\Omega M_+)).$$ Here $\iota_n$ classifies $\oplus_n {\mathcal{L}}$, where ${\mathcal{L}}= \Sigma^\infty_M ({\mathcal{P}}_+)$, and ${\mathcal{P}}\to M$ is a universal bundle in the sense that ${\mathcal{P}}$ is contractible.

Our next main result describes how these results have $K$-theoretic consequences for these ring spectra. First note that the string topology spectrum ${\mathcal{S}}(P)$ is a *nonconnective ring spectrum. Indeed its homology groups are nontrivial through dimension $-n$, where $n$ is the dimension of the manifold $M$. If ${\mathcal{S}}$ is a ring spectrum, let $K_{conn} (S)$ denote the algebraic $K$-theory spectrum of its connective cover $K({\mathcal{S}}_0)$. This $K$-theory spectrum has zero-space $\Omega^\infty K_{conn} ({\mathcal{S}}) =  K_0(\pi_0({\mathcal{S}})) \times BGL({\mathcal{S}})^+$, where the superscript $+$ denotes a group completion that will be described in the text of the paper. We will show that Corollary \[string\] implies the following result about $K$-theory.*

\[ktheory\] Given an $R$-line bundle ${\mathcal{L}}\to M$, there is a homology isomorphism $$\alpha :  Map_{{\mathcal{L}}} (M, BGL(\Sigma^\infty (G_+))) \to \Omega_0^\infty K_{conn}({\mathcal{S}}(P)).$$ The subscript $0$ denotes the path component of the basepoint in $\Omega^\infty K_{conn}({\mathcal{S}}(P))$. $Map_{{\mathcal{L}}} (M, BGL(\Sigma^\infty (G_+))$ is the homotopy colimit of the mapping spaces $Map_{\oplus_n {\mathcal{L}}} (M, BGL_n(\Sigma^\infty (G_+))$.

This theorem can be viewed as a statement about the group completion of the above mapping spaces. However we point out that this is *not the same as the mapping space to the group completion, which would be the zero space of the mapping spectrum $Map_0 (M, K(\Sigma^\infty (G_+))$. This spectrum calculates the ${K(\Sigma^\infty (G_+))}$-cohomology of $M$. However, as we will show below, we can define a homomorphism of $K$-theory groups,*

$$\label{kgcoho}
\gamma: K_{conn}^{-q}({\mathcal{S}}(P)) \to {K(\Sigma^\infty (G_+))}^{-q}(M)$$

which gives a partial geometric understanding of the ${K(\Sigma^\infty (G_+))}$-cohomology theory in terms of the algebraic $K$-theory of the string topology spectrum. The situation when $q=0$ was studied in detail by Lind in [@lind].

We conclude by observing two important applications of Theorem \[ktheory\].

\[application\] Let $M$ be a closed manifold. There are homology equivalences $$\begin{aligned}
  Map_{\Sigma^\infty_M ({\mathcal{P}}_+)} (M, BGL(\Sigma^\infty (\Omega M_+))  &\to \Omega^\infty_0 K_{conn}(LM^{-TM})   \notag  \\
  Map_{{\mathbb{S}}} (M, BGL({\mathbb{S}}))   &\to    \Omega_0^\infty K_{conn}(DM)  \notag\end{aligned}$$ where ${\mathbb{S}}$ is the sphere spectrum, and ${LM^{-TM}}$ is the Thom spectrum of the virtual bundle $-TM$ over $M$, pulled back over $LM$ via the map $e : LM \to M$ that evaluates a loop at the basepoint of the circle. $DM$ denotes the Spanier-Whitehead dual of the manifold $M$, which is an $E_\infty$-ring spectrum.

We point out that in these cases, the map $\gamma$ defined above (\[kgcoho\]) gives homomorphisms $$\gamma : K_{conn}^{-q}(LM^{-TM})  \to A(M)^{-q}(M)    \quad \text{and} \quad \gamma : K_{conn}^{-q}(D(M)) \to A(point)^{-q}(M).$$

The algebraic $K$-theory of nonconnective spectra was defined in terms of Waldhausen categories of modules by Blumberg and Mandell in [@blumbergmandell]. When $X$ is simply connected, they related the Waldhausen category defining $K(D(X))$ to the Waldhausen category defining $A(X) = K(\Sigma^\infty (\Omega X_+))$. It would be interesting to relate Corollary \[application\] regarding the $K_{conn}$-theory to their results.

This paper is organized as follows. In Section 1 we will prove Theorem \[main\] and derive Corollaries \[bgln\] and \[string\]. In Section 2 we describe the $K$-theoretic implications of Thoerem \[main\], and in particular we prove Theorem \[ktheory\].

Automorphisms of $R$-module bundles
===================================

Let $R$ be a ring spectrum. Let ${\mathcal{E}}\to X$ be an $R$-module bundle of rank $n$, in the sense of Lind [@lind]. This is a parameterized spectrum over $X$ in the sense of May and Sigurdsson [@maysigurd], where for each $x \in X$, the fiber ${\mathcal{E}}_x$ is an $R$-module spectrum of rank $n$. We denote the category of such bundles by $R-mod_n (X)$. Again, this category was defined in [@lind]. It was shown there that equivalence classes of such bundles are classified by homotopy classes of maps $X \to BGL_n(R)$. Fix a particular map $\gamma_{\mathcal{E}}: X \to BGL_n(R)$ classifying ${\mathcal{E}}$. This choice defines a basepoint in the mapping space $\gamma_{\mathcal{E}}\in Map_{\mathcal{E}}(X, BGL_n(R))$. The endomorphisms of ${\mathcal{E}}$ in $R-mod_n(X)$ is a parameterized spectrum which we denote by $End_M^R({\mathcal{E}}) \to X$. For every $x \in X$ this defines a fiber spectrum $End_M^R({\mathcal{E}})_x$ which is equivalent to the ring of endomorphisms $End^R(\vee_n R)$. $End_M^R({\mathcal{E}}) $ is a parameterized ring spectrum under composition. By taking a fibrant replacement if necessary, we can take sections to produce an ordinary spectrum $$End^R({\mathcal{E}}) = \Gamma_M(End^R_M ({\mathcal{E}})).$$ The parameterized ring structure on $End_M^R({\mathcal{E}})$ defines a ring spectrum structure on $End^R({\mathcal{E}})$.

\[haut\] We define the group-like monoid ${hAut^R}({\mathcal{E}})$ to be the units of the ring spectrum of endomorphisms, $${hAut^R}({\mathcal{E}}) = GL_1(End^R({\mathcal{E}})).$$

We are now ready to prove Theorem \[main\].

Consider the fiber bundle of infinite loop spaces given by taking the zero spaces of the fibrant model of $End^R_M({\mathcal{E}})$: $$\Omega^\infty (End^R(\vee_n R)) \to    \Omega^\infty_M End^R_M({\mathcal{E}}) \to M.$$ By restricting to path components of those $R$-module endomorphisms that consist of equivalences, we get a subbundle, which we will call $ {\mathcal{G}}L_n({\mathcal{E}})$: $$GL_n(R) \to  {\mathcal{G}}L_n({\mathcal{E}})  \to M.$$ Notice that ${hAut^R}({\mathcal{E}})$ can be described as the space of sections of this bundle, $ {hAut^R}({\mathcal{E}}) = \Gamma_M({\mathcal{G}}L_n({\mathcal{E}}))$. We now observe that the fiber homotopy type of the bundle ${\mathcal{G}}L_n ({\mathcal{E}}) \to M$ has another description. The homotopy class of map $\gamma_{\mathcal{E}}: M \to BGL_n(R)$ that classifies the $R$-module bundle ${\mathcal{E}}$, also classifies a principal $GL_n(R)$-bundle over $M$: $$GL_n(R) \to P_{\mathcal{E}}\to M.$$ Here we are replacing the group-like topological monoid $GL_n(R)$ by a topological group, which by abuse of notation, we continue to refer to as $GL_n(R)$. As was shown in [@lind], the relationship between $P_{\mathcal{E}}$ and the parameterized spectrum ${\mathcal{E}}$ is that there is an equivalence, $$\Omega^\infty_M {\mathcal{E}}\simeq  P_{\mathcal{E}}\times_{GL_n(R)} \Omega^\infty (\vee_n R),$$ where we are continuing the abuse of notation to allow $\Omega^\infty (\vee_n R)$ to refer to a weakly homotopy equivalent infinite loop space that carries an action of the group $GL_nR$. (See [@maysigurd] and [@lind] for details.)

Now consider the corresponding adjoint bundle, $GL_n(R) \to P_{\mathcal{E}}^{Ad} \to X$. In this notation $P_{\mathcal{E}}^{Ad} $ is the homotopy orbit space $P_{\mathcal{E}}^{Ad} = P_{\mathcal{E}}\times_{GL_n(R)} GL_n(R)$ where $GL_n(R)$ is acting on itself by the adjoint action (conjugation). A standard observation about adjoint bundles implies that $P_{\mathcal{E}}^{Ad} $ is the bundle whose fiber over $x \in M$ is the space of $GL_n(R)$-equivariant automorphisms of the fiber of $P_{\mathcal{E}}$ at $x$. The section space $\Gamma_M(P_{\mathcal{E}}^{Ad})$ is therefore the group of equivariant automorphisms of $P_{\mathcal{E}}$ living over the identity on $M$. This is known as the gauge group, ${\mathcal{G}}(P_{\mathcal{E}})$.

From this viewpoint it becomes clear that there is a map of fibrations $ P_{\mathcal{E}}^{Ad} \to {\mathcal{G}}L_n({\mathcal{E}})$ over $M$ (after taking appropriate fibrant and cofibrant replacements), which is an equivalence on the fibers. Therefore there is an equivalence of their spaces of sections, as group-like $A_\infty$-spaces. $$\label{autgl}
 \phi:  {\mathcal{G}}(P_{\mathcal{E}}) = \Gamma_M( P_{\mathcal{E}}^{Ad})  \simeq   \Gamma_M({\mathcal{G}}L_1({\mathcal{E}}))  =  hAut^R_X({\mathcal{E}}).$$ Now a well known theorem of Atiyah and Bott [@atiyahbott] says that the classifying space of the gauge group ${\mathcal{G}}(P)$ of a principal bundle $G \to P \to M$ is equivalent to the mapping space, $B{\mathcal{G}}(P) \simeq Map_P(X, BG)$. Theorem \[main\] now follows by applying this Atiyah-Bott equivalence to the principal bundle $GL_n(R) \to P_{\mathcal{E}}\to X.$

We now consider an application of Theorem \[main\] to an important special case. Let ${\mathcal{L}}\to X$ be an $R$-line bundle, and let $\oplus_n {\mathcal{L}}\to X$ be the Whitney-sum of $n$-copies of ${\mathcal{L}}$. This is an $R$-module bundle of rank $n$, which is defined to be the pullback under the diagonal map $\Delta^n : X \to X^n$ of the exterior $n$-fold product ${\mathcal{L}}^n \to X^n$. Consider the endomorphism spectrum $End^R({\mathcal{L}})$. As noted above, this is a ring spectrum. We first need the following observation:

\[GLN\] There is an equivalence of group-like monoids $$\lambda : {hAut^R}(\oplus_n {\mathcal{L}}) \simeq  GL_n(End^R({\mathcal{L}})).$$

Given a ring spectum $S$, recall that $GL_n(S)$ is defined to be the group-like monoid of units in the endomorphism ring, $$GL_n(S) =  GL_1(End^S(\vee_n S)).$$ Notice that there is a natural equivalence $End^S(\vee_n S) = \prod_n (\vee_n S)$. Also notice that $\vee_n End^R({\mathcal{L}}) \simeq Mor^R({\mathcal{L}}, \oplus_n {\mathcal{L}})$, 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 and let $P_{\mathcal{L}}\to M$ denote the principal $GL_1R$-bundle defined by the classifying map of ${\mathcal{L}}$, $\gamma_{\mathcal{L}}: M \to BGL_1(R)$. Let $\oplus_n P_{\mathcal{L}}$ denote the principal bundle classified by $$M {\xrightarrow}{\Delta} \prod_n M  {\xrightarrow}{\gamma_{\mathcal{L}}^n } \prod_n BGL_1(R) {\xrightarrow}{\mu} BGL_n(R).$$ Here $\mu$ is the usual block addition. This is the principal $GL_n(R)$-bundle associated to the $n$-fold Whitney sum $\oplus_n {\mathcal{L}}$.

Combining equivalence (\[autgl\]) with Lemma \[GLN\] defines an equivalence of group-like $A_\infty$-spaces, $$\begin{CD}
 \Psi_n :  {\mathcal{G}}(\oplus_n  P_{\mathcal{L}})   @>\phi > \simeq >   hAut^R(\oplus_n {\mathcal{L}}) @>\lambda >\simeq >  GL_n(End^R({\mathcal{L}})).
  \end{CD}$$

Notice that these equivalences respect the standard inclusions of wreath products, that makes the following diagrams commute:

$$\label{wreath}
   \begin{CD}
   \Sigma_k \int {\mathcal{G}}(\oplus_n P_{\mathcal{L}})      @>{\hookrightarrow}>>   {\mathcal{G}}((\oplus_{nk} P_{\mathcal{L}}) \\
   @V 1 \times \Psi_n^k VV     @VV\Psi_{nk} V \\
   \Sigma \int (GL_n(End^R({\mathcal{L}})))  @>>{\hookrightarrow}>  GL_{nk}(End^R({\mathcal{L}})).
   \end{CD}$$

Using the Barratt-Eccles $E_\infty$-operad, we can conclude the following:

The disjoint unions $\coprod_{n \geq 0} B{\mathcal{G}}(\oplus_n P_{\mathcal{L}})$ and $\coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))$ have $E_\infty$-algebra structures, and the equivalence $$\sqcup  B\Psi_n  :  \coprod_{n \geq 0} B{\mathcal{G}}(\oplus_n P_{\mathcal{L}})  \to  \coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))$$ is an equivalence of $E_\infty$-spaces.

We are now ready to deduce $K$-theoretic information from the above analysis.

If $X$ is an $E_\infty$ space, let $X^+$ denote the corresponding infinite loop space given by its group completion. In other words, $$X^+ = \Omega B(X).$$

Given a ring spectrum $S$, let $S_0$ denote its connective cover. We denote by $K_{conn}(S)$ the algebraic $K$-theory spectrum $K(S_0)$. A connected component of the zero space of this spectrum is given by $\Omega_0^\infty K_{conn}(S) = \left(\coprod_n BGL_n(S)\right)_0^+$, the component of the basepoint in the group completion. We note that the usual algebraic $K$-theory spectrum $K(S)$ is defined as the $K$-theory of the Waldhausen-category of perfect modules over $S$ [@blumbergmandell]. When $S$ is not connective, these spectra are not necessarily equivalent, where they are for connective ring spectra.

We can now conclude the following.

\[firstK\] If $R$ is a ring spectrum and $R \to {\mathcal{L}}\to M$ is an $R$-line bundle over a closed manifold $M$, then each path component of the zero space of the $K$-theory spectrum of the endomorphism ring has the following homotopy type: $$\Omega_0^\infty (K_{conn}(End^R({\mathcal{L}}))) \simeq   \left(\coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))\right)_0^+ \simeq  \left(\coprod_{n \geq 0} B{\mathcal{G}}(\oplus_n P_{\mathcal{L}})\right)_0^+.$$

We now proceed with the proof of Theorem \[ktheory\] as stated in the introduction.

As is usual, let $BGL (End ({\mathcal{L}}))$ denote the homotopy colimit of the inclusion maps $BGL_n (End ({\mathcal{L}})) {\hookrightarrow}BGL_{n+1} (End ({\mathcal{L}}))$ induced by the usual inclusions of the groups $GL_n (End ({\mathcal{L}})) {\hookrightarrow}GL_{n+1}(End({\mathcal{L}}))$. Then by the group-completion theorem of McDuff-Segal [@mcduffsegal], there is a homology equivalence, $$\alpha : {\mathbb{Z}}\times BGL (End({\mathcal{L}})) \to \left(\coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))\right)^+  \simeq {\mathbb{Z}}\times \Omega_0^\infty  (K_{conn}(End^R({\mathcal{L}}))).$$ Furthermore, by Proposition \[firstK\] there is an equivalence $${\mathbb{Z}}\times {\operatorname{hocolim}}_n B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \simeq {\mathbb{Z}}\times BGL (End({\mathcal{L}})).$$ The maps in the colimit are induced by the inclusions of gauge groups, $j_n:  {\mathcal{G}}(\oplus_n P_{\mathcal{L}}) {\hookrightarrow}{\mathcal{G}}(\oplus_{n+1}P_{\mathcal{L}})$ given by thinking of an automorphism of $\oplus_n P$ as an automorphism of $\oplus_{n+1} P$ by taking the direct sum with the identity map on the last factor.

Now for any principal bundle $G \to P \to M$, consider the Atiyah-Bott equivalence [@atiyahbott] mentioned in the proof of Theorem \[main\]: $$\beta : B{\mathcal{G}}(P) \simeq  Map_P(M, BG).$$ This equivalence is given by the observation that the equivariant mapping space $Map^G(P, EG)$ is a contractible space with a free action of the gauge group ${\mathcal{G}}(P) = Aut^G_M(P)$ given by precomposition. The orbit space of this action is then $Map_P (M, BG)$. The examples relevant here are the principal bundles $GL_n(R) \to \oplus_n P_{\mathcal{L}}\to M$. In this case we have an equivalence $$\beta_n:  B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) {\xrightarrow}{\simeq}   Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR).$$ With respect to these equivalences, the inclusions $j_n : B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \to B{\mathcal{G}}(\oplus_{n+1} P_{\mathcal{L}})$ are given up to homotopy by maps $$q_n : Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR) \to Map_{\oplus_{n+1}{\mathcal{L}}}(M, BGL_{n+1}R)$$ defined by sending $\phi : M \to BGL_n(R)$ to the composition $$q_n(\phi) : M {\xrightarrow}{\Delta} M \times M  {\xrightarrow}{\phi \times \gamma_{\mathcal{L}}}  BGL_n(R) \times BGL_1(R) {\xrightarrow}{\mu} BGL_{n+1}(R)$$ where $\gamma_{\mathcal{L}}$ is the fixed basepoint in $Map_{\mathcal{L}}(M,  BGL_1(R))$, and $\mu$ is the usual block pairing.

If we denote the homotopy colimit of these maps $hocolim_n Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR)$ by $Map_{\mathcal{L}}(M, BGL(R))$, we then have a homology equivalence $${\mathbb{Z}}\times Map_{\mathcal{L}}(M, BGL(R)) {\xrightarrow}{\simeq} {\mathbb{Z}}\times {\operatorname{hocolim}}_n B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \simeq {\mathbb{Z}}\times BGL (End({\mathcal{L}})) \to  {\mathbb{Z}}\times \Omega_0^\infty K_{conn}(End ({\mathcal{L}})),$$ which is the statement of Theorem \[ktheory\].

We now proceed to prove Corollary \[application\], describing $K_{conn}$ of both the string topology spectrum, and of the Spanier-Whitehead dual of a manifold. In the case of the string topology spectrum ${\mathcal{S}}(M) = {LM^{-TM}}$, this is the string topology of the universal principal bundle. That is, a principal bundle over $M$ fiber homotopy equivalent to the path-loop fibration $\Omega M \to {\mathcal{P}}_M \to M$. In the case of the Spanier-Whitehead dual, $DM$, one sees that if one considers the trivial principal bundle, when $G = \{id\}$, then the string topology spectrum of this bundle is simply the dual, $DM = Map(M, {\mathbb{S}})$. Both of these examples are analyzed in more detail in [@cjgauge]. Corollary \[application\] follows by applying Theorem \[ktheory\] to these special cases.

[99]{}

M. Ando, A. J. Blumberg, D. J. Gepner, M. J. Hopkins, and C. Rezk. *Units of ring spectra and Thom spectra* preprint, arXiv:0810.4535

M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins, and C. Rezk, *An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology*, preprint (2012).

M.F. Atiyah, *Thom complexes* , Proc. London Math. Soc. **(3)** , no. 11 (1961), 291–310.

M. Atiyah, and R. Bott *The Yang-Mills equations over Riemann surfaces*, Phil. Trans. R. Soc. Lond. A **308, 523-615 (1982)**

A. Blumberg and M. Mandell, *Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces*, J. Topology 4 (2011), no. 2, 327-342, preprint: arXiv:0912.1670

G. Carlsson and R.L. Cohen *The free loop space and the cyclic groups,* Commentari Math. Helvetici. **62 (1987), 423 - 449.**

M. Chas and D. Sullivan, *String Topology*. preprint: math.GT/9911159.

R.L. Cohen and J.D.S. Jones, *A homotopy theoretic realization of string topology*, Math. Annalen, **vol. 324, 773-798 (2002). preprint: math.GT/0107187**

R.L. Cohen and J.D.S. Jones, *Gauge theory and string topology*, preprint: http://arxiv.org/abs/1304.0613

R.L. Cohen and J.R. Klein, *Umkehr maps*, Homology, Homotopy, and Applications, **vol. 11 (1), (2009), 17-33. preprint: arXiv:0711.0540**

K. Gruher and P. Salvatore, *Generalized string topology operations* Proc. Lond. Math. Soc. (3) **96 (2008), 78Ð106.**

J.A. Lind, *Bundles of spectra and Algebraic $K$-theory*, preprint arXiv:1304567

J. P. May and J. Sigurdsson, [Parametrized homotopy theory]{}. Mathematical Surveys and Monographs, vol. 132, Amer. Math. Soc., 2006 Mathematical Surveys and Monographs, [**vol. 132**]{}, Amer. Math. Soc., 2006

D. McDuff, G.B. Segal, *Homology fibrations and the “group completion" theorem*, Invent. Math., **31**, (1976), 279 - 284.

[^1]: The first author was partially supported by a grant from the NSF.
