# A theorem of Gugenheim on twisted tensor products

Suppose $A$ is a DGA algebra and $C$ a DGA coalgebra. An $(A,C)$-bimodule is an object $M$ that is both a right $A$-module and a left $C$-comodule in the evident compatible way. An $(A,C)$-bundle is an $(A,C)$-bimodule with a differential whose underlying $(A,C)$-bimodule is $C\otimes A$. Thus, for example, a twisted tensor product $C\otimes_t A$ is an $(A,C)$-bundle.

In the article Differential homological algebra by Husemoller, Moore and Stasheff, there is a Remark 1.7 (page 26 here). This says that in the article On a theorem of E.H. Brown by Gugenheim (available here), it is proven that every $(A,C)$-bundle is a twisted tensor product. I scanned the paper by Gugenheim but couldn't pin down where this is done.

I would highly appreciate if someone that is familiar with this article or the result cited in Remark 1.7 can pin down where this is proven and, if possible, provide other comprehensive references.

Context (Update 24/12) I would like to show that every minimal free resolution $C\otimes A$ of the trivial module $k$ of a graded connected $k$-algebra $A$ arises as a twisted tensor product $C\otimes_t A$ where $t :C\longrightarrow A$ is a $\gamma$-cochain (in the sense of Alain Proute's PhD thesis) between the minimal $A_\infty$-coalgebra $C=\text{Tor}$ and $A$.

I have a candidate for this $t$, obtained as follows. Choose a homotopy retraction $(i,p,h)$ of $BA\otimes_\beta A$ onto $C\otimes A$ satisfying the side conditions where $i$ is a honest inclusion of cycles, and produce a minimal $A_\infty$-coalgebra structure on $C$ along with an $A_\infty$-quasi-isomorphism $f : C\longrightarrow BA$ following say the formulas of Markl here, which greatly simplify since $BA$ is a honest coalgebra. Now consider the composition of $\Omega f: \Omega C \longrightarrow \Omega BA$ with the counit $\Omega BA\longrightarrow A$, which are both quasi-isomorphism. This corresponds to a cochain $t = \beta f : C\longrightarrow A$, following Proute. In this way one obtains a quasi-isomorphism $f\otimes_\beta 1 : C\otimes_t A\longrightarrow BA\otimes_\beta A$ which one can check is $i$ because of the side conditions and certain compatibility relations between $\beta$, $h$ and $\Delta_{BA}$.

Note that the first step produces a map $\Phi$ from minimal free resolutions of $k$ with an homotopy retraction data from $BA\otimes_\beta A$ to minimal $A_\infty$-coalgebra structures on $\text{Tor}$ along with the data of an $A_\infty$-quasi-isomorphism to $BA$, while the second step gives a map $\Psi$ from minimal structures along with such an $A_\infty$-quasi-isomorphism to $BA$ to minimal free resolutions of $k$ along with a quasi-isomorphism to $BA\otimes_\beta A$. The above says that $\Psi\Phi$ is the 'identity', modulo the fact I have to figure out how to recover the homotopy data and not only $i$.