Transferring TTP-structures via contraction

Abstract

Let $A \otimes _tC$ be a twisted tensor product of an algebra $A$ and a coalgebra $C$, along a twisting cochain $t:C \rightarrow A$. By means of what is called the tensor trick and under some nice conditions, Gugenheim, Lambe and Stasheff proved in the early 90s that $A \otimes C$ is homology equivalent to the objects $M \otimes _{t^\prime}C$ and $A \otimes _{t''}N$, where $M$ and $N$ are strong deformation retracts of $A$ and $C$, respectively. In this paper, we attack this problem from the point of view of contractions. We find explicit contractions from $A\otimes _t C$ to $M \otimes_{t'}C$ and $A\otimes_{t''}N$. Applications to the comparison of resolutions which split off of the bar resolution, as well as to some homological models for central extensions are given.