Let $A \otimes _tC$ be a {\em twisted tensor product} of an algebra $A$ and a coalgebra $C$, along a {\em twisting cochain} $t:C \rightarrow A$. By means of what is called the {\em tensor trick} and under some nice conditions, Gugenheim, Lambe and Stasheff proved in the early 90s that $A \ot _tC$ is homology equivalent to the objects $M \ot _{t'}C$ and $A \ot _{t''}N$, 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 $A\ot _t C$ to $M \ot _{t'}C$ and $A\ot_{t''}N$. 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.
Homology, Homotopy and Applications, Vol. 7(2005), No. 2, pp. 41-54