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# Homology, Homotopy and Applications

## Volume 7 (2005)

### Number 2

### Proceedings of a Special Session of a Joint RSME-AMS Meeting at Sevilla University

### Transferring *TTP*-structures via contraction

Pages: 41 – 54

DOI: http://dx.doi.org/10.4310/HHA.2005.v7.n2.a2

#### Authors

#### Abstract

Let $A \otimes_t C$ 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 $A \otimes_t C$ is homology equivalent to the objects $M \otimes_{t'} C$ 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.

#### Keywords

homology, homological perturbation theory, twisted tensor product, $A_{\infty}$-structures

#### 2010 Mathematics Subject Classification

Primary 55S10. Secondary 05Exx.