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

Authors

V. Álvarez (Applied Maths Department, E.T.S.I. Informática, University of Seville, Spain)

J. A. Armario (Applied Maths Department, E.T.S.I. Informática, University of Seville, Spain)

M. D. Frau (Applied Maths Department, E.T.S.I. Informática, University of Seville, Spain)

P. Real (Applied Maths Department, E.T.S.I. Informática, University of Seville, Spain)

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.

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Published 1 January 2005