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On the homotopy transfer of \(A_\infty\) structures. (English) Zbl 1424.18017
Let \(V\) and \(W\) be chain complexes and \(f:V\to W\), \(g:W\to V\) chain maps such that \(gf\) is homotopic to the identity automorphism of \(V\), via a chain homotopy \(h\). Assume moreover that \(V\) is equipped with an \(A_\infty\)-structure.
The author of the paper under review takes on the painstaking task of filling in the missing details in the proof of the main result of [M. Markl, Suppl. Rend. Circ. Mat. Palermo (2) 79, 139–151 (2006; Zbl 1112.18007)] that provides an explicit \(A_\infty\)-structure on \(W\), explicit extensions of the chain maps \(f\), \(g\) into \(A_\infty\)-morphisms, and an explicit extension of the homotopy \(h\) into an \(A_\infty\)-homotopy. The author’s approach is based on the translation into the lingo of coalgebras and codifferentials, with the aim of simplifying sign factors. In the last section he shows that, when some additional assumptions, called the side conditions, on the initial data are satisfied, the formulas for the extended structures reduce to the ones provided by the classical homological perturbation lemma.

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
55S99 Operations and obstructions in algebraic topology
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