Transferring algebra structures on complexes

TL;DR

This paper presents a homological method for transferring algebra structures on complexes via homotopy equivalences, with applications to constructing dg algebra structures on resolutions, enhancing their explicitness and symmetry.

Contribution

It introduces a novel approach to transfer algebra structures using homotopy techniques, including the Perturbation Lemma, and applies it to build permutation-invariant dg algebra structures.

Findings

A new method for transferring algebra structures on complexes.

Construction of a permutation-invariant dg algebra on a resolution.

Application of the homotopy transfer to Koszul complexes using scaled de Rham maps.

Abstract

We discuss a homological method for transferring algebra structures on complexes along suitably nice homotopy equivalences, including those obtained after an application of the Perturbation Lemma. We study the implications for the Homotopy Transfer Theorems under such homotopy equivalences. As an application, we discuss how to use the homotopy on a Koszul complex given by a scaled de Rham map to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant.