#### Volume 14, issue 5 (2014)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Homological perturbation theory for algebras over operads

### Alexander Berglund

Algebraic & Geometric Topology 14 (2014) 2511–2548
##### Abstract

We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads $\mathsc{O}$. To solve this problem, we introduce thick maps of $\mathsc{O}$–algebras and special thick maps that we call pseudo-derivations that serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory.

As an application, we derive explicit formulas for transferring $\Omega \left(\mathsc{C}\right)$–algebra structures along contractions, where $\mathsc{C}$ is any connected cooperad in chain complexes. This specializes to transfer formulas for ${\mathsc{O}}_{\infty }$–algebras for any Koszul operad $\mathsc{O}$, in particular for ${A}_{\infty }$–, ${C}_{\infty }$–, ${L}_{\infty }$– and ${G}_{\infty }$–algebras. A key feature is that our formulas are expressed in terms of the compact description of $\Omega \left(\mathsc{C}\right)$–algebras as coderivation differentials on cofree $\mathsc{C}$–coalgebras. Moreover, we get formulas not only for the transferred structure and a structure on the inclusion, but also for structures on the projection and the homotopy.