Which homotopy algebras come from transfer?

Abstract

We characterize $A_{\infty }$-structures that are transfers over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak $A_{\infty }$-morphisms over an arbitrary commutative ring. We then generalize our results to ${\mathcal{P}}_{\infty }$-structures over a field of characteristic zero, for any quadratic Koszul operad $\mathcal{P}$.

15 pages; final version. To appear in Proceedings of the American Mathematical Society