Which homotopy algebras come from transfer?

Abstract

We characterize A ∞ A_\infty -structures that are equivalent to a given transferred structure 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 \text{ A ∞ A_\infty -morphisms} over an arbitrary commutative ring. We then generalize our results to P ∞ \mathcal {P}_\infty -structures over a field of characteristic zero, for any quadratic Koszul operad P \mathcal {P} .