From gravity to string topology

Abstract

The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic $A_\infty$ algebra equipped with a scalar product of degree $-d$. In particular, it acts on the cyclic Hochschild complex of any Poincare duality algebra of degree $d$, and that action factors through a quotient dg properad $ST_{3-d}$ of ribbon graphs which is in focus of this paper. We show that its cohomology properad $H^\bullet(ST_{3-d})$ is highly non-trivial and that it acts canonically on the reduced equivariant homology $\bar{H}_\bullet^{S^1}(LM)$ of the loop space $LM$ of any simply connected $d$-dimensional closed manifold $M$. By its very construction, the string topology properad $H^\bullet(ST_{3-d})$ comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces $M_{g,n}$ of stable algebraic curves of genus $g$ with marked points. This result gives rise to new universal operations in string topology as well as reproduces in a unified way several known constructions: we show that (i) $H^\bullet(ST_{3-d})$ is also a properad under the properad of involutive Lie bialgebras in degree $3-d$ whose induced action on $\bar{H}_\bullet^{S^1}(LM)$ agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) $H^\bullet(ST_{3-d})$ is a properad under the properad of homotopy involutive Lie bialgebras in degree $2-d$; (iii) E. Getzler's gravity operad injects into $H^\bullet(ST_{3-d})$ implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on $\bar{H}_\bullet^{S^1}(LM)$.