From gravity to string topology

The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic A∞ 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 ST3−d of ribbon graphs which is in focus of this paper. We show that its cohomology properad H∙(ST3−d) is highly non-trivial and that it acts canonically on the reduced equivariant homology H¯S1∙(LM) of the loop space LM of any simply connected d-dimensional closed manifold M. By its very construction, the string topology properad H∙(ST3−d) comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces Mg,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∙(ST3−d) is also a properad under the properad of involutive Lie bialgebras in degree 3−d whose induced action on H¯S1∙(LM) agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) H∙(ST3−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∙(ST3−d) implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on H¯S1∙(LM).