Algebra/Topology Seminar

Manuel Rivera: An algebraic model for the infinitesimal bialgebra of string topology

Abstract: The homology of the free loop space of a manifold has a rich non trivial algebraic structure organized by an action of the homology of a compactification of the moduli space of Riemann surfaces with input and output boundary. The Hochschild homology of a Frobenius algebra has an analogous algebraic structure organized by a moduli space of graphs called Sullivan diagrams. In both the free loop space and Frobenius algebra stories we encounter an involutive infinitesimal bialgebra which induces an involutive Lie bialgebra (originally discovered by Chas-Sullivan and Goresky-Hingston) on the S^1-equivariant and cyclic Hochschild homologies respectively. The two algebra structures (the Chas-Sullivan loop product and the Hochschild cup product) have been identified over the rationals and for simply connected manifolds by Felix, Thomas, Vigué-Poirrier and by Merkulov. It is strongly believe that these identifications can be extended to all the algebraic structure present in both stories. However, there are several subtleties concerning Poincare duality at the chain level that one has to take into account when identifying further operations such as the Goresky-Hingston coproduct on the free loop space with its analogous structure on the Hochschild side. I will describe how to associate functorially to a manifold an algebraic object that expresses Poincaré duality at the chain level as two operations which are formally Frobenius compatible. Then, I will outline how this construction can be used to show how the Goresky-Hingston coproduct on the homology of free loop space can be recovered algebraically. As a corollary, we will obtain that such coalgebra structure is an invariant of the homotopy type of the underlying manifold, settling a longstanding question posed by D. Sullivan. Time permitting, I will finish by describing current projects regarding algebraic models for string topology operations for the non simply connected case and by posing certain questions concerning geometric variants of string topology which are not homotopy invariants.