String topology coproduct: geometric and algebraic aspects

String topology coproduct: geometric and algebraic aspects - Manuel Rivera

Manuel Rivera
University of Miami
May 11, 2017
The string topology coproduct is an intersection type operation, originally described by Goresky-Hingston and Sullivan, which considers transverse self-intersections on chains of loops in a smooth manifold and splits loops at these intersection points. The geometric chain level construction of string topology operations involves deforming chains to achieve certain transversality conditions and these deformations introduce higher homotopy terms for algebraic compatibilities and properties. The chain level theory for the coproduct is much more subtle than other operations and thus expected to depend on finer information of the underlying manifold. I will describe joint work with Dingyu Yang (IAS) in which we propose a framework and tools for constructing explicitly the coproduct at the chain level and the higher homotopies for certain algebraic structures around it. I will also announce an algebraic result, joint with Zhengfang Wang (BiCMR), that describes how combine the algebraic Hochschild complex analogues of the loop product and loop coproduct into a single operation in an unbounded chain complex which extends to a cyclic A-infinity algebra.