Algebra/Topology seminar

Speaker: Andrea Bianchi

Title: String topology and graph cobordisms.

Abstract: String topology, introduced by Chas and Sullivan, is the study of the homology of mapping spaces of the form M^X, where M is a closed oriented d-dimensional manifold and X is a space. Fixing M and letting X vary, the homology groups H_*(M^X) carry additional algebraic structure coming from (contravariant) functoriality in X and from Poincare' duality of M; the most famous example is the Chas-Sullivan product. We introduce a symmetric monoidal infty-category GrCob of "graph cobordisms between spaces"; we define compatible local coefficient systems xi_d on the morphism spaces of GrCob, and use the twisted homology of the morphism space GrCob(Y,X) to define higher string operations H_*(M^X)--->H_*(M^Y). We assemble all such operations into a "graph field theory" associated with M, i.e. a contravariant symmetric monoidal functor out of the linearisation of GrCob given by xi_d. We recover some basic operations, including the Chas-Sullivan product, as special cases.The construction of the graph field theory can in fact be carried out for any oriented Poincare' duality space M, and is natural in M with respect to orientation-preserving equivalences; in particular all string operations we obtain are automatically homotopy invariant. The main technical input to the construction is a recent result by Barkan-Steinebrunner, giving a universal property for the category of graph cobordisms between finite sets in terms of commutative Frobenius algebras.