| Let a family Lbb∈ B of closed submanifolds of a fixed closed manifold M be given. We consider the free loop space LM and the spaces PM (La, Lb), defined as the space of paths gamma : [0,1] → M with gamma(0) ∈ La, gamma(1) ∈ Lb. We construct string topology operations involving these spaces, which may be described by saying that the family (h*( LM), {lcub}h*(PM ( La, Lb)){rcub} a,b∈B ) has a form of open-closed topological quantum field theory , extending the known string topology TQFT on h*(LM). Here, h * is a generalized homology theory supporting orientations for M and the Lb. To construct the operations, we introduce the notion of fat B -graph, generalizing fat graphs to the open-closed setting.; The second result is a description of the classifying space of structure-preserving diffeomorphisms of an open-closed surface. This is a surface whose topological boundary has a decomposition into string boundary and free boundary, together with a labeling of the free boundary components by elements of B . The theorem is that there is a category FatB whose objects are fat graphs with extra structure, satisfying FatB ≃⨆ SBDiff B (S; [∂]), where S ranges over most isomorphism types of open-closed surfaces, and DiffB (S; [∂]) stands for the group of structure-preserving diffeomorphisms of S which preserve each connected component of the boundary. |
