Open-Closed String Topology via Fat Graphs
Abstract
Given a smooth closed manifold M with a family Li of closed submanifolds, we consider the free loop space LM and the spaces PM(Li,Lj) of open strings (paths g:[0,1]->M with g(0) in Li, and g(1) in Lj). We construct string topology operations resulting in an open-closed TQFT on the family (h*(LM),h*(PM(Li,Lj)) which extends the known string topology TQFT on h*(LM). Here, h* is a multiplicative generalized homology theory supporting orientations for M and the Li. To construct the operations, we introduce the notion of open-closed fat graph, generalizing fat graphs to the open-closed setting.
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