Poincar\'e duality for loop spaces

Abstract

We prove a Poincar\'e duality theorem with products between Rabinowitz Floer homology and cohomology, for both closed and open strings. This lifts to a duality theorem between open-closed TQFTs. Specializing to the case of cotangent bundles, we define extended loop homology and cohomology and explain from a unified perspective pairs of dual results which have been observed over the years in the context of the search for closed geodesics. These concern critical levels, relations to the based loop space, manifolds all of whose geodesics are closed, Bott index iteration, level-potency, and homotopy invariance. We extend the loop cohomology product to include constant loops. We prove a relation conjectured by Sullivan between the loop product and the extended loop homology coproduct as a consequence of associativity for the product on extended loop homology.Comment: 87 pages, 14 figure