Poincaré duality for loop spaces

Abstract

We show that Rabinowitz Floer homology and cohomology carry the structure of a graded Frobenius algebra for both closed and open strings. We prove a Poincaré duality theorem between homology and cohomology that preserves this structure. This lifts to a duality theorem between graded open-closed TQFTs. We use in a systematic way the formalism of Tate vector spaces. Specializing to the case of cotangent bundles, we define Rabinowitz loop homology and cohomology and explain from a unified perspective pairs of dual results that 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, and level-potency. Moreover, the graded Frobenius algebra structure gives meaning and proof to a relation conjectured by Sullivan between the loop product and coproduct.

arXiv comment86 pages, 18 figures. The main result is now phrased as an isomorphism of graded topological Frobenius algebras in the setup of Tate vector spaces. This is the final version of the paper, to be published in Compositio Mathematica

References in corpus (10)

• Functors and Computations in Floer homology with Applications Part II
• String Topology
• Floer homology and the heat flow
• Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
• Topological quantum field theory structure on symplectic cohomology
• Symplectic cohomology and Viterbo's theorem
• The homotopy invariance of the string topology loop product and string bracket
• Symplectic and contact differential graded algebras
• Multiplicative structures on cones and duality
• Loop coproduct in Morse and Floer homology

Cited by in corpus (3)

• Multiplicative structures on cones and duality
• Loop coproduct in Morse and Floer homology
• Eight flavours of cyclic homology