Poincar\'e duality for loop spaces

TL;DR

This paper establishes a Poincaré duality for Rabinowitz Floer homology and cohomology, revealing their Frobenius algebra structure and implications for loop spaces and TQFTs.

Contribution

It proves a duality theorem preserving Frobenius algebra structures in Rabinowitz Floer homology and cohomology, extending to open-closed TQFTs and unifying various results in loop space theory.

Findings

Rabinowitz Floer homology and cohomology form graded Frobenius algebras.

A Poincaré duality theorem links homology and cohomology while preserving algebraic structures.

Specialized results for cotangent bundles relate to closed geodesics and loop space properties.

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 Poincar\'e 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.