Poincaré duality for loop spaces

doi:10.1112/S0010437X25102819

Poincaré duality for loop spaces

2026-03-30

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DOI：10.1112/S0010437X25102819

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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 & eacute; duality theorem between homology and cohomology that preserves this structure. This lifts to a duality theorem between graded open-closed topological quantum field theories (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.

Keywords:

Symplectic homology

string topology

Poincar & eacute

duality

topological vector spaces

topological quantum field theories

Frobenius algebras

Poincaré duality for loop spaces

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