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Poincaré duality for loop spaces

Part of: Topological linear spaces and related structures Quantum field theory; related classical field theories Symplectic geometry, contact geometry Classical Topics in Algebraic Topology Homotopy theory

Published online by Cambridge University Press:  30 March 2026

Kai Cieliebak ,
Nancy Hingston  and
Alexandru Oancea
Kai Cieliebak
Affiliation:
Universität Augsburg, Universitätsstrasse 14, D-86159 Augsburg, Germany kai.cieliebak@math.uni-augsburg.de
Nancy Hingston
Affiliation:
Department of Mathematics and Statistics, College of New Jersey, Ewing, NJ 08628, USA 4 Jeffrey Lane, Princeton Junction, NJ 08550, USA hingston@tcnj.edu
Alexandru Oancea
Affiliation:
Institut de recherche mathématique avancée (IRMA), Université de Strasbourg, 7 Rue Descartes, 67084 Strasbourg Cedex, France oancea@unistra.fr
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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 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 homologystring topologyPoincaré dualitytopological vector spacestopological quantum field theoriesFrobenius algebras

MSC classification

Primary: 53D40: Floer homology and cohomology, symplectic aspects 55P50: String topology
Secondary: 46A20: Duality theory 55M05: Duality 81T05: Axiomatic quantum field theory; operator algebras 81T30: String and superstring theories; other extended objects (e.g., branes)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 12 , December 2025 , pp. 3140 - 3212
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Copyright
© The Author(s), 2026. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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