Poincaré duality for loop spaces
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866915984694050816 |
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| author | Cieliebak, Kai Hingston, Nancy Oancea, Alexandru |
| author_facet | Cieliebak, Kai Hingston, Nancy Oancea, Alexandru |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2008_13161 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Poincaré duality for loop spaces Cieliebak, Kai Hingston, Nancy Oancea, Alexandru Symplectic Geometry Algebraic Topology 53D40, 55P50, 46A20, 55M05, 81T05, 81T30 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. |
| title | Poincaré duality for loop spaces |
| topic | Symplectic Geometry Algebraic Topology 53D40, 55P50, 46A20, 55M05, 81T05, 81T30 |
| url | https://arxiv.org/abs/2008.13161 |