{"paper":{"title":"Poincar\\'e duality for loop spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Rabinowitz Floer homology and cohomology satisfy Poincaré duality that preserves their graded Frobenius algebra structure.","cross_cats":["math.AT"],"primary_cat":"math.SG","authors_text":"Alexandru Oancea, Kai Cieliebak, Nancy Hingston","submitted_at":"2020-08-30T13:13:13Z","abstract_excerpt":"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.\n  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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The constructions and duality rely on the systematic use of the formalism of Tate vector spaces to manage the infinite-dimensional aspects of the loop spaces and Floer complexes.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Poincaré duality holds for Rabinowitz Floer homology and cohomology as graded Frobenius algebras, extending to open-closed TQFT duality, with applications to cotangent bundles and loop spaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Rabinowitz Floer homology and cohomology satisfy Poincaré duality that preserves their graded Frobenius algebra structure.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e9d6e04ee454a8812f976f68ccfa1fda17a9ba9345b06bbe662c41b6c3031127"},"source":{"id":"2008.13161","kind":"arxiv","version":3},"verdict":{"id":"9a0e8c10-7329-4046-9bd8-9bf4d71451fc","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T14:55:52.142707Z","strongest_claim":"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.","one_line_summary":"Poincaré duality holds for Rabinowitz Floer homology and cohomology as graded Frobenius algebras, extending to open-closed TQFT duality, with applications to cotangent bundles and loop spaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The constructions and duality rely on the systematic use of the formalism of Tate vector spaces to manage the infinite-dimensional aspects of the loop spaces and Floer complexes.","pith_extraction_headline":"Rabinowitz Floer homology and cohomology satisfy Poincaré duality that preserves their graded Frobenius algebra structure."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2008.13161/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":68,"sample":[{"doi":"","year":2008,"title":"A. Abbondandolo, A. Portaluri, and M. Schwarz. The homology of path spaces and Floer homology with conormal boundary conditions. J. Fixed Point Theory Appl., 4(2):263–293, 2008","work_id":"37a825d6-af1e-41e1-a926-96c553616924","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"A. Abbondandolo and M. Schwarz. On the Floer homology of cotan gent bun- dles. Comm. Pure Appl. Math. , 59(2):254–316, 2006","work_id":"7760b7fd-a4e5-4da6-9be7-9e4dac15ae6d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"A. Abbondandolo and M. Schwarz. Floer homology of cotangent b undles and the loop product. Geom. Topol., 14(3):1569–1722, 2010","work_id":"acb4b1db-21fa-4e0e-92cd-7feca001d476","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"A. Abbondandolo and M. Schwarz. On product structures in Floe r homology of cotangent bundles. In Global differential geometry , volume 17 of Springer Proc. Math., pages 491–521. Springer, Heidelberg, ","work_id":"535f5370-ceb6-4e28-a974-621ab76ec97e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"A. Abbondandolo and M. Schwarz. Corrigendum: On the Floer hom ology of cotangent bundles. Comm. Pure Appl. Math. , 67(4):670–691, 2014","work_id":"46833f26-34c8-4567-a81c-93af966ec508","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":68,"snapshot_sha256":"9a5ff9ef6d71e7c2e920c940b0b6d9ca7bfc102fc36be6115ad4b9af1b75ab6e","internal_anchors":1},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}