{"paper":{"title":"The Geometry of Loop Spaces III: Isometry Groups of Contact Manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Wodzicki-Chern-Simons forms on the loop space of a circle bundle detect that the isometry group has infinite fundamental group for large Chern class.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Satoshi Egi, Steven Rosenberg, Yoshiaki Maeda","submitted_at":"2020-11-03T15:48:05Z","abstract_excerpt":"Let $M_p$ be a circle bundle with first Chern class $p[\\omega]$ over a closed $4n$-dimensional integral symplectic manifold $\\bigl(\\overline{M},\\omega\\bigr)$. Equivalently, $M_p$ is a closed contact $(4n+1)$-manifold whose Reeb orbits are all closed and have the same period. For a metric $g$ on $M_p$ compatible with the symplectic structure and the geometry of the circle fiber, we use Wodzicki-Chern-Simons forms on the loop space $LM_p$ to prove that $\\pi_1({\\rm Isom}(M_p,g))$ is infinite for ${|p| \\gg 0}$. We also give the first high-dimensional examples of nonvanishing Wodzicki-Pontryagin fo"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we use Wodzicki-Chern-Simons forms on the loop space LM_p to prove that π₁(Isom(M_p,g)) is infinite for |p| ≫ 0. We also give the first high dimensional examples of nonvanishing Wodzicki-Pontryagin forms.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Existence of a metric g on M_p that is simultaneously compatible with the symplectic structure on the base and with the geometry of the circle fiber, so that the Wodzicki forms on LM_p are well-defined and detect non-trivial isometries (abstract, first paragraph).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves π₁(Isom(M_p,g)) infinite for |p|≫0 in certain contact (4n+1)-manifolds via Wodzicki-Chern-Simons forms on LM_p, plus first high-dim nonvanishing Wodzicki-Pontryagin forms.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Wodzicki-Chern-Simons forms on the loop space of a circle bundle detect that the isometry group has infinite fundamental group for large Chern class.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"47a30c873e8c0bb24f0844bfca35acf4d30d2b6f8be0be3e3176bfc8c8875c98"},"source":{"id":"2011.01800","kind":"arxiv","version":9},"verdict":{"id":"a8517832-803d-4e35-805a-a08c9bad491c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T14:29:31.534187Z","strongest_claim":"we use Wodzicki-Chern-Simons forms on the loop space LM_p to prove that π₁(Isom(M_p,g)) is infinite for |p| ≫ 0. We also give the first high dimensional examples of nonvanishing Wodzicki-Pontryagin forms.","one_line_summary":"Proves π₁(Isom(M_p,g)) infinite for |p|≫0 in certain contact (4n+1)-manifolds via Wodzicki-Chern-Simons forms on LM_p, plus first high-dim nonvanishing Wodzicki-Pontryagin forms.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Existence of a metric g on M_p that is simultaneously compatible with the symplectic structure on the base and with the geometry of the circle fiber, so that the Wodzicki forms on LM_p are well-defined and detect non-trivial isometries (abstract, first paragraph).","pith_extraction_headline":"Wodzicki-Chern-Simons forms on the loop space of a circle bundle detect that the isometry group has infinite fundamental group for large Chern class."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2011.01800/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"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"}