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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","authors_text":"Satoshi Egi, Steven Rosenberg, Yoshiaki Maeda","cross_cats":[],"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2020-11-03T15:48:05Z","title":"The Geometry of Loop Spaces III: Isometry Groups of Contact Manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.01800","kind":"arxiv","version":9},"verdict":{"created_at":"2026-05-24T14:29:31.534187Z","id":"a8517832-803d-4e35-805a-a08c9bad491c","model_set":{"reader":"grok-4.3"},"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","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.","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. 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