Toward a topological description of Legendrian contact homology of unit conormal bundles

Okamoto, Yukihiro

doi:10.2140/agt.2025.25.951

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Toward a topological description of Legendrian contact homology of unit conormal bundles

Yukihiro Okamoto

Algebraic & Geometric Topology 25 (2025) 951–1027

DOI: 10.2140/agt.2025.25.951

Abstract

For a smooth compact submanifold $K$ of a Riemannian manifold $Q$ , its unit conormal bundle $Λ_{K}$ is a Legendrian submanifold of the unit cotangent bundle of $Q$ with a canonical contact structure. Using pseudoholomorphic curve techniques, the Legendrian contact homology of $Λ_{K}$ is defined when, for instance, $Q = ℝ^{n}$ . Aiming at giving another description of this homology, we define a graded $ℝ$ -algebra for any pair $(Q, K)$ with orientations from a perspective of string topology and prove its invariance under smooth isotopies of $K$ . We conjecture that it is isomorphic to the Legendrian contact homology of $Λ_{K}$ with coefficients in $ℝ$ in all degrees. This is a reformulation of a homology group, called string homology, introduced by Cieliebak, Ekholm, Latschev and Ng when the codimension of $K$ is $2$ , though the coefficient is reduced from the original $ℤ [π_{1} (Λ_{K})]$ to $ℝ$ . We compute our invariant (i) in all degrees for specific examples, and (ii) in the $0^{th}$ degree when the normal bundle of $K$ is a trivial $2$ -plane bundle.

Keywords

string topology, embedded submanifolds, Legendrian contact homology

Mathematical Subject Classification

Primary: 53D42

Secondary: 55P50, 57R17

References

Publication

Received: 23 January 2023

Revised: 21 November 2023

Accepted: 31 January 2024

Published: 16 May 2025

Authors

Yukihiro Okamoto
Research Institute for Mathematical Sciences Kyoto University Kyoto Japan	Department of Mathematical Sciences Tokyo Metropolitan University Tokyo Japan

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Volume 25, issue 2 (2025)