Toward a topological description of Legendrian contact homology of unit conormal bundles
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912388193714176 |
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| author | Okamoto, Yukihiro |
| author_facet | Okamoto, Yukihiro |
| contents | 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 pseudo-holomorphic curve techniques, the Legendrian contact homology of $Λ_K$ is defined when, for instance, $Q=\mathbb{R}^n$. In this paper, aiming at giving another description of this homology, we define a graded $\mathbb{R}$-algebra for any pair $(Q,K)$ with orientations from a perspective of string topology and prove its invariance under smooth isotopies of $K$. The author conjectures that it is isomorphic to the Legendrian contact homology of $Λ_K$ with coefficients in $\mathbb{R}$ 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 original $\mathbb{Z}[π_1(Λ_K)]$ to $\mathbb{R}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_13705 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Toward a topological description of Legendrian contact homology of unit conormal bundles Okamoto, Yukihiro Symplectic Geometry Geometric Topology 53D42, 55P50, 57R17 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 pseudo-holomorphic curve techniques, the Legendrian contact homology of $Λ_K$ is defined when, for instance, $Q=\mathbb{R}^n$. In this paper, aiming at giving another description of this homology, we define a graded $\mathbb{R}$-algebra for any pair $(Q,K)$ with orientations from a perspective of string topology and prove its invariance under smooth isotopies of $K$. The author conjectures that it is isomorphic to the Legendrian contact homology of $Λ_K$ with coefficients in $\mathbb{R}$ 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 original $\mathbb{Z}[π_1(Λ_K)]$ to $\mathbb{R}$. 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. |
| title | Toward a topological description of Legendrian contact homology of unit conormal bundles |
| topic | Symplectic Geometry Geometric Topology 53D42, 55P50, 57R17 |
| url | https://arxiv.org/abs/2212.13705 |