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

Yukihiro Okamoto
Dec 2022
摘要
For a smooth compact submanifold $K$ of a Riemannian manifold $Q$, its unitconormal bundle $\Lambda_K$ is a Legendrian submanifold of the unit cotangentbundle of $Q$ with a canonical contact structure. Using pseudo-holomorphiccurve techniques, the Legendrian contact homology of $\Lambda_K$ is definedwhen, for instance, $Q=\mathbb{R}^n$. In this paper, aiming at giving anotherdescription of this homology, we define a graded $\mathbb{R}$-algebra for anypair $(Q,K)$ with orientations from a perspective of string topology and proveits invariance under smooth isotopies of $K$. This is a reformulation of ahomology group, called string homology, introduced by Cieliebak, Ekholm,Latschev and Ng when the codimension of $K$ is $2$, though the coefficient isreduced from original $\mathbb{Z}[\pi_1(\Lambda_K)]$ to $\mathbb{R}$. Wecompute 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. Wealso give a prospect of proving that our invariant is isomorphic to theLegendrian contact homology of $\Lambda_K$ with coefficients in $\mathbb{R}$ inall degrees.
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