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

TL;DR

This paper introduces a new algebraic approach to describe Legendrian contact homology of unit conormal bundles using string topology, providing invariance results and computations for specific cases.

Contribution

It defines a graded real algebra for pairs (Q,K) that offers an alternative to pseudo-holomorphic curve methods and conjecturally matches Legendrian contact homology.

Findings

Invariants computed for specific examples in all degrees.

Invariance under smooth isotopies of K established.

Explicit calculations in the 0-th degree for trivial normal bundles.

Abstract

For a smooth compact submanifold $K$ of a Riemannian manifold $Q$, its unit conormal bundle $\Lambda_K$ 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 $\Lambda_K$ 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 $\Lambda_K$ 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 $\mathbb{Z}[\pi_1(\Lambda_K)]$ 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.