Legendrian contact homology for attaching links in higher dimensional subcritical Weinstein manifolds

TL;DR

This paper presents a method to simplify the computation of Legendrian contact homology for links in higher-dimensional Weinstein manifolds by reducing it to well-understood 1-jet space cases, with applications to loop space homology.

Contribution

It introduces a reduction technique for Legendrian contact homology in subcritical Weinstein manifolds under specific geometric assumptions, simplifying calculations by relating them to 1-jet space invariants.

Findings

Reduced Legendrian contact homology computation to 1-jet spaces

Applied the method to compute the homology of the free loop space of 3^2

Provided a framework for future generalizations

Abstract

Let $\Lambda$ be a link of Legendrian spheres in the boundary of a subcritical $2n$-dimensional Weinstein manifold $X$. We show that, under some geometrical assumptions, the computation of the Legendrian contact homology of $\Lambda$ can be reduced to a computation of Legendrian contact homology in 1--jet spaces. Since the Legendrian contact homology in 1--jet spaces is well studied, this gives a simplified way to compute the Legendrian contact homology of $\Lambda$. We restrict to the case when the attaching spheres of the subcritical handles of $X$ do not interact with each other, and we assume that there are no handles of index $n-1$. Moreover, we will only consider mod 2 coefficients for now. The more general situation will be addressed in a forthcoming paper. As an application we compute the homology of the free loop space of $\mathbb{CP}^2$.