Legendrian Ambient Surgery and Legendrian Contact Homology

TL;DR

This paper introduces Legendrian ambient surgery, a method for modifying Legendrian submanifolds in contact manifolds, and describes how it affects their contact homology via Chekanov-Eliashberg algebra isomorphisms.

Contribution

It develops a new surgical technique called Legendrian ambient surgery and relates the contact homology of the original and modified Legendrian submanifolds.

Findings

Constructs Legendrian embeddings after surgery within small neighborhoods.

Establishes algebra isomorphisms relating Chekanov-Eliashberg algebras pre- and post-surgery.

Shows a correspondence between augmentations of the algebras before and after surgery.

Abstract

Let $L \subset Y$ be a Legendrian submanifold of a contact manifold, $S\subset L$ a framed embedded sphere bounding an isotropic disc $D_S \subset Y \setminus L$, and use $L_S$ to denote the manifold obtained from $L$ by a surgery on $S$. Given some additional conditions on $D_S$ we describe how to obtain a Legendrian embedding of $L_S$ into an arbitrarily small neighbourhood of $L \cup D_S \subset Y$ by a construction that we call Legendrian ambient surgery. In the case when the disc is subcritical, we produce an isomorphism of the Chekanov-Eliashberg algebra of $L_S$ with a version of the Chekanov-Eliashberg algebra of $L$ whose differential is twisted by a count of pseudo-holomorphic discs with boundary-point constraints on $S$. This isomorphism induces a one-to-one correspondence between the augmentations of the Chekanov-Eliashberg algebras of $L$ and $L_S$.