Legendrian Ambient Surgery and Legendrian Contact Homology

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 $L_S$.