Homological invariants of codimension 2 contact submanifolds

TL;DR

This paper introduces new homological invariants for codimension 2 contact submanifolds, extending contact homology concepts to higher dimensions and providing tools to distinguish embeddings and obstruct cobordisms.

Contribution

It constructs a novel invariant deforming contact homology algebra, with applications to contact topology and embedding classification.

Findings

Examples of non-isotopic contact embeddings in overtwisted and tight manifolds.

New obstructions to relative symplectic and Lagrangian cobordisms.

Demonstration of invariants distinguishing formal isotopy from contact isotopy.

Abstract

Codimension 2 contact submanifolds are the natural generalization of transverse knots to contact manifolds of arbitrary dimension. In this paper, we construct new invariants of codimension 2 contact submanifolds. Our main invariant can be viewed as a deformation of the contact homology algebra of the ambient manifold. We describe various applications of these invariants to contact topology. In particular, we exhibit examples of codimension 2 contact embeddings into overtwisted and tight contact manifolds which are formally isotopic but fail to be isotopic through contact embeddings. We also give new obstructions to certain relative symplectic and Lagrangian cobordisms.