Instanton Floer homology and contact structures

TL;DR

This paper introduces a novel contact invariant in sutured instanton Floer homology for 3-manifolds with boundary, demonstrating its properties and potential applications in understanding Stein fillings and fundamental groups.

Contribution

It is the first contact invariant in the instanton Floer setting, extending Floer homology techniques to contact 3-manifolds with boundary.

Findings

Invariant vanishes for overtwisted contact structures.

Invariant is nonzero for boundary manifolds embedding into Stein fillable manifolds.

Proposes a strategy linking fundamental groups to Stein fillings.

Abstract

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds -- with or without boundary -- defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by the authors in [1].