Stein fillings and SU(2) representations

TL;DR

This paper establishes a link between Stein structures on 4-manifolds and nontrivial SU(2) representations of 3-manifold groups, using sutured instanton homology invariants to distinguish contact structures.

Contribution

It proves that contact invariants are linearly independent for Stein-induced contact structures with distinct Chern classes, leading to new results on SU(2) representations of 3-manifold groups.

Findings

Contact invariants are linearly independent for certain Stein-induced contact structures.

Existence of nontrivial SU(2) representations for 3-manifold groups.

Stein domains bounding 3-manifolds imply nontrivial fundamental group representations.

Abstract

We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold with distinct Chern classes modulo torsion then their contact invariants in sutured instanton homology are linearly independent. As a corollary, we show that if a 3-manifold bounds a Stein domain that is not an integer homology ball then its fundamental group admits a nontrivial homomorphism to SU(2). We give several new applications of these results, proving the existence of nontrivial and irreducible SU(2) representations for a variety of 3-manifold groups.