Stein fillings and $\mathrm{SU}(2)$ 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 $SU\left(2\right)$. We give several new applications of these results, proving the existence of nontrivial and irreducible $SU\left(2\right)$ representations for a variety of $3$–manifold groups.