Volume 22, issue 7 (2018)

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Stein fillings and $\mathrm{SU}(2)$ representations

John A Baldwin and Steven Sivek

Geometry & Topology 22 (2018) 4307–4380

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.

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contact structures, Stein fillings, instanton Floer homology
Mathematical Subject Classification 2010
Primary: 53D40, 53D10
Secondary: 57R17, 57M27, 57R58
Received: 11 October 2017
Revised: 8 March 2018
Accepted: 8 April 2018
Published: 6 December 2018
Proposed: Ciprian Manolescu
Seconded: András I Stipsicz, Cameron Gordon
John A Baldwin
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
Steven Sivek
Department of Mathematics
Imperial College London
United Kingdom