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
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.

Keywords
contact structures, Stein fillings, instanton Floer homology
Mathematical Subject Classification 2010
Primary: 53D40, 53D10
Secondary: 57R17, 57M27, 57R58
Publication
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
Authors
 John A Baldwin Department of Mathematics Boston College Chestnut Hill, MA United States https://www2.bc.edu/john-baldwin/ Steven Sivek Department of Mathematics Imperial College London London United Kingdom http://wwwf.imperial.ac.uk/~ssivek