Stein fillings and SU(2) representations

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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Keywords

contact structures, Stein fillings, instanton Floer homology

Mathematical Subject Classification 2010

Primary: 53D40, 53D10

Secondary: 57R17, 57M27, 57R58

References

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

Volume 22, issue 7 (2018)

John A Baldwin and Steven Sivek

Abstract