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On fillable contact structures up to homotopy

Author: Paolo Lisca
Journal: Proc. Amer. Math. Soc. 129 (2001), 3437-3444
MSC (2000): Primary 57M50, 57R57; Secondary 53C15, 57R15
Published electronically: April 24, 2001
MathSciNet review: 1845023
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Abstract: Let $Y$ be a closed, oriented $3$-manifold. The set $\mathcal{F}_Y$of homotopy classes of positive, fillable contact structures on $Y$ is a subtle invariant of $Y$, known to always be a finite set. In this paper we study $\mathcal{F}_Y$ under the assumption that $Y$ carries metrics with positive scalar curvature. Using Seiberg-Witten gauge theory, we prove that two positive, fillable contact structures on $Y$are homotopic if and only if they are homotopic on the complement of a point. This implies that the cardinality of $\mathcal{F}_Y$ is bounded above by the order of the torsion subgroup of $H_1(Y;{\mathbb Z})$. Using explicit examples we show that without the geometric assumption on $Y$ such a bound can be arbitrarily far from holding.

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Additional Information

Paolo Lisca
Affiliation: Dipartimento di Matematica, Università di Pisa I-56127 Pisa, Italy

Keywords: Contact structures, gauge theory, positive scalar curvature, symplectic fillings, Seiberg--Witten equations
Received by editor(s): November 29, 1999
Received by editor(s) in revised form: April 12, 2000
Published electronically: April 24, 2001
Additional Notes: The author’s research was partially supported by MURST
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2001 American Mathematical Society