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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On fillable contact structures up to homotopy
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by Paolo Lisca PDF
Proc. Amer. Math. Soc. 129 (2001), 3437-3444 Request permission

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
  • Email: lisca@dm.unipi.it
  • 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
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 3437-3444
  • MSC (2000): Primary 57M50, 57R57; Secondary 53C15, 57R15
  • DOI: https://doi.org/10.1090/S0002-9939-01-05964-0
  • MathSciNet review: 1845023