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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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


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:
  • 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:
  • MathSciNet review: 1845023