## On fillable contact structures up to homotopy

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- by Paolo Lisca PDF
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**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.## References

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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