Journal of Symplectic Geometry

Contact Structures on Open 3-Manifolds

James J. Tripp

Full-text: Open access

Abstract

In this paper, we study contact structures on any open $3$-manifold $V$ that is the interior of a compact $3$-manifold. To do this, we introduce new proper contact isotopy invariants called the slope at infinity and the division number at infinity. We first prove several classification theorems for $T^2 \times [0, \infty)$, $T^2 \times \R$, and $S^1 \times \R^2$ using these concepts. The only other classification result on an open $3$-manifold is Eliashberg's classification on $\R^3.$ Our investigation uncovers a new phenomenon in contact geometry: There are infinitely many tight contact structures on $T^2 \times [0,1)$ that cannot be extended to a tight contact structure on $T^2 \times [0, \infty)$. Similar results hold for $T^2 \times \R$ and $S^1 \times \R^2$. Finally, we show that if every $S^2 \subset V$ bounds a ball or an $S^2$ end, then there are uncountably many tight contact structures on $V$ that are not contactomorphic, yet are isotopic. Similarly, there are uncountably many overtwisted contact structures on $V$ that are not contactomorphic, yet are isotopic. These uncountability results generalize work by Eliashberg when $V = S^1 \times \R^2$.

Article information

Source
J. Symplectic Geom., Volume 4, Number 1 (2006), 93-116.

Dates
First available in Project Euclid: 2 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.jsg/1154549060

Mathematical Reviews number (MathSciNet)
MR2240214

Zentralblatt MATH identifier
1117.53058

Citation

Tripp, James J. Contact Structures on Open 3-Manifolds. J. Symplectic Geom. 4 (2006), no. 1, 93--116. https://projecteuclid.org/euclid.jsg/1154549060


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