Mathematics – Symplectic Geometry
Scientific paper
2004-08-03
Mathematics
Symplectic Geometry
18 pages, 3 figures, additions to intro, clearer statement of thm 1.1 (same proof), Modifications made to section 6 giving sho
Scientific paper
In this paper, we study contact structures on any open 3-manifold V which is the interior of a compact 3-manifold. To do this, we introduce proper contact isotopy invariants called the slope at infinity and the division number at infinity. We first prove several classification theorems for T^2 x [0, \infty), T^2 x R, and S^1 x R^2 using these concepts. This investigation yields infinitely many tight contact structures on T^2 x [0,\infty), T^2 x R, and S^1 x R^2 which admit no precompact embedding into another tight contact structure on the same space. Finally, we show that if V is irreducible and has an end of nonzero genus, 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.
Tripp James
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