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- W2070086361 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 $V = S^1 times R^2$." @default.
- W2070086361 created "2016-06-24" @default.
- W2070086361 creator A5054397434 @default.
- W2070086361 date "2006-01-01" @default.
- W2070086361 modified "2023-10-01" @default.
- W2070086361 title "Contact Structures on Open 3-Manifolds" @default.
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- W2070086361 doi "https://doi.org/10.4310/jsg.2006.v4.n1.a5" @default.
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