Surgery diagrams for contact 3-manifolds
Surgery diagrams for contact 3-manifolds
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发表时间：
2003-07
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通讯作者：
Fan Ding;H. Geiges;A. Stipsicz
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作者：
Fan Ding;H. Geiges;A. Stipsicz
在之前的两篇论文中，两位第一名的作者引入了接触3流形中沿Legendrian结的接触r-surgery的概念。他们还展示了如何（至少在原则上）将任何接触r-手术转换为一系列接触±1手术，并以此证明任何（闭合）接触3流形可以通过一系列这样的手术从3球上的标准接触结构中得到。在本文中，我们给出了该结果的一个较短的证明，并给出了一个更显式的将一个接触r-手术转化为正负1个手术的算法。我们用它给出了3球和S^1\ * S^2上的所有接触结构，以及任意闭合的、可定向的3流形上的所有过扭接触结构的显式手术图。这相当于一个新的Lutz-Martinet定理的证明，即这种流形上的每一个2平面场的同伦类都用一个接触结构来表示。
In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact plus or minus 1 surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on the 3-sphere by a sequence of such surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact r-surgery into plus or minus 1 surgeries. We use this to give explicit surgery diagrams for all contact structures on the 3-sphere and S^1\times S^2, as well as all overtwisted contact structures on arbitrary closed, orientable 3-manifolds. This amounts to a new proof of the Lutz-Martinet theorem that each homotopy class of 2-plane fields on such a manifold is represented by a contact structure.