Pre-taut sutured manifolds and essential laminations
Pre-taut sutured manifolds and essential laminations
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预拉紧缝合歧管和必要的叠片
DOI：
10.18910/10472
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发表时间：
2001
影响因子：
0.4
通讯作者：
Tsuyoshi Kobayashi
中科院分区：
文献类型：
--
作者：
M. Hirasawa;Tsuyoshi Kobayashi
1989年，D. Gabai和U. Oertel [8]引入了本质层压的概念，本质层压是一种介于不可压缩表面和绷紧叶理之间的混合物体，并推广了两者。我们说一个三维流形是层流的，如果它包含一个本质层。文[8]的一个重要结果是：层流流形的泛覆盖同胚于R。这一事实为研究通过Dehn手术沿着结获得的流形提供了一种强有力的方法，特别是关于性质P猜想（对3中的非平凡结进行非平凡Dehn手术永远不会产生简单连通的流形）和Cabling猜想（非索结上的Dehn手术不能产生可约流形）。例如，参见[4]非环面交替结，[3]，[12] 2-桥结，[17]大多数代数结和[9]具有某种基本缠结分解的结。我们注意到，在[8]中，一个3-流形是层流的当且仅当它包含一个本质分支曲面（定义见§2），而上述作者在[8]之后通过构造本质分支曲面得到了他们的结果。我们注意到缝合流形理论在[14]和[18]中得到了应用。他们的方法之一是在一个纽结的外部（）构造一个封闭的本质分支曲面，并证明在任何非平凡的Dehn填充沿着纽结（）之后，它仍然是本质的（我们称之为持续本质的）。我们可以看到，在[8]中，它具有强形式的性质P，并且布线猜想对于以下情况是正确的。(We假设一个纽结具有强性质P，如果沿着通过非平凡Dehn手术得到的每个流形都具有泛覆盖R。）然而，是否每一个具有强性质P的纽结在其补集中都有一个持续的本质层压，这是一个悬而未决的问题。在[1]、[2]中，M.布里特纳姆在证明纽结的强性质P方面有一个范式转变。而不是建设一个分支表面的补充一个给定的结，他首先建造了一个分支表面，然后嵌入一个结在其补充。更确切地说，他首先构建了一个封闭的分支表面3从任何在-
In 1989, D. Gabai and U. Oertel [8] introduced the concept of the essential lamination, which is a hybrid object lying between incompressible surfaces and taut foliations, and generalizing both. We say that a 3-manifold is laminar if it contains an essential lamination. An important result of [8] is that the universal covers of laminar manifolds are homeomorphic to R. This fact furnishes a strong method for studying the manifolds obtained by Dehn surgery along knots, especially concerning Property P Conjecture (nontrivial Dehn surgery on a nontrivial knot in 3 never yields a simply-connected manifold) and Cabling Conjecture (Dehn surgery on a non-cable knot cannot yield a reducible manifold). For example, see [4] for non-torus alternating knots, [3], [12] for 2-bridge knots, [17] for most algebraic knots and [9] for knots with some kind of essential tangle decompositions. We note that by [8] a 3-manifold is laminar if and only if it contains an essential branched surface (for the definition see §2), and the above authors who followed [8] obtained their results by constructing essential branched surfaces. We note that sutured manifold theory was used in [14] and [18]. One of their approaches is to construct a closed essential branched surface in the exterior ( ) of a knot and show that remains essential after any nontrivial Dehn filling along ∂ ( ) (we call such persistently essential). Then we see, by [8], that has Property P in a strong form and that the cabling conjecture is true for . (We say that a knot has strong Property P if every manifold obtained by a nontrivial Dehn surgery along has universal cover R.) It is, however, an open question whether or not every knot with strong Property P admits a persistently essential lamination in its complement. In [1], [2], M. Brittenham had a paradigm shift in proving strong Property P for knots. Instead of constructing a branched surface in the complement of a given knot, he first constructed a branched surface and then embedded a knot in its complement. More precisely, he first constructed a closed branched surface in 3 from any in-