Foliations and the topology of 3-manifolds
Foliations and the topology of 3-manifolds
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DOI：
10.4310/jdg/1214437784
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发表时间：
1987-11
影响因子：
2.5
通讯作者：
David Gabai
中科院分区：
文献类型：
--
作者：
David Gabai
在这篇文章中，我们讨论了3流形的拓扑结构与其所具有的叶形之间的密切关系。我们将介绍和陈述主要结果，然后用它和它的证明思想来陈述一些几何和拓扑推论。几乎所有结果的细节都可以在[G4]中找到。给定一个紧致的，连通的，定向的3流形，什么时候存在一个余维为1的横向定向叶理，它横向于dM并且没有Reeb分量？如果这样的7存在，dM必然是环面（可能是空的）并，M要么是SxS（7是积叶积），要么是不可约的。第一个条件遵循欧拉特征原因，后一个条件基本遵循Reeb [Re]和Novikov [N]的工作，尽管最初是Rosenberg [Ro]观察到的。我们的主要结果表明，当Ü2(M, dM) # 0时，这些条件是充分的。如果M上存在这样的叶化7，那么从Thurston [Ti]的工作可以得出，任何紧叶L都是Thurston范数最小化曲面[即，对于任何适当嵌入的T，对于[T] = [L] G H2{M, dM)（或H2{M），如果我们讨论H2（M)上的范数)，其中S‘表示类[L] G Ü2（M1dM）的^-（球面和圆盘分量）],\x{L’）\ < IxCni。我们的主要结果表明，对于满足上述必要条件的3流形M，任何范数最小化曲面都可以实现为没有Reeb分量的叶状的紧致叶。
In this announcement we discuss the close relationship between the topology of 3-manifolds and the foliations that is possesses. We will introduce and state the main result, then use it and the ideas of its proof to state some geometric and topological corollaries. Details to almost all the results can be found in [G4]. Given a compact, connected, oriented 3-manifold, when does there exist a codimension-1 transversely oriented foliation 7 which is transverse to dM and has no Reeb components? If such an 7 exists dM necessarily is a (possibly empty) union of tori and M is either SxS (and 7 is the product foliation) or irreducible. The first condition follows by Euler characteristic reasons while the latter basically follows from the work of Reeb [Re] and Novikov [N] although first observed by Rosenberg [Ro]. Our main result says that such conditions are sufficient when Ü2(M, dM) # 0. If such a foliation 7 exists on M then it follows from the work of Thurston [Ti] that any compact leaf L is a Thurston norm minimizing surface [i.e., \x{L')\ < IxCni for any properly embedded T with [T] = [L] G H2{M, dM) (or H2{M) if we were discussing the norm on H2(M)), where S' denotes ^-(sphere and disc components)] for the class [L] G Ü2(M1dM). Our main result says that for a 3-manifold M satisfying the above necessary conditions any norm minimizing surface can be realized as a compact leaf of a foliation without Reeb components.