FOLIATIONS AND THE TOPOLOGY OF 3-MANIFOLDS. II
FOLIATIONS AND THE TOPOLOGY OF 3-MANIFOLDS. II
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DOI：
10.4310/jdg/1214441488
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发表时间：
2008
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通讯作者：
David Gabai
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作者：
David Gabai
在本文及其续文中，我们研究了以下问题：设M是一个紧定向的不可约3流形，其边界为环面。如果TV是通过沿着一条基本曲线α填充dM而得到的（即，N是通过沿着a向dM附加一个2手柄，然后用一个3单元盖住得到的S而得到的），那么N是否具有紧叶理？本文考虑H2(M) Φ 0的情况，并在[3]中研究了在s中的结点k上通过零坐标系手术得到N的情况。利用充盈流形上叶的存在性，我们得到了一些拓扑推论。我们现在（为了清楚起见）陈述本文主要结果（定理1.7,1.8）的一个稍微不那么一般的版本。定理。设M为边界为环面的阿向Haken 3流形，H2(M) Φ 0。设S为任意Thurston范数最小化曲面，表示一类H2(M)。那么，在dM中沿一条基本的简单闭合曲线填充M得到的流形N，除l-例外（不超过同位素）外，具有紧绷的有限深度叶理！使得S是&的叶，填充的核心横向于F。将我们的主要结果与Alexander， Reeb， Novikov和Thurston的工作（参见[2,2.5和2.8]）和一些三维拓扑相结合，我们得到以下结果。推论2.14。M是M1的连通和？，其中每个Mi要么是s上的一个定向环面，要么是球束。如果k是M中的一个结，它不在3单元格中，那么k由它的补决定。
In this paper and its continuation [3] we investigate the following question: Let M be a compact oriented irreducible 3-manifold whose boundary is a torus. If TV is obtained by filling dM along an essential curve α (i.e., N is obtained by attaching a 2-handle to dM along a and then capping off the resulting S with a 3-cell), then does N possess a taut foliation? In this paper we consider the case when H2(M) Φ 0 and in [3] we study the case when N is obtained by zero frame surgery on a knot k in S. Using the existence of foliations on the filled manifolds we obtain a number of topological corollaries. We now state (for reasons of clarity) a slightly less general version of the main result (Theorems 1.7,1.8) of this paper. Theorem. Let M be an atoroidal Haken 3-manifold whose boundary is a torus and H2(M) Φ 0. Let S be any Thurston norm minimizing surface representing a class of H2(M). Then with at most l-exception (up to isotopy) the manifold N obtained by filling M along an essential simple closed curve in dM possesses a taut finite depth foliation !F such that S is a leaf of & and the core of the filling is transverse to !F. Combining our main result with the work of Alexander, Reeb, Novikov, and Thurston (see [2, 2.5 and 2.8]) and some 3-dimensional topology we obtain the following results. Corollary 2.14. Let M be a connected sum of M1 ? , Mr where each Mi is either an oriented torus or sphere bundle over S. If k is a knot in M which does not lie in a 3-cell, then k is determined by its complement.