Incompressible surfaces via branched surfaces
Incompressible surfaces via branched surfaces
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DOI：
10.1016/0040-9383(84)90031-4
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
W. Floyd;U. Oertel
中科院分区：
文献类型：
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作者：
W. Floyd;U. Oertel
一个包含一个2边不可压缩曲面的紧致、不可约、可定向的3-流形称为H&en流形。虽然哈肯流形可以包含无限多个非同位素的不可压缩曲面，但哈肯证明了可以通过某些剪切和粘贴操作从有限的曲面集合生成不可压缩的α-不可压缩曲面的所有同位素类（参见[3]或[7]）。然而，这些操作也产生不是不可压缩的曲面。在本文中，我们使用分支曲面，以产生确切的2边，不可压缩，a-不可压缩的表面（直到合线）在哈肯3-流形。我们的方法在某种程度上与哈肯的方法相似，因为我们通过将不可压缩，a-不可压缩曲面在Haken标准形中相对于3-流形的固定柄分解。3-流形M3中的分支曲面zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJihGFEDCBA B是在图1（a）所示的空间4 Y上局部建模的子空间。（在JM附近的图1（c）的空间9+上局部建模）。如图1所示，如果Scan是同位素的，则适当嵌入M中的表面S由B携带，使得它几乎平行于B延伸。（精确的定义将在1美元中给出。如果S由B携带，则在B中的点附近的S的片的数量确定B中的分支轨迹的补的每个分量的整数权重。权重满足某些明显的条件：在图1（a）中，w2+ w3= w1，w2+ w3 = w5，等等。相反，给定B上满足这些可加性条件的一组权重，可以通过沿着分支轨迹粘合B-（分支轨迹）的分量的平行副本来构造由B承载的对应曲面。
A COMPACT, irreducible, orientable 3-manifold which contains a 2-sided incompressible surface is called a H&en manifold. While a Haken manifold may contain an infinite number of nonisotopic, incompressible surfaces, Haken showed that one can generate all isotopy classes of incompressible, a-incompressible surfaces from a finite set of surfaces by certain cut-and-paste operations (See [3] or [7].) However, these operations also yield surfaces which are not incompressible. In this paper we use branched surfaces to produce exactly the 2-sided, incompressible, a-incompressible surfaces (up to isotopy) in a Haken 3-manifold. Our approach parallels Haken’s to some extent, because we obtain a finiteness statement by putting incompressible, a-incompressible surfaces in Haken’s normal form relative to a fixed handle decomposition of the 3-manifold.A branched surface zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA B in a 3-manifold M3 is a subspace locally modelled on the space 4Y shown in Fig. l (a)(locally modelled on the space 9+ of Fig. l (c) near JM). A surface S properly embedded in M is carried by B if Scan be isotoped so that it runs nearly parallel to B, as indicated in Fig. 1.(A precise definition will be given in $1.) If S is carried by B, then the number of sheets of S near a point in B determines an integer weight for each component of the complement of the branch locus in B. The weights satisfy certain obvious conditions: in Fig. l (a), w2+ w3= w,, wi+ w,= w5, etc. Conversely, given a set of weights on B satisfying these additive conditions, one can construct a corresponding surface carried by B by glueing parallel copies of the components of B-(branch locus) along the branch locus.