Taut normal surfaces
Taut normal surfaces
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拉紧法线表面
DOI：
10.1016/0040-9383(95)00008-9
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
Ningyi Wang
中科院分区：
文献类型：
--
作者：
J. L. Tollefson;Ningyi Wang
给定一个紧的3-流形M和一个三角剖分Y，设9%表示由法曲面理论得到的Y的正规化匹配方程的解的紧的、凸的、线性的胞元（在R”中）。这个射影解空间9%是一个重要的但相对未知的对象，与一个3-流形M相关联。法向曲面与某些匹配方程的容许解一一对应，这种解到9%的投影称为相应法向曲面的投影类。将M的拓扑与9%的结构联系起来是一个重要的问题。这个问题的一个方面是要找到的情况下，正常的表面，其投影类属于相同的面对9~共享显着的性质。它也是感兴趣的，以确定是否一个给定的类的表面总是表示为正常的表面，其投影类是一个顶点的9%。只有有限数量的曲面（至多多个副本）的投影类是顶点，这些曲面很容易从匹配方程中的数据构造出来。9%的面与分支曲面密切相关，我们的一些结果可以转化为关于分支曲面的结果。法向曲面F的权是F n T-（1）中的点的数目，并且用wt（F）表示。雅科和Oertel [4]证明了：如果F是闭不可约3-流形M中的最小权内射法曲面，则带有F的9%极小面CF只带有内射曲面。由此可见，有一个内射正常的表面，其投影类是一个顶点的9%，并导致一个算法，用于检测这样的表面。若M不是不可约的，则M存在一个2-球面的完全分解集，其投射类为9%的面的线性无关顶点[6]。文[6]还证明了，如果一个3-流形M有一个可压缩边界，则存在一个本质压缩圆盘，其投影类是一个9%的顶点。本文的目的是研究9~的具有最小权的绷紧法向曲面的面CF。对于一个非零的同调类gEH ~ 2（M，aM; Z），范数x（g）被定义为x-（G）=-x（G的G-球面和圆盘分量）在所有表示g的适当嵌入曲面G上的下确界，x到H ~ 2（M，aM; R）的扩张是利用凸性和连续性来实现的.一个适当地嵌入到M中的定向的不可压缩的不可压缩的曲面F称为拉紧的，如果它的同调类[F]在H，（M，aM; Z）中是非平凡的，F是~--极小化的，并且F的分支没有同调平凡的并.一个张紧曲面F称为Iw-张紧曲面，如果它在所有表示同调类[F]的张紧曲面中具有最小权。如果F是lw-张紧的，则F的n个成对不相交的拷贝，记为nF，是表示类n [F]的lw-张紧曲面（参见[11]中的引理1）。
GIVEN a compact 3-manifold M with a triangulation Y, let 9% denote the compact, convex, linear cell (in R”) of solutions to the normalized matching equations for Y from normal surface theory. This projective solution space 9% is an important yet relatively unknown object associated to a 3-manifold M. Normal surfaces are in a one-to-one correspondence with the admissible solutions to certain matching equations and the projection of such a solution to 9% is called the projective class of the corresponding normal surface. It is an important problem to relate the topology of M to the structure of 9%. One aspect of this problem is to find circumstances under which normal surfaces whose projective classes belong to the same face of 9~ share significant properties. It is also of interest to determine whether or not a given class of surfaces is always represented by a normal surface whose projective class is a vertex of 9%. There are only a finite number of surfaces (up to multiple copies) whose projective classes are vertices and these are easily constructed from the data in the matching equations. The faces of 9% are closely related to branched surfaces and some of our results can be translated into results about branched surfaces. The weight of a normal surface F is the number of points in F n T--(l) and is denoted by wt (F). Jaco and Oertel [4] show that if F is a least weight injective normal surface in a closed irreducible 3-manifold M then the minimal face CF of 9% carrying F carries only injective surfaces. From this it follows that there is an injective normal surface whose projective class is a vertex of 9% and one is led to an algorithm for detecting such surfaces. If M is not irreducible then there exists a complete decomposing set of 2-spheres for M whose projective classes are linearly independent vertices of a single face in 9%[6]. It is also shown in [6] that if a 3-manifold M has a compressible boundary then there exists an essential compression disk whose projective class is a vertex of 9%. Our goal in this paper is to study the faces CF of 9~ associated with taut normal surfaces having minimal weight. For a nonzero homology class gEH2 (M, 8M; Z) the norm x (g) is defined [ll] as the infimum, over all properly embedded surfaces G representing g, of x-(G)=-x (G-spherical and disk components of G).The extension of x to all of H,(M, aM; R) is made using convexity and continuity. An oriented, incompressible, &incompressible surface F properly embedded in M is said to be taut if its homology class [F] is nontrivial in H,(M, aM; Z), F is~--minimizing, and there is no homologically trivial union of components of F. A taut surface F is said to be Iw-taut if it has minimal weight among all taut surfaces representing the homology class [F]. If F is lw-taut then n pairwise disjoint copies of F, denoted by nF, is an lw-taut surface representing the class n [F](see Lemma 1 in [ll]).