Moves for Flow-Spines and Topological Invariants of 3-Manifolds
Moves for Flow-Spines and Topological Invariants of 3-Manifolds
复制标题
流脊柱的移动和 3 流形的拓扑不变量
DOI：
10.3836/tjm/1270129457
复制
发表时间：
1992
影响因子：
0.6
通讯作者：
I. Ishii
中科院分区：
文献类型：
--
作者：
I. Ishii
封闭的3个manifold $ m $的脊柱$ p $是$ m $的2维多面体，因此，普通社区的$ P $是同型3球。沿其脊柱$ p $的歧管$ m $，我们获得了一个3球$ b^{3} $，并在其边界上进行了标识。封闭式表面，由H. seifert引入3维情况下。封闭的表面（参见[1]，[3]）。三角剖分。流动旋转的DS-DIAGR具有电子周期。封闭的3个术语入院无限多流量，在本文中，我们将给出两个流动的条件，以代表相同的歧管，也就是说通过三种类型的操作序列将其称为“移动”。 s 1）。 \ s 1中的E-DATA的图形表示。E-DATA不仅确定了3个manifold $ m $，还决定了$ M $的一类非偏差流（请参阅\ s 1）。数据分为两种类型，不更改非单个流的移动，而将第一个类型的移动变化。
A spine $P$ for a closed 3-manifold $M$ is a 2-dimensional polyhedron in $M$ such that the complement of the regular neighborhood of $P$ is homeomorphic to the 3-ball. Cutting off a closed 3-manifold $M$ along its spine $P$, we get a 3-ball $B^{3}$ with an identification on its boundary. This is a polyhedral representation of $M$, which is first considered by M. Dehn in the case of closed surfaces, and introduced by H. Seifert in the 3-dimensional case. A DS-diagram is a polyhedral representation of a special class, which was first introduced in [3]. A spine corresponding to a DS-diagram forms a closed fake surface (cf. [1], [3]). A spine which forms a closed fake surface is called a standard or a simple spine. As is pointed out in [12], a standard spine is the dual ofa singular triangulation. A flow-spine introduced in [7] is a standard spine of a more special class, which is generated by a pair of a non-singular flow and its local section. It was shown in [4] and [7] that a DS-diagram for a flow-spine has an E-cycle. An E-cycle is a cycle of the graph of a DS-diagram which represents a kind of symmetry of a polyhedral representation. (See \S 1 for precise.) A closed 3-manifold admits infinitely many flow-spines. In this paper, we shall give conditions for two flow-spines to represent the same manifold, that is, it will be shown that any two flow-spines of a 3-manifold can be transformed from one to another by a finite sequence of operations of three types which we call “moves”. A flow-spine is completely determined by a data on the E-cycle, which will be called an E-data (cf. \S 1). An E-data is the one called a singularity-data in [7]. Our moves of flow-spines are described in terms of E-data. For an easy description of moves of E-data, we introduce the graphic representation of an E-data in \S 1. An E-data determines not only a 3-manifold $M$ but also a class of non-singular flows on $M$ (see \S 1). Moves of E-data are divided into two types, moves which do not change the class of non-singular flows and those which change the class. Moves of the first type are called regular moves and discussed in \S 2. The second type consists of only