Dehn-Thurston coordinates for curves on surfaces
Dehn-Thurston coordinates for curves on surfaces
复制标题
曲面上曲线的 Dehn-Thurston 坐标
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
R. Strong
中科院分区：
文献类型：
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作者：
F. Luo;R. Strong
1.1.本文研究了曲面上一维子流形的同伦类空间。这个主题是由Max Dehn在他1938年的论文[de]中提出的。在这项工作中，Dehn为研究曲面的映射类群和曲面上一维子流形的同伦类空间奠定了基础。根据Dehn的观点，最重要的真一维子流形是这样的曲线系统，它具有这样的性质：子流形的任何分量都不是零同伦的或通过相对于边界的同伦进入曲面的边界的。Dehn将曲面Σ的算术域CS(Σ)定义为所有等距类曲线系统的集合。Dehn在[De]中的主要研究重点是映射类组在“算术域”CS(Σ)上的作用。1976年，威廉·瑟斯顿独立地重新发现了空间CS(Σ)，并将另一个重要因素投入到“算术域”CS(Σ)的研究中。即两个一维子流形的保序类之间的几何交数。回想一下，如果α和β是真的一维子流形的保序类，则它们的几何交数(由I(α，β)表示)是它们的代表之间的交集的最小数目，即，
1.1. We study the space of isotopy classes of 1-dimensional submanifolds in a surface in this paper. The subject was originated by Max Dehn in his 1938 paper [De]. In this work, Dehn laid the foundation for the studies of the mapping class group of a surface and the space of isotopy classes of 1-dimensional submanifolds in a surface. According to Dehn, the most important proper 1-dimensional submanifolds are the curve systems which have the property that no component of the submanifold is null homotopic or homotopic into the boundary of the surface by homotopies relative to the boundary. Dehn defined the arithmetic field of a surface Σ, denoted by CS(Σ), to be the set of all isotopy classes of curve systems. Dehn’s main focus of study in [De] was the action of the mapping class group on the ”arithmetic field” CS(Σ). In 1976, William Thurston independently rediscovered the space CS(Σ) and put one more vital ingredient into the study of ”arithmetic field” CS(Σ). Namely, the geometric intersection numbers between two isotopy classes of 1-dimensional submanifolds. Recall that if α and β are isotopy classes of proper 1-dimensional submanifolds, then their geometric intersection number, denoted by I(α, β), is the minimal number of intersections between their representatives, i.e.,