A finite set of generators for the homeotopy group of a 2-manifold
A finite set of generators for the homeotopy group of a 2-manifold
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DOI：
10.1017/s030500410003824x
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发表时间：
1964-10
影响因子：
0.8
通讯作者：
W. Lickorish
中科院分区：
文献类型：
--
作者：
W. Lickorish
空间X的同伦群Λx是X到自身的所有同胚的群，模那些同胚的子群是单位元的同位素。本文将X看作是一个闭的定向2-流形，并具有一个多面体结构，然后将Λx的定义限制在考虑分段线性同胚和合痕的情况下。虽然对多面体范畴的这种限制对下面的讨论并不是真正必要的，但它确实倾向于简化一些论证。在（2）中，X的同胚以下列方式与X中的每一条简单闭（多面体）曲线c相关联。首先，设A是欧氏平面中的一个环，由（r，θ）参数化，其中1 ≤ r ≤ 2，θ是一个模2 π的真实的数。我们定义了一个同胚H：A → A by H则固定在A的边界上。如果现在e：A → X是一个方向保持嵌入，并且eA是c在X中的邻域，则eHe−1| eA可以通过X-eA上的恒等式扩展为同胚h：X → X。任何与h同素的分段线性同胚hc称为关于c的扭，或者如果c不指定，则称为扭。
The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to be a closed oriented 2-manifold, together with a polyhedral structure, and the definition of Λx is then restricted to the consideration of piecewise-linear homeomorphisms and isotopies. Although this restriction to the polyhedral category is not really essential to what follows, it does tend to simplify some of the arguments. In (2) a homeomorphism of X was associated with every simple closed (polyhedral) curve c in X in the following way. First, let A be an annulus in the Euclidean plane parametrized by (r, θ) where 1 ≤ r ≤ 2 and θ is a real number mod 2 π. We define a homeomorphism H: A → A by H is then fixed on the boundary of A. If now e: A → X is an orientation-preserving embedding, and eA is a neighbourhood of c in X, then eHe−1|eA can be extended by the identity on X − eA to a homeomorphism h:X → X. Any piecewise linear homeomorphism hc which is isotopic to h will be called a twist about c or, if c is not specified, just a twist.