Lifting and projecting homeomorphisms
Lifting and projecting homeomorphisms
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DOI：
10.1007/bf01304911
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发表时间：
1972-12
影响因子：
0.6
通讯作者：
J. Birman;H. Hilden
中科院分区：
文献类型：
--
作者：
J. Birman;H. Hilden
1. 介绍。设X是一个路径连通和局部路径连通的拓扑空间，G是X的所有自同胚的群，D是映射同位素到恒等式的子群。定义X的同伦群为G/D群。设2为pc， I. pc覆盖X的空间，具有投影p.研究了2与X的同伦群之间的关系。（定理3）证明了在2、X和p上足够强的约束条件下，X的同构群与~的同构群的一个因子群同构，随着l~和X上的约束的减弱，结果也就变弱。本文所研究的情况是作者在早期的研究[1]中首先注意到的。2-流形的同伦群在黎曼曲面理论和3-流形的分类中占有重要的地位。在[1]中表明，利用ff属的任何闭合紧致可定向曲面与。移除的（2g+ 2）点可视为（2ff+ 2）穿孔球的2层覆盖物，并利用该穿孔球的同伦群的已知性质。这种关系的发展表明，更一般的空间的其他覆盖也可能是感兴趣的，从而推动了目前的研究。在本文的最后（第4节），简要讨论了表面拓扑的新应用。此应用程序的详细工作将在[2]中找到，[2]应与本工作同时出现。我们首先回顾一些关于覆盖空间的著名结果。设Xo off X，设20 ~ pl （x0）然后由覆盖空间投影p: fi2-+ X导出一个从：rl（2,20）到gl （X, x0）的单态p。[例如，见b[9]的第72页。为简便起见，我们用~表示~ rl (X, 20)，用~表示：rl (X, xo)，用xc表示~的子群p。
1. Introduction. Let X be a pathwise connected and locally pathwise connected topological space, G the group of all self-homeomorphisms of X, and D the sub~ oup of maps isotopic to the identity. The homeotopy group of X is defined to be the group G/D. Let 2 be a pc, I. pc covering space of X, with projection p. The relationship between the homeotopy groups of 2 and X is studied. It is shown (Theorem 3) that under sufficiently strong restrictions on 2, X and p the homeotopy group of X is isomorphic to a factor group of the homeotopy group of~, with weaker results as one weakens the restrictions on l~ and X. The situation studied here first came to the authors' attention in an earlier investigation [1]. The homeotopy~ oups of 2-manifolds play an important role in the theory of Riemann surfaces, and also in the classification of 3-manifolds. It was sho~ m in [1] that one could gain considerable insight into the structure of the homeotopy groups of surfaces by utilizing the fact that any closed compact orientable surface of genus ff with.(2g+ 2) points removed can be regarded as a 2-sheeted covering of a (2ff+ 2)-punctured sphere, and making use of the known properties of the homeotopy group of the punctured sphere. The development of this relationship suggested that other coverings of more general spaces might also be of interest, thus motivating the present investigation. At the conclusion of this paper (Section 4) a new application to surface topology is discussed briefly. A detailed workingout of this application will be found in [2], which should appear concurrently with the present work.2. We begin by reviewing some well-known results about covering spaces. Let Xo ff X, and suppose 2o~ pl (x0). Then the covering space projection p: fi2-+ X induces a monomorphism p. from: rl (2, 2o) to gl (X, x0).[See, for example, page 72 of [9].] For simplicity in notation, we will write~ for~ rl (X, 20),~ for: rl (X, xo), and xc for the subgroup p.~ of~.