Actions of Tori on Orbifolds
Actions of Tori on Orbifolds
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Tori 在 Orbifolds 上的动作
DOI：
10.1007/bf02411354
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发表时间：
1991
影响因子：
0.7
通讯作者：
É. Salem
中科院分区：
文献类型：
--
作者：
A. Haefliger;É. Salem
本文研究了n维环面Tn在m维轨道上的光滑有效作用。这样的作用自然发生在单连通流形上的黎曼叶理的研究中（见[9]）。Seifert、Orlik-Raymond（[11]）、Fintushel（[4]和[5]）等几位作者使用的基本技术，对于研究圆或环面在流形上的作用，很容易推广，并有望澄清，以适用于更一般的情况下orbifolds。在[2] F. Bonahon和L. Siebenmann对圆在3-orbifolds上的局部自由作用作了细致的研究。在第1节给出基本定义之后，我们在第2节通过研究轨道的不变管状邻域的泛覆盖来研究它们的一般结构。它们用三个不变量来描述：1）格r= In在Rn中的子群r 0，2）中心扩张0--* r/r 0--. A~ D~ 1，其中D是有限群，3）一个忠实表示p（切片表示）到等距群O（B）中，其中K是由2）构造的李群G的极大紧子群，而B是一个维数为m-n+的欧几里得球在第4节中研究了轨道空间上的管状邻域是如何粘在一起的，在那里我们使用了一个基本结果，G.施瓦茨（参见第3节）。在轨道空间W中，H3（W，Zn）（在上同调意义下）的一个元素是粘滞的障碍.通过元素H2（W，Zn）来参数化不同的胶合。在第5节中，我们对轨道的基本群进行了一般性的观察，并给出了一个理论公式，根据轨道空间W上的数据来计算它。
In this paper, we study smooth effective actions of a torus T n of dimension n on an orbifold of dimension m. Such actions occur naturally in the study of Riemannian foliations on simply connected manifolds (see [9]). The basic techniques used by several authors, Seifert, Orlik-Raymond ([11]), Fintushel ([4] and [5]), etc.., for the study of actions of circles or tori on manifolds are easily generalized and hopefully clarified to apply to the more general case of orbifolds. In [2] F. Bonahon and L. Siebenmann have made a careful study of locally free actions of the circle on 3-orbifolds. After giving in Section 1 the basic definitions, we study in Section 2 the general structure of invariant tubular neighborhoods of orbits by passing to their universal coverings. They are described in terms of three invariants: 1) a subgroup r0 of the lattice r= I n in R n, 2) a central extension0--* r/r0--. A~ D~ 1, where D is a finite group, 3) a faithful representation p (the slice representation) of K into the group of isometries O (B), where K is the maximal compact subgroup of a Lie group G constructed from 2), and B is a Euclidean ball of dimension m-n+ dim K.The way tubular neighborhoods are glued together above the orbit space is studied in Section 4 where we use a basic result whose proof was given to us by G. Schwarz (cf. Section 3). There is an obstruction for the gluing which is an element of H3 (W, Z n)(in the sense of (~ ech cohomology), where W is the orbit space. The different gluings are parameterized by elements of H2 (W, Zn). In Section 5, we make general observations about the fundamental group of an orbifold and give a theoretical recipe to compute it in terms of data on the orbit space W.