Convex polytopes, Coxeter orbifolds and torus actions
Convex polytopes, Coxeter orbifolds and torus actions
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DOI：
10.1215/s0012-7094-91-06217-4
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发表时间：
1991-03
影响因子：
2.5
通讯作者：
Michael W. Davis;T. Januszkiewicz
中科院分区：
文献类型：
--
作者：
Michael W. Davis;T. Januszkiewicz
0。导言。N维凸多面体是简单的，如果每个顶点相交的余维1面的个数为n.本文研究了以简单凸多面体为轨道空间的流形上的某些群作用.设P“表示这样一个简单的多面体，我们考虑两种情况：(1)群是Z，M”是n维的；(2)群是T“，M是2n维的，M2”/T是P“。直到群的自同构，在第二种情况下，动作必须局部同构于Z.On的标准表示。在第一种情况下，我们称M是P“的”小覆盖“；在第二种情况下，我们称M是P上的”环流形“。第一个例子分别由Z和T”在RP“和CP”上的自然作用提供。在这两种情况下，轨道空间都是n-单形。与P“的一个小覆盖相联系，有一个同态2”Z‘Z，其中m是P“的余维--一个面的个数.同态2指定了每个余维--一个面的一个迷向子群.我们称它为小覆盖的”特征函数“.类似地，Pn上的环面流形的特征函数是一个映射Z--)7/”.一个基本结果是P“上的小覆盖和环面流形是按它们的特征函数分类的(见命题1.7和1.8)。这些流形的代数拓扑非常漂亮。它们的同调和上同调群的计算与交换代数中的一些著名结构和凸多面体的组合理论密切相关。下面我们将讨论其中的一些结构。设f表示P“的/-面数，h表示f(T1)中”的系数。则(fo，f，)称为P“的f-向量，(ho，h，)称为P”的h-向量。F向量和h向量显然是相互决定的。由McMullen提出的上界定理断言，对于所有n维余维为1的m个面的凸多面体，不等式h<；(，-，{-x)都成立。1971年，McMullen猜想了一个整数序列(H0，h，)是简单凸多面体的h-向量的充要条件。这些条件的充分性由Billera和Lee证明，必要性由Stanley证明(更多细节和参考文献见[Bronsted])。研究与实践
0. Introduction. An n-dimensional convex polytope is simple if the number of codimension-one faces meeting at each vertex is n. In this paper we investigate certain group actions on manifolds, which have a simple convex polytope as orbit space. Let P" denote such a simple polytope. We have two situations in mind. (1) The group is Z, M" is n-dimensional and (2) The group is T", M is 2n-dimensional and M2"/T" P". Up to an automorphism of the group, the action is required to be locally isomorphic to the standard representation of Z. on in the second case. In the first case, we call M a "small cover" of P"; in the second, it is a "toric manifold" over P. First examples are provided by the natural actions of Z. and T" on RP" and CP", respectively. In both cases the orbit space is an n-simplex. Associated to a small cover of P", there is a homomorphism 2" Z’ Z, where m is the number of codimension-one faces of P". The homomorphism 2 specifies an isotropy subgroup for each codimension-one face. We call it a "characteristic function" of the small cover. Similarly, the characteristic function ofa toric manifold over pn is a map Z --) 7/". A basic result is that small covers and toric manifolds over P" are classified by their characteristic functions (see Propositions 1.7 and 1.8). The algebraic topology of these manifolds is very beautiful. The calculation of their homology and cohomology groups is closely related to some well-known constructions in commutative algebra and the combinatorial theory of convex polytopes. We discuss some of these constructions below. Let f denote the number of/-faces of P" and let h denote the coefficient of "in f(t 1). Then (fo, f,) is called the f-vector and (ho, h,) the h-vector of P". The f-vector and the h-vector obviously determine one another. The Upper Bound Theorem, due to McMullen, asserts that the inequality h < (,-,{-x), holds for all n-dimensional convex polytopes with m faces of codimension one. In 1971 McMullen conjectured simple combinatorial conditions on a sequence (h0, h,) of integers necessary and sufficient for it to be the h-vector of a simple convex polytope. The sufficiency of these conditions was proved by Billera and Lee and necessity by Stanley (see [Bronsted] for more details and references). Research on