Commuting elements, simplicial spaces and filtrations of classifying spaces
Commuting elements, simplicial spaces and filtrations of classifying spaces
复制标题
通勤元素、单纯空间和分类空间的过滤
DOI：
10.1017/s0305004111000570
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发表时间：
2008
影响因子：
0.8
通讯作者：
Enrique Torres
中科院分区：
文献类型：
--
作者：
A. Adem;F. Cohen;Enrique Torres
设G表示一个拓扑群。本文利用自由群的下降中心级数构造具有几何实现B（q， G）的同态简化空间，从而对分类空间BG进行过滤。特别地，在这种情况下，由G中所有有序交换n元元的空间构造出了一个单空间。讨论了这些构造的基本性质，包括当群G是有限群或紧连通李群时的同伦类型和上同调。对于有限群，该构造得到了一个具有单态的覆盖空间，这与群论中等价于Feit-Thompson奇阶定理的一个微妙结果有关。这里的技术也产生了hm （π， G）的基数的计数公式，其中π是有限生成的自由群的任何下降的中心序列商。另一个应用是确定了有限群的交换n元组在G上得到的空间B（2， G）的结构，使得每个非中心元素的中心子都是阿贝尔的（称为传递交换群），这在Suzuki关于有限群结构的工作中起了关键作用。
Abstract Let G denote a topological group. In this paper the descending central series of free groups are used to construct simplicial spaces of homomorphisms with geometric realizations B(q, G) that provide a filtration of the classifying space BG. In particular this setting gives rise to a single space constructed out of all the spaces of ordered commuting n–tuples of elements in G. Basic properties of these constructions are discussed, including the homotopy type and cohomology when the group G is either a finite group or a compact connected Lie group. For a finite group the construction gives rise to a covering space with monodromy related to a delicate result in group theory equivalent to the odd-order theorem of Feit–Thompson. The techniques here also yield a counting formula for the cardinality of Hom(π, G) where π is any descending central series quotient of a finitely generated free group. Another application is the determination of the structure of the spaces B(2, G) obtained from commuting n-tuples in G for finite groups such that the centralizer of every non–central element is abelian (known as transitively commutative groups), which played a key role in work by Suzuki on the structure of finite simple groups.
环空间的几何
DOI：
--
发表时间：
2009
期刊：
影响因子：
--
作者：
H. Irie;T. Otofuji;K.Fukaya;伊藤秀史;S.Koike;T. Funaki;金銅誠之;Yoshiaki Maeda
通讯作者：
Yoshiaki Maeda