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Commuting elements, simplicial spaces and filtrations of classifying spaces

Published online by Cambridge University Press:  05 September 2011

ALEJANDRO ADEM ,
FREDERICK R. COHEN  and
ENRIQUE TORRES GIESE
ALEJANDRO ADEM
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver BC V6T1Z2, Canada. e-mail: adem@math.ubc.ca
FREDERICK R. COHEN
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A. e-mail: cohf@math.rochester.edu
ENRIQUE TORRES GIESE
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A. e-mail: etorresg@umich.edu
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 1 , January 2012 , pp. 91 - 114
Copyright
Copyright © Cambridge Philosophical Society 2011

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