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This article gives a natural decomposition of the suspension of generalized moment-angle complexes or partial product spaces which arise as polyhedral product functors described below. The geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of Goresky-MacPherson concerning complements of certain subspace arrangements, as well as Davis-Januszkiewicz and Buchstaber-Panov concerning moment-angle complexes. Since the stable decompositions here are geometric, they provide corresponding homological decompositions for generalized moment-angle complexes for any homology theory.

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Abstract. Let G denote a topological group. In this article 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.

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Let F n = x 1 , . . . , x n denote the free group with generators {x 1 , . . . , x n }. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F n to the general linear group over the integers. In particular among them are the automorphisms χ k,i which conjugate the generator x k by the generator x i leaving the x j fixed for j = k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by χ k,i for i < k is given here. Partial results are obtained for the group generated by all χ k,i . 20F28; 20F40, 20J06