Commuting Elements, Simplicial Spaces and Filtrations of Classifying Spaces

Let Γ denote the q-th stage of the descending central series of the free group on n generators Fn. For each q ≥ 2 and every topological group G, a simplicial space B∗(q, G) is constructed where Bn(q, G) = Hom(Fn/Γ , G) and the realizations B(q, G) = |B∗(q, G)| filter the classifying space BG. In particular for q = 2 this yields a single space B(2, G) assembled from all the n–tuples of commuting elements in G. Homotopy properties of the B(q, G) are considered for finite groups, including their description as homotopy colimits. Cohomology calculations are provided for compact Lie groups. The spaces B(2, G) are described in detail for transitively commutative groups. Stable homotopy decompositions of the B(q, G) are also provided with a formula giving the cardinality of Hom(Fn/Γ , G) for finite discrete groups G (and thus the cardinality of Hom(Z, G) in case q = 2) in terms of the ranks of the homology groups for the associated filtration quotients of B(q, G). Specific calculations for H1(B(q, G);Z) are shown to be delicate in case G is finite of odd order in the sense that resulting topological properties (which are not yet fully understood) are equivalent to the Feit–Thompson theorem.