  Using spaces of homomorphisms and the descending central series of the free
groups, simplicial spaces are constructed for each integer q>1 and every
topological group G, with realizations B(q,G) that 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. Cohomology calculations are provided for compact
Lie groups. The spaces B(2,G) are described in detail for transitively
commutative groups.
