Commuting elements, simplicial spaces, and filtrations of classifying spaces

TL;DR

This paper constructs a filtration of the classifying space BG using simplicial spaces derived from commuting elements and homomorphism spaces, analyzing their properties for various groups.

Contribution

It introduces a new family of simplicial spaces B(q,G) that filter BG, connecting commuting elements, homomorphisms, and homotopy properties for different group types.

Findings

B(2,G) spaces built from commuting n-tuples

Homotopy properties analyzed for finite groups

Cohomology computed for compact Lie groups

Abstract

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.