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The classifying space
of a
topological group
can be filtered
by a sequence of subspaces
,
,
using the descending central series of free groups. If
is
finite, describing them as homotopy colimits is convenient when applying
homotopy theoretic methods. In this paper we introduce natural subspaces
defined for a fixed
prime
. We show that
is stably homotopy
equivalent to a wedge of
as
runs over the primes
dividing the order of
.
Colimits of abelian groups play an important role in understanding the homotopy type of these spaces.
Extraspecial
–groups
are key examples, for which these colimits turn out to be finite. We prove that for extraspecial
–groups
of order
,
,
does not have the
homotopy type of a
space, thus answering in a negative way a question posed by Adem. For a finite group
, we compute the
complex
–theory
of
modulo torsion.
Keywords
homotopy colimit, classifying space, $K$–theory, descending
central series
