In mathematics, more specifically in topology, the Volodin space
X
of a ring R is a subspace of the classifying space
B
G
L
(
R
)
given by
X
=
⋃
n
,
σ
B
(
U
n
(
R
)
σ
)
where
U
n
(
R
)
⊂
G
L
n
(
R
)
is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and
σ
a permutation matrix thought of as an element in
G
L
n
(
R
)
and acting (superscript) by conjugation. The space is acyclic and the fundamental group
π
1
X
is the Steinberg group
St
(
R
)
of R. In fact, Suslin's 1981 paper explains that X yields a model for the Quillen's plus-construction
B
G
L
(
R
)
/
X
≃
B
G
L
+
(
R
)
in algebraic K-theory.
