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.
