In mathematics, more specifically in topology, the Volodin space of a ring R is a subspace of the classifying space given by
where is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and a permutation matrix thought of as an element in and acting (superscript) by conjugation. The space is acyclic and the fundamental group is the Steinberg group of R. In fact, showed that X yields a model for Quillen's plus-construction in algebraic K-theory.
An analogue of Volodin's space where GL(R) is replaced by the Lie algebra was used by to prove that, after tensoring with Q, relative K-theory K(A, I), for a nilpotent ideal I, is isomorphic to relative cyclic homology HC(A, I). This theorem was a pioneering result in the area of trace methods.