Volodin space

In mathematics, more specifically in topology, the Volodin space $$X$$ of a ring R is a subspace of the classifying space $$BGL(R)$$ given by
 * $$X = \bigcup_{n, \sigma} B(U_n(R)^\sigma)$$

where $$U_n(R) \subset GL_n(R)$$ is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and $$\sigma$$ a permutation matrix thought of as an element in $GL_n(R)$ and acting (superscript) by conjugation. The space is acyclic and the fundamental group $$\pi_1 X$$ is the Steinberg group $$\operatorname{St}(R)$$ of R. In fact, showed that X yields a model for Quillen's plus-construction $$BGL(R)/X \simeq BGL^+(R)$$ in algebraic K-theory.

Application
An analogue of Volodin's space where GL(R) is replaced by the Lie algebra $$\mathfrak{gl}(R)$$ 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.