Volodin space

In mathematics, more specifically in topology, the Volodin space formula_1 of a ring "R" is a subspace of the classifying space formula_2 given by
where formula_4 is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and formula_5 a permutation matrix thought of as an element in formula_6 and acting (superscript) by conjugation. The space is acyclic and the fundamental group formula_7 is the Steinberg group formula_8 of "R". In fact, showed that "X" yields a model for Quillen's plus-construction formula_9 in algebraic K-theory.
Application.
An analogue of Volodin's space where GL("R") is replaced by the Lie algebra formula_10 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.