In mathematics, more specifically in topology, the Volodin space
of a ring R is a subspace of the classifying space
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.
[1] The space is acyclic and the fundamental group
of R. In fact, Suslin (1981) showed that X yields a model for Quillen's plus-construction
An analogue of Volodin's space where GL(R) is replaced by the Lie algebra
was used by Goodwillie (1986) 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.