Volodin space

In mathematics, more specifically in topology, the Volodin space ${\displaystyle X}$ of a ring R is a subspace of the classifying space ${\displaystyle BGL(R)}$ given by

${\displaystyle X=\bigcup _{n,\sigma }B(U_{n}(R)^{\sigma })}$

where ${\displaystyle U_{n}(R)\subset GL_{n}(R)}$ is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and ${\displaystyle \sigma }$ a permutation matrix thought of as an element in ${\displaystyle GL_{n}(R)}$ and acting (superscript) by conjugation.^[1] The space is acyclic and the fundamental group ${\displaystyle \pi _{1}X}$ is the Steinberg group ${\displaystyle \operatorname {St} (R)}$ of R. In fact, Suslin (1981) showed that X yields a model for Quillen's plus-construction ${\displaystyle BGL(R)/X\simeq BGL^{+}(R)}$ in algebraic K-theory.