Volodin space
In mathematics, more specifically in topology, the Volodin space X {\displaystyle X} of a ring R is a subspace of the classifying space B G L ( R ) {\displaystyle BGL(R)} given by X = ⋃ n , σ B ( U n ( R ) σ ) {\displaystyle X=\bigcup _{n,\sigma }B(U_{n}(R)^{\sigma })} where U n ( R ) ⊂ G L n ( R ) {\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 σ {\displaystyle \sigma } a permutation matrix thought of as an element in G L n ( R ) {\displaystyle GL_{n}(R)} and acting (superscript) by conjugation. The space is acyclic and the fundamental group π 1 X {\displaystyle \pi _{1}X} is the Steinberg group St ( R ) {\displaystyle \operatorname {St} (R)} of R. In fact, Suslin (1981) showed that X yields a model for Quillen's plus-construction B G L ( R ) / X ≃ B G L + ( R ) {\displaystyle BGL(R)/X\simeq BGL^{+}(R)} in algebraic K-theory.