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 $U_n(R)\subset G{L}_n(R)$ 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 $G{L}_n(R)$ and acting (superscript) by conjugation. The space is acyclic and the fundamental group ${\pi }_{1}X$ is the Steinberg group $\mathrm{St}(R)$ of R. In fact, Suslin (1981) showed that X yields a model for Quillen's plus-construction $BGL(R)/X\simeq BG{L}^{+}(R)$ in algebraic K-theory.

Application

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

References

• , (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859–887)