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 $GLn(R)$ 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 $\mathfrak{𝔤}\mathfrak{𝔩}(R)$ 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

• Goodwillie, Thomas G. (1986), "Relative algebraic K-theory and cyclic homology", Annals of Mathematics, Second Series 124 (2): 347–402, doi:10.2307/1971283 
• Weibel, Charles (2013). "The K-book: an introduction to algebraic K-theory". http://www.math.rutgers.edu/~weibel/Kbook.html. 
• Suslin, A. A. (1981), "On the equivalence of K-theories", Comm. Algebra 9 (15): 1559–66, doi:10.1080/00927878108822666 
• Volodin, I. (1971), "Algebraic K-theory as extraordinary homology theory on the category of associative rings with unity", Izv. Akad. Nauk. SSSR 35 (4): 844–873, doi:10.1070/IM1971v005n04ABEH001121, Bibcode: 1971IzMat...5..859V , (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859–887)