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.
Application An analogue of Volodin's space where GL(R) is replaced by the Lie algebra g l ( R ) {\displaystyle {\mathfrak {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.
Notes
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, JSTOR 1971283, MR 0855300 Weibel, Charles (2013). "The K-book: an introduction to algebraic K-theory". 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 Ser. Mat., 35 (4): 844–873, Bibcode:1971IzMat...5..859V, doi:10.1070/IM1971v005n04ABEH001121, MR 0296140, (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859–887)