THE EQUIVALENCE OF TWO ALGEBARAIC K-THEORIES

For a ring R with 1, the higher K-theory of Quillen is defined by the higher homotopy groups of the plus construction of the general linear group of R. On the other hand, the Volodin K-theory is defined by the higher homotopy groups of the Volodin space. In this paper we show that these two K-theories are equivalent. We show that the Volodin space is a homotopy fiber of the acyclic map from BGL(R) to its plus construction.