Abstract
This paper investigates the Hurewicz homomorphism h n:K nA→H n(E(A); Z) between the algebraic K-theory of a ring A and the homology of the linear group E(A) generated by elementary matrices over A. The main theorem asserts that for any n≥2, the kernel of h n is a torsion group of finite exponent, and provides an upper bound, independent of A, for its exponent. The proof of this uses the fact that BE(A) + is an infinite loop space, because its basic idea is to observe that the result follows from the study of the kernel of the Hurewicz homomorphism in the range of stability. The discussion of the problem involves then the description of the relationship between the Hurewicz map and the k-invariants of the space BE(A) +. Finally, some partial information on the cokernel of h n is also obtained.
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