We investigate the k-invariants for the K-theory and étale K-theory spaces of schemes. We give numerical estimates of the orders of k-invariants for the K-theory and étale K-theory spaces of regular and proper models over of smooth, proper and geometrically irreducible curves defined over global fields F.

This paper presents some new results on algebraic K-theory with finite coefficients. The argument is based on a topological construction of a space $F_mK(R)$, for any ring $R$ and any integer $m\geq 2$, having the property that the ordinary homotopy theory of $F_mK(R)$ is isomorphic to the algebraic K-theory of $R$ with coefficients in $\bb {Z}/m$: $\pi_n(F_mK(R))\cong K_n(R;\bb {Z}/m)$ for $n\geq 1$. This space $F_mK(R)$ is called the $\mod m$ K-theory space of $R$. The paper is devoted to the investigation of several properties of the groups $K_n(R;\bb {Z}/m)$, for $n\in\bb {Z}$, and to some calculations of the integral homology of finite K-theory spaces.

This paper is devoted to the complete calculation of the additive structure of the 2-torsion of the integral cohomology of the infinite special linear group SL(ℤ) over the ring of integers ℤ. This enables us to determine the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups.

This paper shows that for the Moore spectrum MG associated with any abelian group G, and for any positive integer n, the order of the Postnikov k-invariant kn+1(MG) is equal to the exponent of the homotopy group πnMG. In the case of the sphere spectrum S, this implies that the exponents of the homotopy groups of S provide a universal estimate for the exponent of the kernel of the stable Hurewicz homomorphism hn: πnX → En(X) for the homology theory E*(—) corresponding to any connective ring spectrum E such that π0E is torsion-free and for any bounded below spectrum X. Moreover, an upper bound for the exponent of the cokernel of the generalized Hurewicz homomorphism hn: En(X) → Hn(X; π0E), induced by the 0-th Postnikov section of E, is obtained for any connective spectrum E. An application of these results enables us to approximate in a universal way both kernel and cokernel of the unstable Hurewicz homomorphism between the algebraic K-theory of any ring and the ordinary integral homology of its linear group.

For any prime number p, let Γn, p denote the congruence subgroup of SLn(ℤ) of level p, i.e. the kernel of the surjective homomorphism fp: SLn(ℤ) → SLn(p) induced by the reduction mod p (Fp is the field with p elements). We define

using upper left inclusions Γn, p ↪ Γn+1, p. Recall that the groups Γn, p are homology stable with M-coefficients, for instance if M = ℚ, ℤ[1/p], or ℤ/q with q prime and q ╪ p: Hi(Γn, p; M) ≅ Hi(Γp; M) for n ≥ 2i + 5 from [7] (but the homology stability fails if M = ℤ or ℤ/p).

The purpose of this paper is to give universal bounds for the order of the Postnikov k-invariants of infinite loop spaces. This is done by giving universal bounds for the order of the k-invariants of m-connected r-fold loop spaces in dimensions ≦ r + 2m. An application of the result provides information on the Hurewicz homomorphism between the algebraic K-theory of aring and the homology of its general linear group.