The exponent of the homotopy groups of Moore spectra and the stable Hurewicz homomorphism, by Dominique Arlettaz

This paper shows that, for any positive integer n, the order of the (n+1)-dimensional Postnikov k-invariant of the Moore spectrum associated with any abelian group G is equal to the exponent of its n-th homotopy group. 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 relating the homotopy groups of a bounded below spectrum X to its E-homology groups, for any connective ring spectrum E such that its 0-th homotopy group is torsion-free. Moreover, an upper bound for the exponent of the cokernel of the generalized Hurewicz homomorphism between the E-homology groups of X and the ordinary homology groups of X with coefficient in the 0-th homotopy group 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.

Dominique Arlettaz <>