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The Exponent of the Homotopy Groups of Moore Spectra and the Stable Hurewicz Homomorphism

Published online by Cambridge University Press:  20 November 2018

Dominique Arlettaz
Dominique Arlettaz*
Affiliation:
Dominique Arlettaz Institut de mathématiques Université de Lausanne CH—1015 Lausanne Switzerland e-mail: dominique.arlettaz@ima. unil. ch
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Abstract

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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.

Keywords

55N2055P4219D5555Q4555S45
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 3 , 01 June 1996 , pp. 483 - 495
Copyright
Copyright © Canadian Mathematical Society 1996

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