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

  • Dominique Arlettaz (a1)

Abstract

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 k n+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 : πnXEn (X) for the homology theory E *(—) corresponding to any connective ring spectrum E such that π 0 E 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; π 0 E), 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.

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References

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

  • Dominique Arlettaz (a1)

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