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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

References

1. Adams, J.E., Stable homotopy and generalised homology, The University of Chicago Press, 1974.Google Scholar
2. Arlettaz, D., The Hurewicz homomorphism in algebraic K-theory, J. Pure Appl. Algebra 71 (1991), 1–12.Google Scholar
3. Arlettaz, D., The order of the differentials in the Atiyah-Hirzebruch spectral sequence, k-Theory 6 (1992), 347–361.Google Scholar
4. Arlettaz, D., Exponents for extraordinary homology groups, Comment. Math. Helv. 68 (1993), 653–672.Google Scholar
5. Borel, A., Cohomologie réelle stable de groupes S-arithmétiques classiques, C. R. Acad. Sci. Paris Ser. A 274 (1972), 1700–1702.Google Scholar
6. Sah, C.H., Homology of classical groups made discrete III, J. Pure Appl. Algebra 56 (1989), 269–312.Google Scholar
7. Scherer, J., Exponents for high-dimensional Gamma groups, Exposition. Math. 13 (1995), 455–468.Google Scholar
8. Suslin, A.A., Homology of GLn, characteristic classes andMilnor K-theory, in Algebraic K-theory, number theory, geometry and analysis, Lecture Notes in Math., Springer 1046 (1984), 357–375.Google Scholar
9. Switzer, R.M., Algebraic topology - homotopy and homology, Die Grundlehren der mathematischen Wissenschaften, Springer 212 (1975).Google Scholar
10. Vick, J.W., Poincare duality and Postnikov factors, Rocky Mountain J. Math 3 (1973), 483–499.Google Scholar