Hostname: page-component-7f64f4797f-8cpv7 Total loading time: 0 Render date: 2025-11-04T15:32:49.768Z Has data issue: false hasContentIssue false
  • English
  • Français

The transfer and Whitehead's conjecture

Published online by Cambridge University Press:  24 October 2008

Nicholas J. Kuhn  and
Stewart B. Priddy
Nicholas J. Kuhn
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, U.S.A.
Stewart B. Priddy
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60201, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we present a proof of G. W. Whitehead's conjecture about symmetric products of the sphere spectrum S. Our methods are based on the transfer, the Steinberg module, and the structure of the Hecke algebra. Our results are valid for all primes and extend those of the first author for p = 2 [7]. As originally stated, the conjecture asserts that

is zero on p-primary components in positive degrees [11]. By considering the quotients L(k) = Σ-kSPpkS/SPk-1-S, this is seen to be equivalent to the exactness of

on homotopy groups, where ∂k is the connecting map and ε is the inclusion of the bottom cell. Here and throughout all spaces and spectra are localized at p.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 3 , November 1985 , pp. 459 - 480
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Adams, J. F.. Infinite Loop Spaces. Annals of Math. Studies, vol. 90 (Princeton University Press, 1978).CrossRefGoogle Scholar
[2]Cohen, F. R., Lada, T. J. and May, J. P.. The Homology of Iterated Loop Spaces, Springer Lecture Notes in Math., vol. 533 (1976).CrossRefGoogle Scholar
[3]Iwahori, N.. On the structure of a Hecke ring of Chevalley group over a finite field. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 10 (1964), 215–236.Google Scholar
[4]Kahn, D. S.. On the stable decomposition of Ω∞S∞ A, Springer Lecture Notes in Math., vol. 658 (1978), 206–213.Google Scholar
[5]Kahn, D. S. and Priddy, S. B.. The transfer and stable homotopy theory. Math. Proc. Cambridge Philos. Soc. 83 (1978), 103–111.CrossRefGoogle Scholar
[6]Kahn, D. S. and Priddy, S. B.. On the transfer in the homology of symmetric groups. Math. Proc. Cambridge Philos. Soc. 83 (1978), 91–101.CrossRefGoogle Scholar
[7]Kuhn, N. J.. A Kahn-Priddy sequence and a conjecture of G. W. Whitehead. Math. Proc. Cambridge Philos. Soc. 92 (1982), 467–483.CrossRefGoogle Scholar
[8]Kuhn, N. J.. Spacelike resolutions of spectra. Proc. of the Northwestern Homotopy Theory Conference, A.M.S. Contemporary Math. Series, 19 (1983), 153–165.CrossRefGoogle Scholar
[9]Kuhn, N. J. (with an appendix by P. Landrock). The modular Hecke algebra and Steinberg representation of finite Chevalley groups. J. Algebra 91 (1984), 125–141.CrossRefGoogle Scholar
[10]Kuhn, N. J.. Chevalley group theory and the transfer in the homology of symmetric groups. Topology, to appear.Google Scholar
[11]Milgram, R. J. (ed.). Problems presented to the 1970 A.M.S. Summer Colloquium in Algebraic Topology. Algebraic Topology, Proc. Symp. Pure Math. XXII, A.M.S. (1971), 187–201.CrossRefGoogle Scholar
[12]Mitchell, S. A. and Priddy, S. B.. Stable splittings derived from the Steinberg module. Topology 22 (1983), 285–298.CrossRefGoogle Scholar
[13]Quillen, D.. The Adams conjecture. Topology 19 (1971), 67–80.CrossRefGoogle Scholar