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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.
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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.

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

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References

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