The transfer and Whitehead's conjecture

Abstract

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 -k SP pk S/SP k-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 .