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 spectrumS.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 forp= 2 [7]. As originally stated, the conjecture asserts thatis zero onp-primary components in positive degrees [11]. By considering the quotientsL(k)= Σ-kSPpkS/SPk-1-S, this is seen to be equivalent to the exactness ofon homotopy groups, where ∂kis the connecting map and ε is the inclusion of the bottom cell. Here and throughout all spaces and spectra are localized atp.