Extended powers of spectra and a generalized Kahn-Priddy theorem

Abstract

lN[7], Kahn and Priddy showed that, localized at a prime p, there is a map of infinite loop spaces QBCp-,QOSo that is a projection onto a direct factor in the category of topological spaces. Here QX = R “Z “X where a cc is the 0th space functor from the category of spectra to the category of spaces, right adjoint to the suspension spectrum functor C”, BC, is the classifying space of the symmetric group Zp, and QoSo is the bzsepoint component of Qs’. This splitting in fact “deloops” once[8], so that QyPB&-+QS’ has a right inverse, where 8’ is the simply connected cover of 5’. Note that QS’ is the fiber of the structure map QS’+S’ associated to the infinite loop space S’. In this paper we prove a generalization of the Kahn-Priddy theorem in this strong form, in which S’ is replaced by an arbitrary connected infinite loop space. When p = 2, our theorem corresponds to work by Finkelstein and Kahn announced in [5]. Our proof is quite short and noncomputational. We make use of manipulations of the adjoint pair (Zn;, a=). This allows us to work stably so that we can make use of the splitting[6], valid for any connected space X: