An Orientation map for height p - 1 real E theory

Abstract

Let p be an odd prime and let EO = E[superscript hC] [subscript p-1] be the Cp αxed points of height p - 1 Morava E theory. We say that a spectrum X has algebraic EO theory if the splitting of K[subscript *](X) as an K[subscript *][Cp]-module lifts to a topological splitting of EO [subscript grave] X. We develop criteria to show that a spectrum has algebraic EO theory, in particular showing that any connective spectrum with mod p homology concentrated in degrees 2k(p - 1) has algebraic EO theory. As an application, we answer a question posed by Hovey and Ravenel [10] by producing a unital orientation MW [subscript 4p-4] --> EO analogous to the MSU orientation of KO at p = 2 where MW [subscript 4p-4] is the Thom spectrum of the (4p - 4)-connective Wilson space.