SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp

Introduction. In this paper, we derive formulæ in Brown-Peterson homology at the prime 2 related to the family of elements φn ∈ MSp8n−3 of N. Ray, whose central rôle in the structure of MSp has been highlighted by recent work of V. Vershinin and other Russian topologists. In effect, we give explicit “chromatic” representatives for these elements, which were known to be detected in KO and mod 2 KU-homology, and are thus “v1-periodic” in the parlance of [4] and [5]. In future work we will investigate further the v1 periodic part of MSp and discuss the relationship of our work with that of B. Botvinnik. I would like to thank Nigel Ray for many helpful discussions and large amounts of advice on MSp (including severe warnings!) over many years; in particular, §5 in this paper was prompted by his suggestions about the detection of φn in the classical Adams spectral sequence. I would also like to thank Boris Botvinnik, Vassily Gorbunov and Vladimir Vershinin for discussions on the material of earlier versions of this paper both during and after the J. F. Adams Memorial Symposium and in particular for bringing to my attention Bŭhstaber’s article [2] which contains related results.