Products on 𝑀𝑈-modules

Abstract

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of M U MU -modules such as B P BP , K ( n ) K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over M U [ 1 2 ] ∗ MU[\frac {1}{2}]_* that are concentrated in degrees divisible by 4 4 ; this guarantees that various obstruction groups are trivial. We extend these results to the cases where 2 = 0 2=0 or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2 2 -local M U ∗ MU_* -modules as M U MU -modules.