What we still don’t know about loop spaces of spheres

Abstract

We describe a program for computing the Morava K-theory of certain iterated loop spaces of spheres, based on a hypothetical duality with the computation of that of the Eilenberg-MacLane spaces by Wilson and the author [RW80]. Under this duality the duals of the bar spectral sequence and facts about K(n)∗(BZ/p) used in [RW80] are the Eilenberg-Moore spectral sequence and facts about K(n)∗(Ω2S2m+1) respectively. The program depends on the existence of a new geometric structure in those loop spaces dual to the cup product. We get a precise answer under these hypotheses. Iterated loop spaces of spheres have played a central role in homotopy theory for many years. They have been thoroughly studied but there are still some open questions concerning them. We will denote ΩS by Lm,s. Its homology has long been known for many years and is given in [CLM76]. Its BP homology is known only for m ≤ 1; see [Rav93]. Its Morava K-theory was computed for m = 1 by Yamaguchi [Yam88] and for m = 2 by Tamaki (unpublished). In §5 we will describe a speculative program to compute the Morava K-theory more generally in a way that is analogous to the Ravenel-Wilson computation [RW80] of the Morava K-theory of EilenbergMacLane spaces, which is reviewed in §4. It depends on a hypothetical new geometric structure on the stable Snaith summands of iterated loop spaces.