In [6] and [9] the second author and Bill Richter showed that the natural ‘degree’ filtration on the homology of ΩSU(n) has a geometric realization, and that this filtration stably splits (as conjectured by M. Hopkins and M. Mahowald). The purpose of the present paper is to prove the real and quaternionic analogues of these results. To explain what this means, consider the following two ways of viewing the filtration and splitting for ΩSU(n). Whenn= ∞, ΩSU=BU. The filtration isBU(1)⊆BU(2)⊆… and the splittingBU≅ V1≤<∞is a theorem of Snaith[14]. The result for ΩSU(n) may then be viewed as a ‘restriction’ of the result forBU. On the other hand there is a well-known inclusion ℂPn−1. This extends to a map ΩΣℂPn−1→ΩSU(n), and the filtration (or splitting) may be viewed, at least algebraically, as a ‘quotient’ of the James filtration (or splitting) of ΩΣℂPn−1. It is now clear what is meant by the ‘real and quaternionic analogues’. In the quaternionic case, we replaceBUbyBSp=Ω(SU/SP), ΩSU(n) by Ω(SU(2n)/SP(n))and ℂPn−1by ℍPn−1. The integral homology of Ω(SU(2n)/SP(n)) is the symmetric algebra on the homology of ℍPn−1, and may be filtered by the various symmetric powers. We show that this filtration can be realized geometrically, and that the spaces of the filtration are certain (singular) real algebraic varieties (exactly as in the complex case). The strata of the filtration are vector bundles over filtrations of Ω(SU(2n−2)/SP(n−1)), and the filtration stably splits. See Theorems (1·7) and (2·1) for the precise statement. In the real case we replaceBUby Ω(SU/SO), Ω(SU(n)/SO(n)) and ℂPn−1by ℝPn−1. Here integral homology must be replaced by mod 2 homology, and splitting is only obtained after localization at 2. (Snaith's splitting ofBOin [14] can be refined [2, 8] so as to be exactly analogous to the splitting ofBU:BO≅V1≤<∞MO(k).)
Algebraic vector bundles over real algebraic varieties