Ω SU ( n ) does not Split in 2 Suspensions , for n ≥ 3

Solving a conjecture of Hopkins and Mahowald, the second author [Ri] showed that Mitchell’s [Mi3] filtration {Fn,k}k=1 of ΩSU(n) splits stably, analogous to the Snaith [Sn2] splitting of BU . Crabb and Mitchell [C-M] then gave similar splittings of ΩU(n)/O(n) and ΩU(2n)/Sp(n). The first filtration Fn,1 is the inclusion CPn−1 ⊂ ΩSU(n), which was actually known to split off by the work of James [Ja], which was refined by Miller [Mi2]. James split ΣCPn−1 off SU(n), with a map J : SU(n) −→ ΩΣ ( ΣCPn−1 ) , whose loop Ω (J) : ΩSU(n) −→ ΩN+1ΣN+1CPn−1 splits CPn−1 stably off ΩSU(n). Cohen and Peterson [C-P], using Dyer-Lashof operations, showed that N > 2 for any such map J. However there is no Dyer-Lashof obstruction to factoring Ω (J) through Ω2Σ2CPn−1. That is, the image of Ω (J) consists only of monomials in H∗ ( CPn−1;Z/2 ) . The question arose: does there exist a map ρ : ΩSU(n) −→ Ω2Σ2CPn−1 which splits off CPn−1 stably? The existence of such a map ρ would have implied the stable splitting of ΩSU(n), provided the Mitchell-Segal [Mi3, Se1] group completion model ∐ k Fn,k ⊂ ΩU(n) has a C2-structure (cf. May [Ma3]). Following Snaith [Sn2], the k splitting map could have been constructed as the composite