Stable splittings derived from the Steinberg module

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inaccessible. Examples include Mahowald’s maps ~,[5] based on Snaith’s splittings and, more recently, certain maps used in Kuhn’s proof of the Whitehead conjecture[6, 71. These latter maps are based on our splitting of B(Z/2)k. Our main result shows that the suspension spectrum of a product of lens spaces B(E/P)~ can be split using the Steinberg idempotent of F,,[GL,(ff,,)]. Let Sp”(S”) denote the n-fold symmetric product of the sphere spectrum. We recall Sp”(S”) = K(Z) by the Dold-Thorn theorem. Let D(k) be the cofiber of the diagonal map d: SpP”-‘(S’)+SpJ’“(S”). Then D(co) = K(Z/p). Let M(k) = X-‘-o(k)/D(k 1). In mod-p cohomology H*(M(k)) has a basis consisting of admissible Steenrod operations of length exactly k.