Given two natural numbers n and k, consider the Thom space over the classifying space of a rank n elementary abelian 2-group associated to k copies of its real reduced regular representation. The Steinberg module of the general linear group gives rise to a stable summand of this Thom space, denoted by $L(n,k)$. Takayasu (1999) showed the existence of a cofibre sequences $$\Sigma^kL(n-1,2k+1) \to L(n,k) \to L(n,k+1),$$ which generalizes the stable splitting of Mitchell and Priddy. A cofiber sequence of the same form was proved by Arone and Mahowald by combining Goodwillie calculus with the James fibration.
I will describe in this talk how to derive the existence of the above cofibre sequences from the vanishing of some extension groups in the category of modules over the mod 2 Steenrod algebra.
This is joint work with Lionel Schwartz.
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