The cohomology of the Steenrod algebra and representations of the general linear groups

Abstract

Let Trk be the algebraic transfer that maps from the coinvariants of certain GLk-representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer trk, : πS* ((BVk)+) → π (S0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Trk, is an isomorphism for k = 1,2,3 and that Tr = ⊕k Trk is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree d and apply Sq0 repeatedly at most (k - 2) times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the GLk-representations. As a consequence, every finite Sq0-family in the coinvariants has at most (k - 2) nonzero elements. Two applications are exploited. The first main theorem is that Trk is not an isomorphism for k ≥ 5. Furthermore, for every k > 5, there are infinitely many degrees in which Trk. is not an isomorphism. We also show that if Trℓ detects a nonzero element in certain degrees of Ker(Sq0). then it is not a monomorphism and further, for each k>l, Trk is not a monomorphism in infinitely many degrees. The second main theorem is that the elements of any Sq0-family in the cohomology of the Steenrod algebra, except at most its first (k - 2) elements, are either all detected or all not detected by Trk, for every k. Applications of this study to the cases k = 4 and 5 show that Tr4 does not detect the three families g, D3 and p′, and that Tr5 does not detect the family {hn+19n| n ≥ 1}.