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

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: π _*S((BV ₖ)₊) → π _*S(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 Tr detects a nonzero element in certain degrees of Ker(Sq⁰), then it is not a monomorphism and further, for each k>, 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 ₙ₊₁gₙ |\; n≥ 1\.