Let Tr_k be the algebraic transfer that maps from the coinvariants of certain GL_k-representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr_k, : πS* ((BV_k)+) → π (S₀). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Tr_k, is an isomorphism for k = 1,2,3 and that Tr = ⊕_k Tr_k 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 GL_k-representations. As a consequence, every finite S_q⁰-family in the coinvariants has at most (k - 2) nonzero elements. Two applications are exploited. The first main theorem is that Tr_k is not an isomorphism for k ≥ 5. Furthermore, for every k > 5, there are infinitely many degrees in which Tr_k. is not an isomorphism. We also show that if Tr_ℓ detects a nonzero element in certain degrees of Ker(S_q⁰). then it is not a monomorphism and further, for each k>l, Tr_k is not a monomorphism in infinitely many degrees. The second main theorem is that the elements of any S_q⁰-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 Tr_k, 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 {h_n+19n| n ≥ 1}.

Hung N.H.V.

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