Let Tr_k: F₂ ⊗_GLk PH_i(B double-struck V sign_k) → Ext_A^k,k+iy (double struck F sign₂, double struck F sign₂) be the alge-braic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer tr_k: π*^S((B double-struck V sign_k)+) → π*^S (S ⁰). It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that Tr_k is an isomorphism for k = 1, 2, 3. However, Singer showed that Tr₅ is not an epimorphism. In this paper, we prove that Tr₄ does not detect the nonzero element gs∈ Ext_A^{4, 12.2s} (double struck F sign₂, double struck F sign₂) for every s ≥ 1. As a consequence, the localized (Sq⁰)^-1Tr₄ given by inverting the squaring operation Sq⁰ is not an epimorphism. This gives a negative answer to a prediction by Minami.

Bruner R.R., Ha L.M., Hung N.H.V.

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