Let Tr_k : ��_2 (⊗ over GL_k) PH_i(B��_k) → Ext^(k,k+i)_A(��_2,��_2) be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer tr_k : π_∗^S((B��_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 Tr_5 is not an epimorphism. In this paper, we prove that Tr_4 does not detect the non zero element g_s ∈ Ext^(4,12·2^s)_A(��_2,��_2) for every s ≥ 1. As a consequence, the localized (Sq^0)^(−1)Tr_4 given by inverting the squaring operation Sq^0 is not an epimorphism. This gives a negative answer to a prediction by Minami.
Algebraic Geometry | Geometry and Topology
Bruner, R. R., Hà, L. M., & Hưng, N. H. V. On the behavior of the algebraic transfer. Trans. Amer. Math. Soc. 357 (2005), 473-487. https://doi.org/10.1090/S0002-9947-04-03661-X