On the behavior of the algebraic transfer

Let $Tr_k:\mathbb{F}_2\underset{GL_k}{\otimes} PH_i(B\mathbb{V}_k)\to Ext_{\mathcal{A}}^{k,k+i}(\mathbb{F}_2, \mathbb{F}_2)$ be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\mathbb{V} _k)_+) \to \pi_*^S(S^0)$. 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 nonzero element $g_s\in Ext_{\mathcal{A}}^{4,12\cdot 2^s}(\mathbb{F}_2, \mathbb{F}_2)$ for every $s\geq 1$. 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.