On the behavior of the algebraic transfer

Let Trk : F 2 ⊗ PH i (BV k ) → Ext k,k+i A (F 2 ,F 2 ) be the alge-GL k braic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer tr k : π S * ((BV 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 nonzero element g s ∈ Ext 4,12.2s A (F 2 ,F 2 ) for every s > 1. As a consequence, the localized (Sq 0 ) -1 Tr 4 given by inverting the squaring operation Sq° is not an epimorphism. This gives a negative answer to a prediction by Minami.