On the behavior of the algebraic transfer

Abstract

Let T r k : F 2 ⊗ G L k P H i ( B V k ) → E x t A k , k + i ( F 2 , F 2 ) 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 t r k : π ∗ S ( ( B V k ) + ) → π ∗ S ( S 0 ) 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 T r k Tr_k is an isomorphism for k = 1 , 2 , 3 k=1, 2, 3 . However, Singer showed that T r 5 Tr_5 is not an epimorphism. In this paper, we prove that T r 4 Tr_4 does not detect the nonzero element g s ∈ E x t A 4 , 12 ⋅ 2 s ( F 2 , F 2 ) g_s\in Ext_{\mathcal {A}}^{4,12\cdot 2^s}(\mathbb {F}_2, \mathbb {F}_2) for every s ≥ 1 s\geq 1 . As a consequence, the localized ( S q 0 ) − 1 T r 4 (Sq^0)^{-1}Tr_4 given by inverting the squaring operation S q 0 Sq^0 is not an epimorphism. This gives a negative answer to a prediction by Minami.