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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


Dedicated to Professor Huỳnh Mùi on the occasion of his sixtieth birthday. The third author was supported in part by the Vietnam National Research Program, Grant N. The computer calculations herein were done on equipment supplied by NSF grant DMS-0079743. Author's Accepted Manuscript. © Copyright 2004 American Mathematical Society, first published in Trans. Amer. Math. Soc. 357 (2005), 473-487,