2 ) be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that T r k is an isomorphism for k = 1, 2, 3. However, Singer showed that T r 5 is not an epimorphism. In this paper, we prove that T r 4 does not detect the nonzero element gs ∈ Ext 4,12·2 s A (F 2 , F 2 ) for every s ≥ 1. As a consequence, the localized (Sq 0 ) −1 T r 4 given by inverting the squaring operation Sq 0 is not an epimorphism. This gives a negative answer to a prediction by Minami.

No abstract

The algebraic transfer is an important tool to study the cohomology of the Steenrod algebra. In this study, we will construct a version of the algebraic transfer in E 2 -term of May spectral sequence and use this version to study the image of the algebraic transfer. By this method, we obtain the description of the image of ϕ s in some degrees.

We study the algebraic transfer constructed by Singer [19] using the May spectral sequence technique. We show that the two squaring operators defined by Kameko [8] and Nakamura [16] on the domain and range respectively of our E 2 version of the algebraic transfer are compatible. We also prove that the two Sq 0 -families n i ∈ Ext 5,36·2 i A (Z/2, Z/2), i ≥ 0, and k i ∈ Ext 7,36·2 i A (Z/2, Z/2), i ≥ 1, are in the image of the algebraic transfer.

Sub-Hopf algebras of the Steenrod algebra and the Singer transfer LÊ MINH HÀThe Singer transfer provides an interesting connection between modular representation theory and the cohomology of the Steenrod algebra. We discuss a version of "Quillen stratification" theorem for the Singer transfer and its consequences.