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On a conjecture for the algebraic transfer in generic families of internal degrees
  • Đặng Võ Phúc
Đặng Võ Phúc
FPT University, FPT University, FPT University, FPT University

Corresponding Author:[email protected]

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Abstract

Let $\mathbb F_2$ denote the binary field. Write $\mathscr A$ for the Steenrod ring over $\mathbb F_2.$ In this paper, we study Singer’s conjecture \cite{Singer} for the algebraic transfers of ranks 5 and 6 in the generic families of internal degrees. The Singer algebraic transfer stands as a valuable instrument for unraveling the intricate structure of the  cohomology ${\rm Ext}_{\mathscr A}^{s,k} := {\rm Ext}_{\mathscr A}^{s}(\mathbb F_2, \Sigma^{k}\mathbb F_2)$ of the (mod-2) Steenrod ring. Remarkably, we have shown that the indecomposable element $y\in {\rm Ext}_{\mathscr A}^{6,44}$ is not in the image of the sixth algebraic transfer.