On the unresolved conjecture for the algebraic transfers over the binary field

Abstract

Let us consider the binary field $\mathbb Z/2.$ An important problem of algebraic topology is to determine the cohomology ${\rm Ext}_{\mathcal A}^{h, *}(\mathbb Z/2, \mathbb Z/2)$ of the Steenrod ring $\mathcal A.$ This remains open for all homological degrees $h\geq 6.$ The algebraic transfer of rank $h$, defined by W.M. Singer in [Math. Z. \textbf{202} (1989), 493-523], is a $\mathbb Z/2$-linear map that plays a crucial role in describing the Ext groups. The conjecture proposed by Singer himself, namely that the algebraic transfer is one-to-one, has only been verified for ranks $h<5$ and remains an open problem in general. The objective which drives the writing of this article is to study the behavior of the algebraic transfer for ranks $h\in \{6,\, 7,\, 8\}$ across various internal degrees. More precisely, we prove that the algebraic transfer is an isomorphism in certain bidegrees. A noteworthy aspect of our research is the rectification of the results outlined by M. Moetele and M.F. Mothebe in [East-West J. of Mathematics \textbf{18} (2016), 151-170]. This correction focuses on the $\mathcal A$-generators for the polynomial algebra $\mathbb Z/2[t_1, t_2, \ldots, t_h]$ in degree thirteen and the ranks $h$ mentioned above. As direct consequences, we are able to confirm the Singer conjecture for the algebraic transfer in the cases under consideration. Especially, we affirm that \textit{the decomposable element $h_6Ph_2 \in {\rm Ext}_{\mathcal A}^{6, 80}(\mathbb Z/2, \mathbb Z/2)$ does not reside within the image of the sixth algebraic transfer}. This event carries significance as it enables us to either strengthen or refute the Singer conjecture, which is relevant to the behavior of the algebraic transfer. Additionally, we also show that the indecomposable element $q\in {\rm Ext}_{\mathcal A}^{6, 38}(\mathbb Z/2, \mathbb Z/2)$ is not detected by the sixth algebraic transfer. \textit{Prior to this research, no other authors had delved into the Singer conjecture for these cases. The significant and remarkable advancement made in this paper regarding the investigation of Singer's conjecture for ranks $h,\, 6\leq h\leq 8,$ highlights a deeper understanding of the enigmatic nature of ${\rm Ext}_{\mathcal A}^{h, h+\bullet}(\mathbb Z/2, \mathbb Z/2)$}.