Journal of the
Korean Mathematical Society JKMS

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. 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 William Singer, 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 of this work is to study the behavior of the algebraic transfer for ranks $h\in \{6,\, 7,\, 8\}$ in some internal degrees. More precisely, we show that the algebraic transfer is an isomorphism in certain bidegrees. As a consequence, we are able to confirm the Singer conjecture for the algebraic transfer in the cases under consideration. Especially, we affirm that 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. 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 ${\rm Ext}_{\mathcal A}^{h, h+\bullet}(\mathbb Z/2, \mathbb Z/2)$.

Keywords: Adams spectral sequences, Primary cohomology operations, Steenrod algebra, Hit problem, algebraic transfer

MSC numbers: 55Q45, 55S10, 55S05, 55T15, 55R12