A note on the non-trivial elements in the cohomology groups of the Steenrod algebra

Abstract

Let $F_2$ be the prime field of two elements and let $G{L}_s:=GL(s,F_2)$ be the general linear group of rank $s.$ Denote by $\mathcal{A}$ the Steenrod algebra over $F_2.$ The (mod-2) Lambda algebra, $\Lambda ,$ is one of the tools to describe those mysterious "Ext-groups". In addition, the $s$-th algebraic transfer of William Singer \cite{Singer} is also expected to be a useful tool in the study of them. This transfer is a homomorphism $T{r}_s:F_2{\otimes }_{G{L}_s}{P}_{\mathcal{A}}(H_{\ast }(B{\mathbb{V}}_s,F_2))\to {\mathrm{Ext}}_{\mathcal{A}}^{s,s+\ast }(F_2,F_2),$ where ${\mathbb{V}}_s$ denotes the elementary abelian $2$-group of rank $s$, and $H_{\ast }(B{\mathbb{V}}_s)$ is the homology group of the classifying space of ${\mathbb{V}}_s,$ while $P_{\mathcal{A}}(H_{\ast }(B{\mathbb{V}}_s,F_2))$ means the primitive part of $H_{\ast }(B{\mathbb{V}}_s,F_2)$ under the action of $\mathcal{A}.$ It has been shown that $T{r}_s$ is highly non-trivial and, more precisely, that $T{r}_s$ is an isomorphism for $s\le 3.$ In addition, Singer proved that $T{r}_4$ is an isomorphism in some internal degrees. He was also investigated the image of the fifth transfer by using invariant theory. In this note, we use another method to study the image of $T{r}_5.$ More precisely, by direct computations using a representation of $T{r}_5$ over the algebra $\Lambda ,$ we show that $T{r}_5$ detects the non-zero elements $h_0{d}_0\in {\mathrm{Ext}}_{\mathcal{A}}^{5,5+14}(F_2,F_2),h_2{e}_0\in {\mathrm{Ext}}_{\mathcal{A}}^{5,5+20}(F_2,F_2)$ and $h_1{h}_4{c}_0\in {\mathrm{Ext}}_{\mathcal{A}}^{5,5+24}(F_2,F_2).$ The same argument can be used for homological degrees $s\ge 6$ under certain conditions.

10 pages. This article is to update some references and to correct some minor errors in a paper with the same title published in Advances in Mathematics: Scientific Journal

References in corpus (2)

• On the generators of the polynomial algebra as a module over the Steenrod algebra
• Sub-Hopf algebras of the Steenrod algebra and the Singer transfer