No abstract

We write $BV_h$ for the classifying space of the elementary Abelian 2-group $V_h$ of rank $h,$ which is homotopy equivalent to the cartesian product of $h$ copies of $\mathbb RP^{\infty}.$ Its cohomology with $\mathbb Z/2$-coefficients can be identified with the graded unstable algebra $P^{\otimes h} = \mathbb Z/2[t_1, \ldots, t_h]= \bigoplus_{n\geq 0}P^{\otimes h}_n$ over the Steenrod ring $\mathcal A$, where grading is by the degree of the homogeneous terms $P^{\otimes h}_n$ of degree $n$ in $h$ generators with the degree of each $t_i$ being one. Let $GL_h$ be the usual general linear group of rank $h$ over $\mathbb Z/2.$ The algebra $P^{\otimes h}$ admits a left action of $\mathcal A$ as well as a right action of $GL_h.$ A central problem of homotopy theory is to determine the structure of the space of $GL_h$-coinvariants, $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ,$ where ${\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ={\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ denotes the space of primitive homology classes, considered as a representation of $GL_h$ for all $n.$ Solving this problem is very difficult and still unresolved for $h\geq 4.$ The aim of this Note is of studying the dimension of $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ for the case $h = 4$ and the "generic" degrees $n$ of the form $k(2^{s} - 1) + r.2^{s},$ where $k,\, r,\, s$ are positive integers. Applying the results, we investigate the behaviour of the Singer cohomological "transfer" of rank $4$, which is a homomorphism from a certain subquotient of the divided power algebra $\Gamma(a_1^{(1)}, \ldots, a_4^{(1)})$ to mod-2 cohomology groups ${\rm Ext}_{\mathcal A}^{4, 4+n}(\mathbb Z/2, \mathbb Z/2)$ of the algebra $\mathcal A.$ Singer's algebraic transfer is one of the relatively efficient tools in determining the cohomology of the Steenrod algebra.

Fix $\mathbb Z/2$ is the prime field of two elements and write $\mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, \mathbb Z/2)$ the general linear group of rank $d$ over $\mathbb Z/2$ and by $\mathscr P_d$ the polynomial algebra $\mathbb Z/2[x_1, x_2, \ldots, x_d]$ as a connected unstable left $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson "hit problem" of finding the minimal set of $\mathcal A_2$-generators for $\mathscr P_d.$ Equivalently, we need to determine a basis for the $\mathbb Z/2$-vector space $$Q\mathscr P_d := \mathbb Z/2\otimes_{\mathcal A_2} \mathscr P_d \cong \mathscr P_d/\mathcal A_2^+\mathscr P_d$$ in each degree $n\geq 1.$ Note that this space is a representation of $GL_d$ over $\mathbb Z/2.$ The problem for $d= 5$ is not yet completely solved, and unknow in general.In this work, we give an explicit solution to the hit problem of five variables in the generic degree $n = r(2^t -1) + 2^ts$ with $r = d = 5,\ s =8$ and $t$ an arbitrary non-negative integer. An application of this study to the cases $t = 0$ and $t = 1$ shows that the Singer algebraic transfer of rank $5$ is an isomorphism in the bidegrees $(5, 5+(13.2^{0}-5))$ and $(5, 5+(13.2^{1}-5)).$ Moreover, the result when $t\geq 2$ was also discussed. Here, the Singer transfer of rank $d$ is a $\mathbb Z/2$-algebra homomorphism from $GL_d$-coinvariants of certain subspaces of $Q\mathscr P_d$ to the cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A_2}^{d, d+*}(\mathbb Z/2, \mathbb Z/2).$ It is one of the useful tools for studying these mysterious Ext groups.

The cohomology ${\rm Ext}_{\mathscr A}(\mathbb F, \mathbb F) = \{{\rm Ext}^{h, t}_{\mathscr A}(\mathbb F, \mathbb F)\}_{(h, t)\in \mathbb Z^{2}, h\geq 0,\, t\geq 0}$ of the Steenrod algebra $\mathscr A$ over the prime field $\mathbb F$ with 2 elements is an object of great interest. This cohomology could be identified with the $E_2$-term for the (2-local) Adams spectral sequence, whose abutment is the 2-component of the stable homotopy groups of spheres. Let $P^{\otimes h}$ denote the unstable algebra $H^{*}(B\mathbb F^{h}, \mathbb F).$ Writing $(P^{\otimes h})^{*} = H_{*}(B\mathbb F^{h}, \mathbb F)$ for the dual of $P^{\otimes h}$ and $P_{\mathscr A}(P^{\otimes h})^{*}$ for the primitive part consisting of all elements in $(P^{\otimes h})^{*}.$ For the general linear group $GL_h:= GL(h, \mathbb F)$, the subject of the present paper is the algebraic transfer of rank $h,$$$ Tr_h^{\mathscr A}: (\mathbb F\otimes_{GL_h} P_{\mathscr A}(P^{\otimes h})^{*})_n \longrightarrow {\rm Ext}^{h, h+n}_{\mathscr A}(\mathbb F, \mathbb F) $$which is proposed by W.M. Singer \cite{W.S1} as an algebraic version of the geometrical transfer $\pi_*^{S}((B\mathbb F^{h})_{+})\longrightarrow \pi^{S}_*(S^{0})$ to the stable homotopy groups of spheres. Computing the domain of $Tr_h^{\mathscr A}$ in each positive degree $n$ is a hard work. Singer's transfer, which gives important information on the cohomology of the algebra $\mathscr A$, is highly nontrivial and, more precisely, that $Tr_h^{\mathscr A}$ is an isomorphism for $h\leq 3$. Nevertheless, the transfers $Tr_4^{\mathscr A}$ and $Tr_5^{\mathscr A}$ are not isomorphisms. More explicitly, the non-zero elements $g_s\in {\rm Ext}_{\mathscr A}^{4, 12.2^{s}}(\mathbb F, \mathbb F)$ for all $s\geq 1$ and $Ph_1\in {\rm Ext}_{\mathscr A}^{5, 14}(\mathbb F, \mathbb F)$ are not detected by $Tr_4^{\mathscr A}$ and $Tr_5^{\mathscr A},$ respectively. Very little information is known for ranks $\geq 6.$ In this Note, we prove that $Tr_6^{\mathscr A}$ does not detect the non-zero elements $h_5Ph_1\in {\rm Ext}_{\mathscr A}^{6, 46}(\mathbb F, \mathbb F)$ and $h_2h_6g_1\in {\rm Ext}_{\mathscr A}^{6, 92}(\mathbb F, \mathbb F).$ (Noting that the Adams elements $h_s\in {\rm Ext}_{\mathscr A}^{1, 2^{s}}(\mathbb F, \mathbb F)\, (s\geq 0)$ are detected by $Tr_1^{\mathscr A}.$) To make this, we shall explicitly compute the domain of $Tr_6^{\mathscr A}$ in general degree $n = 23.2^{s} - 6$ for all $s\geq 0.$ This research can be understood as a continuation of our recent works \cite{D.P1, D.P2}.

On the generators of the polynomial algebra as a module over the Steenrod algebra Sur les générateurs de l'algèbre polynomiale comme module sur l'algèbre de Steenrod