On mod-2 cohomology of the Steenrod ring and Singer's conjecture for the algebraic transfer

Abstract

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}.

Singer’s algebraic transfer for all ranks s in some degrees and the negation of the dimension results for the graded spaces F2 ⊗ AF2 [x1,...,xs] in degrees s+ 5

“…al. [23,24,25,26], Walker and Wood [56], the present author [30,31,32,33,35,36,38,39,40], Sum [47,48,52,53] and others), but it was a well known unresolved problem for s 5. The problem was systematically studied for s 4 (see [4,18,28,49]).…”

“…Thus, we have seen that Singer's transfer is highly non-trivial. Since then, it has been extensively studied by many algebraists like Bruner, Hà and Hưng [9], Chơn and Hà [13,14,15], Hà [16], Hưng [17], Nam [27], the present author [30,32,33,34,35,36,37,38,39,40], Quỳnh [42], Sum [48,51,52,53] and others. Remarkably, in higher rank, Singer states that the transfer is not a monomorphism in bidegree (5,14) and gives the following prediction.…”

“…Due to Boardman [4] and the fact that s 0 (ϕ s ) * is an algebra homomorphism, deduce that the element c 2 g 2 is in the image of (ϕ 7 ) * . In [40], we have shown that every indecomposable element in the Sq 0 -family…”

Singer's algebraic transfer for all ranks $s\geq 5$ and the negation of the dimension results for the graded spaces $\mathbb F_2\otimes_{\mathscr A} \mathbb F_2[x_1, \ldots, x_s]$ in degrees $s+5$

“…al. [26,27,28,29], Walker and Wood [59], the present author [33,34,35,36,38,39,41,42,43], Sum [50,51,55,56] and others), but it was a well known unresolved problem for s 5. The problem was systematically studied for s 4 (see [4,21,31,52]).…”

“…Thus, we have seen that Singer's transfer is highly non-trivial. Since then, it has been extensively studied by many algebraists like Bruner, Hà and Hưng [9], Chơn and Hà [13,14,15], Hà [16], Hưng [17], Nam [30], the present author [33,35,36,37,38,39,40,41,42,43], Quỳnh [45], Sum [51,54,55,56] and others. Remarkably, in higher rank, Singer states that the transfer is not a monomorphism in bidegree (5,14) and gives the following prediction.…”