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

Abstract

The cohomology ${\rm Ext}^{h}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2) = \{{\rm Ext}^{h, t}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2)=: H^{h, t}(\mathcal A)\}_{(h, t)\in \mathbb Z^{2}, h\geq 0,\, t\geq 0}$ of the Steenrod ring $\mathcal A$ over the prime field $\mathbb Z/2$ with 2 elements is an object of great interest, since it is identified with the $E_2$-term for the (2-local) Adams spectral sequence (ASS), whose abutment is the 2-component of the stable homotopy groups of spheres. The structure of ${\rm Ext}^{h}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2) $ is still largely mysterious for cohomological degrees $h\geq 6.$ Let $P^{\otimes h}$ denote the unstable algebra $H^{*}(B(\mathbb Z/2)^{\times h}, \mathbb Z/2)$ and $GL_h:= GL(h, \mathbb Z/2)$ denote the $h$-dimensional general linear group over $\mathbb Z/2.$ The subject of this manuscript is the algebraic transfer $$ Tr_h^{\mathcal A}: ([\mathbb Z/2\otimes_{\mathcal A}P^{\otimes h}]^{GL_h})^{*} \longrightarrow {\rm Ext}^{h}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2) $$of rank $h$ \cite{Singer} which could be viewed as the $E_2$-level in the ASS of the stable transfer $B(\mathbb Z/2)^{\times h}_{+}\longrightarrow \mathbb S^{0}.$ Here the domain of $Tr_h^{\mathcal A}$ is the primitive part of $H_{*}(B(\mathbb Z/2)^{\times h}, \mathbb Z/2)$ under action of $\mathcal A$ and is dual to the space of $GL_h$-invariants, $[\mathbb Z/2\otimes_{\mathcal A}P^{\otimes h}]^{GL_h}.$ Computing domain and codomain of $Tr_h^{\mathcal A}$ in each positive degree is a hard work. Singer's transfer, which gives important information on $\mathbb Z/2$-cohomology of Steenrod ring, is highly nontrivial and, more precisely, that $Tr_h^{\mathcal A}$ is an isomorphism for $h\leq 3$. Nevertheless, the transfers $Tr_4^{\mathcal A}$ and $Tr_5^{\mathcal A}$ are not isomorphisms. W. Singer conjectured that $Tr_h^{\mathcal A}$ is a monomorphism for every $h>0.$ This currently has not information about ranks $h\geq 6.$ The aim of the present work is to verify Singer's conjecture for $6\leq h\leq 8$ in certain "generic" degrees. Especially, computational techniques show that in the rank 6 case, the critical element $h_5Ph_2\in {\rm Ext}_{\mathcal A}^{6, 12.2^{2}}(\mathbb Z/2, \mathbb Z/2)$ and every decomposable element of $Sq^{0}$-family initiated by $h_5Ph_1$ are not detected by $Tr_6^{\mathcal A}$ and that in the rank 7 case, the indecomposable elements $i\in {\rm Ext}_{\mathcal A}^{7, 30}(\mathbb Z/2, \mathbb Z/2)$, and $j\in {\rm Ext}_{\mathcal A}^{7, 33}(\mathbb Z/2, \mathbb Z/2)$ are not detected by $Tr_7^{\mathcal A}.$ Here $Sq^{0}$ denotes the classical squaring operation ${\rm Ext}^{h}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2)\longrightarrow {\rm Ext}^{h}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ Further, the sixth cohomology groups ${\rm Ext}_{\mathcal A}^{6, (k+\ell).2^s}(\mathbb Z/2, \mathbb Z/2)$ have also been explicitly computed for $k = 6,\, \ell \in \{1,\, 6,\, 9, \, 15,\, 17\}$ and any $s\geq 0.$ Additionally, we claim that the indecomposable elements in the $Sq^{0}$-family $\{t_{s-1}\in {\rm Ext}_{\mathcal A}^{6, 21.2^s}(\mathbb Z/2, \mathbb Z/2):\, s \geq 1\}$ are detected by $Tr_6^{\mathcal A}.$ Besides, the indecomposable elements in the $Sq^{0}$-families $\{C(s-3)\in {\rm Ext}_{\mathcal A}^{6, 7.2^{s}}(\mathbb Z/2, \mathbb Z/2):\, s\geq 3\}$ and $\{G(s-2)\in {\rm Ext}_{\mathcal A}^{6, 15.2^s}(\mathbb Z/2, \mathbb Z/2):\, s \geq 2\}$ will be discussed. We also propose a conjecture on a $\mathbb Z/2$-bases for the indecomposable elements in ${\rm Ext}_{\mathcal A}^{6}(\mathbb Z/2, \mathbb Z/2).$

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