{"title":"A counter-example to Singer's conjecture for the algebraic transfer","authors":"Nguyen Sum","doi":"arxiv-2408.06669","DOIUrl":null,"url":null,"abstract":"Write $P_k:= \\mathbb F_2[x_1,x_2,\\ldots ,x_k]$ for the polynomial algebra\nover the prime field $\\mathbb F_2$ with two elements, in $k$ generators $x_1,\nx_2, \\ldots , x_k$, each of degree 1. The polynomial algebra $P_k$ is\nconsidered as a module over the mod-2 Steenrod algebra, $\\mathcal A$. Let\n$GL_k$ be the general linear group over the field $\\mathbb F_2$. This group\nacts naturally on $P_k$ by matrix substitution. Since the two actions of\n$\\mathcal A$ and $GL_k$ upon $P_k$ commute with each other, there is an inherit\naction of $GL_k$ on $\\mathbb F_2{\\otimes}_{\\mathcal A}P_k$. Denote by $(\\mathbb\nF_2{\\otimes}_{\\mathcal A}P_k)_n^{GL_k}$ the subspace of $\\mathbb\nF_2{\\otimes}_{\\mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of\ndegree $n$. In 1989, Singer [23] defined the homological algebraic transfer\n$$\\varphi_k :\\mbox{Tor}^{\\mathcal A}_{k,n+k}(\\mathbb F_2,\\mathbb F_2)\n\\longrightarrow (\\mathbb F_2{\\otimes}_{\\mathcal A}P_k)_n^{GL_k},$$ where\n$\\mbox{Tor}^{\\mathcal{A}}_{k, k+n}(\\mathbb{F}_2, \\mathbb{F}_2)$ is the dual of\nExt$_{\\mathcal{A}}^{k,k+n}(\\mathbb F_2,\\mathbb F_2)$, the $E_2$ term of the\nAdams spectral sequence of spheres. In general, the transfer $\\varphi_k$ is not\na monomorphism and Singer made a conjecture that $\\varphi_k$ is an epimorphism\nfor any $k \\geqslant 0$. The conjecture is studied by many authors. It is true\nfor $k \\leqslant 3$ but unknown for $k \\geqslant 4$. In this paper, by using a\ntechnique of the Peterson hit problem we prove that Singer's conjecture is not\ntrue for $k=5$ and the internal degree $n = 108$. This result also refutes a\none of Ph\\'uc in [19].","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"55 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.06669","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Write $P_k:= \mathbb F_2[x_1,x_2,\ldots ,x_k]$ for the polynomial algebra
over the prime field $\mathbb F_2$ with two elements, in $k$ generators $x_1,
x_2, \ldots , x_k$, each of degree 1. The polynomial algebra $P_k$ is
considered as a module over the mod-2 Steenrod algebra, $\mathcal A$. Let
$GL_k$ be the general linear group over the field $\mathbb F_2$. This group
acts naturally on $P_k$ by matrix substitution. Since the two actions of
$\mathcal A$ and $GL_k$ upon $P_k$ commute with each other, there is an inherit
action of $GL_k$ on $\mathbb F_2{\otimes}_{\mathcal A}P_k$. Denote by $(\mathbb
F_2{\otimes}_{\mathcal A}P_k)_n^{GL_k}$ the subspace of $\mathbb
F_2{\otimes}_{\mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of
degree $n$. In 1989, Singer [23] defined the homological algebraic transfer
$$\varphi_k :\mbox{Tor}^{\mathcal A}_{k,n+k}(\mathbb F_2,\mathbb F_2)
\longrightarrow (\mathbb F_2{\otimes}_{\mathcal A}P_k)_n^{GL_k},$$ where
$\mbox{Tor}^{\mathcal{A}}_{k, k+n}(\mathbb{F}_2, \mathbb{F}_2)$ is the dual of
Ext$_{\mathcal{A}}^{k,k+n}(\mathbb F_2,\mathbb F_2)$, the $E_2$ term of the
Adams spectral sequence of spheres. In general, the transfer $\varphi_k$ is not
a monomorphism and Singer made a conjecture that $\varphi_k$ is an epimorphism
for any $k \geqslant 0$. The conjecture is studied by many authors. It is true
for $k \leqslant 3$ but unknown for $k \geqslant 4$. In this paper, by using a
technique of the Peterson hit problem we prove that Singer's conjecture is not
true for $k=5$ and the internal degree $n = 108$. This result also refutes a
one of Ph\'uc in [19].