arXiv subjects

Nguyen Khac Tin

Publications and source records attributed to Nguyen Khac Tin.

3 recordsLinked to original sources

arXiv2021

On the hit problem for the Steenrod algebra in some generic degrees and applications

Let $\mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb F_2[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra over the prime field of two elements, $\mathbb F_2.$ We investigate the Peterson hit problem for the polynomial algebra $\mathcal P_{n},$ viewed as a graded left module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ For $n>4,$ this problem is still unsolved, even in the case of $n=5$ with the help of computers. The purpose of this paper is to continue our study of the hit problem by developing a result in \cite{ph31} for $\mathcal P_n$ in the generic degree $r(2^s-1)+m.2^s$ where $r=n=5,\ m=13,$ and $s$ is an arbitrary non-negative integer. Note that for $s=0,$ and $s=1,$ this problem has been studied by Phuc \cite{ph20ta}, and \cite{ph31}, respectively. As an application of these results, we get the dimension result for the polynomial algebra in the generic degree $d=(n-1).(2^{n+u-1}-1)+\ell.2^{n+u-1}$ where $u$ is an arbitrary non-negative integer, $\ell \in \{23, 67 \},$ and $n=6.$ One of the major applications of hit problem is in surveying a homomorphism introduced by Singer, which is a homomorphism $$Tr_n :\text{Tor}^{\mathcal A}_{n, n+d} (\mathbb F_2,\mathbb F_2) \longrightarrow (\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n})_d^{GL(n; \mathbb F_2)}$$ from the homology of the Steenrod algebra to the subspace of $(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n})_d$ consisting of all the $GL(n; \mathbb F_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, $\text{Tor}^{\mathcal A}_{n, n+d}(\mathbb F_2,\mathbb F_2).$ The behavior of the fifth Singer algebraic transfer in degree $5(2^s-1)+13.2^s$ was also discussed at the end of this paper.

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arXiv2021

A note on the hit problem for the Steenrod algebra and its applications

Let $P_{k}=H^{*}((\mathbb{R}P^{\infty})^{k})$ be the modulo-$2$ cohomology algebra of the direct product of $k$ copies of infinite dimensional real projective spaces $\mathbb{R}P^{\infty}$. Then, $P_{k}$ is isomorphic to the graded polynomial algebra $\mathbb{F}_{2}[x_{1},\ldots,x_{k}]$ of $k$ variables, in which each $x_{j}$ is of degree 1, and let $GL_k$ be the general linear group over the prime field $\mathbb{F}_2$ which acts naturally on $P_k$. Here the cohomology is taken with coefficients in the prime field $\mathbb F_2$ of two elements. We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $P_k$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this Note, we explicitly compute the hit problem for $k = 5$ and the degree $5(2^s-1)+24.2^s$ with $s$ an arbitrary non-negative integer. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-$2$ Steenrod algebra, $\mbox{Tor}^{\mathcal{A}}_{k, k+n}(\mathbb{F}_2, \mathbb{F}_2),$ to the subspace of $\mathbb{F}_2\otimes_{\mathcal{A}}P_k$ consisting of all the $GL_k$-invariant classes of degree $n.$ We show that Singer's conjecture for the algebraic transfer is true in the case $k=5$ and the above degrees. This method is different from that of Singer in studying the image of the algebraic transfer. Moreover, as a consequence, we get the dimension results for polynomial algebra in some generic degrees in the case $k=6.$

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arXiv2016

On Singer's conjecture for the fifth algebraic transfer

Let $P_k:= \mathbb{F}_2[x_1,x_2,\ldots ,x_k]$ be the polynomial algebra in $k$ variables with the degree of each $x_i$ being $1,$ regarded as a module over the mod-$2$ Steenrod algebra $\mathcal{A},$ and let $GL_k$ be the general linear group over the prime field $\mathbb{F}_2$ which acts naturally on $P_k$. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $P_k$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-$2$ Steenrod algebra, $\mbox{Tor}^{\mathcal{A}}_{k, k+n}(\mathbb{F}_2, \mathbb{F}_2),$ to the subspace of $\mathbb{F}_2\otimes_{\mathcal{A}}P_k$ consisting of all the $GL_k$-invariant classes of degree $n.$ In this paper, we explicitly compute the hit problem for $k = 5$ and the degree $7.2^s-5$ with $s$ an arbitrary positive integer. Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case $k=5$ and the above degree.

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