Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The squaring operartion and the Singer algebraic transfer   Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots
,x_k]$, 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 regularly on $P_k$. We study the
algebraic transfer constructed by Singer using the technique of the hit
problem. This transfer is a homomorphism from the homology of the mod-2
Steenrod algebra, $\text{Tor}^{\mathcal A}_{k,k+n} (\mathbb F_2,\mathbb F_2)$,
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 extend a result of
Hung on the relation between the Singer algebraic transfer and the squaring
operation on the cohomology of the Steenrod algebra. Using this result, we show
that Singer's conjecture for the algebraic transfer is true in the case $k=5$
and the degree $5(2^{s} -1)$ with $s$ an arbitrary positive integer.

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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex