The negative answer to Kameko's conjecture on the hit problem

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules. A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. In this paper we give explicit formulae for the dimension of ${\Q}(n)$ in degrees less than or equal to $12.$

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“…Theorem 3 (Kameko, 1990;Sum, 2010). Let a, b be monomials in P(n) such that w j (a) = 0 for j > r > 0.…”

Section: Preliminariesmentioning

confidence: 99%

Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$

Mothebe¹,

Kaelo²,

Ramatebele³

2016

JMR

177

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“…Then Q(M) is a graded vector space over F 2 and a basis for Q(M) lifts to a minimal generating set for M as a module over A 2 . Kameko [7] computed dim(Q d (P(n))) for n = 1, 2, 3 and conjectured a best upper bound n k=1 (2 k − 1) for general n. Nguyen Sum [10] showed that this conjecture is true for n = 4. However, in a recent paper [11] he proved that this conjecture turns out to be wrong for any n > 4.…”

Section: Remarksmentioning

confidence: 99%

Criteria for a Symmetrized Monomial in B(3) to Be Non-Hit

Janfada¹

2014

Communications of the Korean Mathematical Society

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“…Then θ commutes with the coboundary map δ, and thus induces an endomorphism on Ext * , * E 0 A (Z/2, Z/2). On H * (BV s ), there is also a squaring map constructed by Kameko [8] in his thesis which has been extremely useful in the study of the hit problem (see for example Sum [21]). It is given explicitly as follows.…”

Section: The Squaring Operationsmentioning

confidence: 99%

On the May spectral sequence and the algebraic transfer II

Chơn¹,

Hà²

2014

Topology and its Applications

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We study the algebraic transfer constructed by Singer [19] using the May spectral sequence technique. We show that the two squaring operators defined by Kameko [8] and Nakamura [16] on the domain and range respectively of our E 2 version of the algebraic transfer are compatible. We also prove that the two Sq 0 -families n i ∈ Ext 5,36·2 i A (Z/2, Z/2), i ≥ 0, and k i ∈ Ext 7,36·2 i A (Z/2, Z/2), i ≥ 1, are in the image of the algebraic transfer.

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The negative answer to Kameko's conjecture on the hit problem

Cited by 60 publications

References 28 publications

Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$

Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$

Criteria for a Symmetrized Monomial in B(3) to Be Non-Hit

On the May spectral sequence and the algebraic transfer II

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