A Note on the Unstability Conditions of the Steenrod Squares on the Polynomial Algebra

Janfada, Ali S.

doi:10.4134/jkms.2009.46.5.907

Cited by 8 publications

(7 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Peterson [32], Wood [63], Singer [48], and Priddy [45] laid the first foundation for the study of the hit problem and pointed out its relationship to several classical problems in the homotopy theory such as cobordism theory of manifolds, modular representation theory of the general linear group, and stable homotopy type of classifying spaces of finite groups. Later, many researchers have been interested in discovering these hit problems (see Boardman [4], Crabb and Hubbuck [10], Janfada and Wood [19], Janfada [20,21], Kameko [22], Mothebe et. al.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Singer’s algebraic transfer for all ranks s in some degrees and the negation of the dimension results for the graded spaces F2 ⊗ AF2 [x1,...,xs] in degrees s+ 5

Phúc¹

2022

Preprint

View full text Add to dashboard Cite

Let Ps:=F2[x1,x2,...,xs] be the graded polynomial algebra over the prime field of two elements,F2, insvariables x1,x2,...,xs, each of degree one. This algebra is considered as a graded moduleover the mod-2 Steenrod algebra, A. The classical "hit problem", initiated by Frank Peterson[Abstracts Amer. Math. Soc. 833 (1987), 55-89], concerned with seeking a minimal set of A-module Ps. Equivalently, when F2 is an A-module concentrated in degree 0, one can write downexplicitly a monomial basis for theZ-graded vector space overF2: QPs:=F2⊗APs=Ps/A+·Ps, where A+ denotes the augmentation ideal of A.The problem is unresolved in general. In thispaper, we study the hit problem for Ps with s>5. More explicitly, we first compute explicitly thedimension of QPs for s= 5 in the generic degree 21·2t−5witht= 1. Note that the problem when t= 0 was solved by Nguyễn Sum [Vietnam J. Math. 49 (2021), 1079-1096]. Next, we shall reject aresult of Meshack Moetele and Mbakiso Fix Mothebe [East-West Journal of Mathematics 18 (2016),151-170] on the dimension of QPs in degreess+ 5 for 8< s<9. We also give an explicit formula forthe dimension of QPs in degree 14 for alls >0 and in degree 15 for all s >0, s,10. In addition, a conjecture on the dimension of QPs in degree 2s−1−s has been given. As applications, weinvestigate William Singer’s conjecture [Math. Z. 202 (1989), 493-523] on the algebraic transfer of rank 5 in degrees 21·2t−5 for all t>0 and of ranks s >0 in internal degreesd, 130}and{D1(t) = (Sq0)t(D1(0))∈Ext5,57·2tA(F2,F2)|t>0} belongs to theimage of the algebraic transfer of rank 5. We also study the behavior of the algebraic transfer of rank 7 in the generic degrees 23·2t−7 fort= 0and`·2t−7for`∈ {9,16}, t63. Our results then claim that the non-zero elements Pc0∈Ext7,23·20A(F2,F2), k0=k∈Ext7,9·22A(F2,F2)and h0} belongs to the image of the transfer.

show abstract

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Singer’s algebraic transfer for all ranks s in some degrees and the negation of the dimension results for the graded spaces F2 ⊗ AF2 [x1,...,xs] in degrees s+ 5

Phúc¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Peterson [31], Wood [62], Singer [47], and Priddy [44] laid the first foundation for the study of the hit problem and pointed out its relationship to several classical problems in the homotopy theory such as cobordism theory of manifolds, modular representation theory of the general linear group, and stable homotopy type of classifying spaces of finite groups. Later, many researchers have been interested in discovering these hit problems (see Boardman [4], Crabb and Hubbuck [10], Janfada and Wood [18], Janfada [19,20], Kameko [21], Mothebe et. al.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Singer's algebraic transfer for all ranks $s\geq 5$ and the negation of the dimension results for the graded spaces $\mathbb F_2\otimes_{\mathscr A} \mathbb F_2[x_1, \ldots, x_s]$ in degrees $s+5$

Phúc¹

2022

Preprint

View full text Add to dashboard Cite

Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree one. This algebra is considered as a graded module over the mod-2 Steenrod algebra, $\mathscr {A}$. The classical "hit problem", initiated by Frank Peterson [Abstracts Amer. Math. Soc., 833 (1987), 55-89], concerned with seeking a minimal set of $\mathscr A$-module $P_s.$ Equivalently, when $\mathbb F_2$ is an $\mathscr A$-module concentrated in degree 0, one can write down explicitly a monomial basis for the $\mathbb Z$-graded vector space over $\mathbb F_2$:$$ QP_s:= \mathbb F_2 \otimes_{\mathscr A} P_s.$$The problem is unresolved in general. The paper is devoted to refute the results of Moetele and Mothebe [East-West J. of Mathematics 18 (2016), 151-170] on the dimension of $QP_s$ in degree $13$ and the dimension of $QP_9$ in degree $14.$ Further, we have computed explicitly the dimension of $QP_s$ in degree $14$ for all $s.$ Then, by the facts of the $s$-th cohomology groups of the Steenrod algebra ${\rm Ext}_{\mathscr A}^{s, s+13}(\mathbb F_2, \mathbb F_2)$ and ${\rm Ext}_{\mathscr A}^{s, s+14}(\mathbb F_2, \mathbb F_2),$ we show that that the algebraic transfer of rank $s> 0$, defined by W. Singer [Math. Z., 202 (1989), 493-523], is an isomorphism in degrees $13$ and $14.$

show abstract

“…By these, it seems likely that an explicit description of Q ⊗q n for general q will appear in the near future. Hit problems for the symmetric polynomials have been investigated by the works of Janfada and Wood (see [11,12,13,14]). The following Kameko maps [20] are often used in studying the hit problem: For each n ≥ 0, the down Kameko map Sq 0 : (P q ) 2n+q → (P q ) n is a surjective linear map defined on monomials by Sq 0 (f ) = g if f = 1≤i≤q x i g 2 and Sq 0 (f ) = 0 otherwise.…”

Section: Outline Of Main Resultsmentioning

confidence: 99%

Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} \mathscr P_AH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application

Phúc¹

2021

Preprint

View full text Add to dashboard Cite

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This is the content of "hit problem" of Frank Peterson. Based on this problem, we are interested in the $q$-th algebraic transfer $Tr_q^{A}$ of W. Singer \cite{W.S1}, which is one of the useful tools for describing mod-2 cohomology of the algebra $A.$ This transfer is a linear map from the space of $GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The present paper is to investigate this algebraic transfer for the cohomological degree $q = 4.$ More specifically, basing the techniques of the hit problem of four variables, we explicitly determine the structure of $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and a representation of the rank 4 transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. Also, we give some conjectures on the dimensions of $k\otimes_{GL_q(k)} ((P_4)_n^{*})$ for the remaining degrees $n.$ As a consequence, Singer's conjecture for the algebraic transfer is true in the rank 4 case. This study and our previous results \cite{D.P11, D.P12} provided a complete picture of the behavior of $Tr_4^{A}.$

show abstract

A Note on the Unstability Conditions of the Steenrod Squares on the Polynomial Algebra

Cited by 8 publications

References 22 publications

Singer’s algebraic transfer for all ranks s in some degrees and the negation of the dimension results for the graded spaces F2 ⊗ AF2 [x1,...,xs] in degrees s+ 5

Singer’s algebraic transfer for all ranks s in some degrees and the negation of the dimension results for the graded spaces F2 ⊗ AF2 [x1,...,xs] in degrees s+ 5

Singer's algebraic transfer for all ranks $s\geq 5$ and the negation of the dimension results for the graded spaces $\mathbb F_2\otimes_{\mathscr A} \mathbb F_2[x_1, \ldots, x_s]$ in degrees $s+5$

Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} \mathscr P_AH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application

Contact Info

Product

Resources

About