scholarly journals On the dimension of the "cohits" space $\mathbb Z_2\otimes_{\mathcal A_2} H^{*}((\mathbb RP(\infty))^{\times t}, \mathbb Z_2)$ and some applications

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Negative Integer ◽  
Central Problem ◽  
Minimal Set ◽  
Prime Field ◽  
Monomial Basis ◽  
Hit Problem

We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with the degree of each $x_i$ being one. Let $\mathcal A_2$ be the Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the $\mathbb Z_2$-graded vector space $\mathbb Z_2\otimes_{\mathcal A_2} P_t.$ This problem, which is called the "hit" problem for Steenrod algebra, has been systematically studied for $t\leq 4.$ The present paper is devoted to the investigation of the structure of the "cohits" space $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ in some certain "generic" degrees. More specifically, we explicitly determine a monomial basis of $\mathbb Z_2\otimes_{\mathcal A_2} P_5$ in degree \mbox{$n_s=5(2^{s}-1) + 42.2^{s}$} for every non-negative integer $s.$ As a result, it confirms Sum's conjecture \cite{N.S2} for a relation between the minimal sets of $\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and degree $n_s$. Based on Kameko's map \cite{M.K} and a previous result by Sum \cite{N.S1}, we obtain a inductive formula for the dimension of $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ in a generic degree given. As an application, we obtain the dimension of $\mathbb Z_2\otimes_{\mathcal A_2} P_6$ in the generic degree $5(2^{s+5}-1) + n_0.2^{s+5}$ for all $s\geq 0,$ and show that the Singer's cohomological transfer \cite{W.S1} is an isomorphism in bidegree $(5, 5+n_s)$.

scholarly journals The hit problem of five variables in the generic degree $5(2^{s}-1) + 42.2^{s}$ and its application

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Central Problem ◽  
Minimal Set ◽  
Prime Field ◽  
Efficient Approach ◽  
Transfer Homomorphism ◽  
Hit Problem

We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with $\deg(x_j) = 1.$ Let $\mathcal A_2$ be the Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the $\mathbb Z_2$-graded vector space $\{(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n\}_{n\geq 0}.$ It is called \textit{the "hit" problem} for Steenrod algebra and has been completely solved for $t\leq 4.$ In this article, we explicitly solve the hit problem of five variables in the "generic" degree $n=5(2^{s}-1) + 42.2^{s}$ for every non-negative integer $s.$ The result confirms Sum's conjecture [15] for the relation between the minimal sets of $\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and degree $n$ above. An efficient approach for surveying the hit problem of five variables has been presented. As an application, we obtain the dimension of $(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n$ for $t = 6$ and degree $5(2^{s+5}-1) + 42.2^{s+5}$ for all $s\geq 0.$ At the same time, we show that the Singer transfer homomorphism is an isomorphism in bidegree $(5, 5+n)$.

scholarly journals The hit problem of five variables in the generic degree $5(2^{s}-1) + 42.2^{s}$ and its application

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Central Problem ◽  
Minimal Set ◽  
Prime Field ◽  
Efficient Approach ◽  
Minimal Sets ◽  
Hit Problem

We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with $\deg(x_j) = 1.$ Let $\mathcal A_2$ be the Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the $\mathbb Z_2$-graded vector space $\{(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n\}_{n\geq 0}.$ It is called \textit{the "hit" problem} for Steenrod algebra and has been completely solved for $t\leq 4.$ In this article, we explicitly solve the hit problem of five variables in the generic degree $5(2^{s}-1) + 42.2^{s}$ for any $s\geq 0.$ The result confirms Sum's conjecture \cite{N.S2} for the relation between the minimal sets of $\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and the above generic degree. An efficient approach to surveying the hit problem of five variables has been presented. As an application, we obtain the dimension of $(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n$ for $t = 6$ and the generic degree $n = 5(2^{s+5}-1) + 42.2^{s+5}$ for all $s\geq 0.$ At the same time, we show that the fifth Singer algebraic transfer is an isomorphism in bidegree $(5, 47.2^{s})$ with $s\geq 0.$

On the hit problem for the unstable $\mathscr A$-module $\mathcal P_5 = H^*((K(\mathbb F_2, 1))^{\times 5}, \mathbb F_2)$ and application

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Algebraic Topology ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Open Problems ◽  
Polynomial Algebras ◽  
Prime Field ◽  
Trivial Element ◽  
Hit Problem

Let us consider the prime field of two elements, $\mathbb F_2.$ One of the open problems in Algebraic topology is the hit problem for a module over the mod 2 Steenrod algebra $\mathscr A$. More specifically, this problem asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$ regarded as a connected unstable $\mathscr A$-module on $m$ variables $x_1, \ldots, x_m,$ each of degree one. The algebra $\mathcal P_m$ is the cohomology with $\mathbb F_2$-coefficients of the product of $m$ copies of the Eilenberg-MacLan space of type $(\mathbb F_2, 1).$ The hit problem has been thoroughly studied for 35 years in a variety of contexts by many authors and completely solved for $m\leq 4.$ Furthermore, it has been closely related to some classical problems in the homotopy theory and applied in studying the $m$-th Singer algebraic transfer $Tr^{\mathscr A}_m$ \cite{W.S1}. This transfer is one of the useful tools for studying the Adams $E^{2}$-term, ${\rm Ext}_{\mathscr A}^{*, *}(\mathbb F_2, \mathbb F_2) = H^{*, *}(\mathscr A, \mathbb F_2).$The aim of this work is to continue our study of the hit problem of five variables. At the same time, this result will be applied to the investigation of the fifth transfer of Singer and the modular representation of the general linear group of rank 5 over $\mathbb F_2.$ More precisely, we grew out of a previous result of us in \cite{D.P3} on the hit problem for $\mathscr A$-module $\mathcal P_5$ in the generic degree $5(2^t-1) + 18.2^t$ with $t$ an arbitrary non-negative integer. The result confirms Sum's conjecture \cite{N.S2} on the relation between the minimal set of $\mathscr A$-generators for the polynomial algebras $\mathcal P_{m-1}$ and $\mathcal P_{m}$ in the case $m=5$ and the above generic degree. Moreover, by using our result \cite{D.P3} and a presentation in the $\lambda$-algebra of $Tr_5^{\mathscr A}$, we show that the non-trivial element $h_1e_0 = h_0f_0\in {\rm Ext}_{\mathscr A}^{5, 5+(5(2^0-1) + 18.2^0)}(\mathbb F_2, \mathbb F_2)$ is in the image of the fifth transfer and that $Tr^{\mathscr A}_5$ is an isomorphism in the bidegree $(5, 5+(5(2^0-1) + 18.2^0)).$ In addition, the behavior of $Tr^{\mathscr A}_5$ in the bidegree $(5, 5+(5(2^t-1) + 18.2^t))$ when $t\geq 1$ was also discussed. This method is different from that of Singer in studying the image of the algebraic transfer.

scholarly journals The "hit" problem of five variables in the generic degree and its application

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Efficient Method ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Left Module ◽  
Prime Field ◽  
Hit Problem

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 $1$. We are interested in the {\it Peterson "hit" problem} of finding a minimal set of generators for $P_s$ as a graded left module over the mod-2 Steenrod algebra, $\mathscr {A}$. For $s\geqslant 5,$ it is still open.In this paper, we study the hit problem of five variables in a generic degree. By using this result, we survey Singer's conjecture for the fifth algebraic transfer in the respective degrees. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer

scholarly journals On generators of the unstable $\mathscr A$-module $H^{*}((K(\mathbb F_2, 1))^{\times 5})$ in a generic degree and applications

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Algebraic Topology ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Modular Representations ◽  
Minimal Set ◽  
Polynomial Algebras ◽  
Prime Field ◽  
Important Open Problem ◽  
Hit Problem ◽  
The Relationship

Let us consider the prime field of two elements, $\mathbb F_2.$ It is well-known that the classical "hit problem" for a module over the mod 2 Steenrod algebra $\mathscr A$ is an interesting and important open problem of Algebraic topology, which asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, regarded as a connected unstable $\mathscr A$-module on $m$ variables $x_1, \ldots, x_m,$ each of degree 1. The algebra $\mathcal P_m$ is the $\mathbb F_2$-cohomology of the product of $m$ copies of the Eilenberg-MacLan complex $K(\mathbb F_2, 1).$ Although the hit problem has been thoroughly studied for more than 3 decades, solving it remains a mystery for $m\geq 5.$ The aim of this work is of studying the hit problem of five variables. More precisely, we develop our previous work \cite{D.P3} on the hit problem for $\mathscr A$-module $\mathcal P_5$ in a degree of the generic form $n_t:=5(2^t-1) + 18.2^t,$ for any non-negative integer $t.$ An efficient approach to solve this problem had been presented. Moreover, we provide an algorithm in MAGMA for verifying the results and studying the hit problem in general. As an consequence, the calculations confirmed Sum's conjecture \cite{N.S2} for the relationship between the minimal sets of $\mathscr A$-generators of the polynomial algebras $\mathcal P_{m-1}$ and $\mathcal P_{m}$ in the case $m=5$ and degree $n_t.$ Two applications of this study are to determine the dimension of $\mathcal P_6$ in the generic degree $5(2^{t+4}-1) + n_1.2^{t+4}$ for all $t > 0$ and describe the modular representations of the general linear group of rank 5 over $\mathbb F_2.$ As a corollary, the cohomological "transfer", defined by W. Singer \cite{W.S1}, is an isomorphism at the bidegree $(5, 5+n_0).$ Singer's transfer is one of the relatively efficient tools to approach the structure of mod-2 cohomology of the Steenrod algebra.

scholarly journals The "hit" problem of five variables in the generic degree and its application

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Efficient Method ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Prime Field ◽  
Graded Module ◽  
Transfer Homomorphism ◽  
Hit Problem

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, $\mathscr {A}$. We are interested in the "hit" problem of finding a minimal set of generators for $\mathscr A$-module $P_s.$ This problem is unresolved for every $s\geqslant 5.$ In this paper, we study the hit problem of five variables in a generic degree, from which we investigate Singer's conjecture [Math. Z. 202 (1989), 493-523] for the transfer homomorphism of rank $5$ in degrees given. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer.

scholarly journals On Peterson's open problem and representations of the general linear groups

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
General Linear ◽  
Algebra Homomorphism ◽  
Linear Groups ◽  
Minimal Set ◽  
Prime Field ◽  
General Linear Groups ◽  
Ext Groups ◽  
Hit Problem

Fix $\mathbb Z/2$ is the prime field of two elements and write $\mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, \mathbb Z/2)$ the general linear group of rank $d$ over $\mathbb Z/2$ and by $\mathscr P_d$ the polynomial algebra $\mathbb Z/2[x_1, x_2, \ldots, x_d],$ which is viewed as a connected unstable $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson "hit problem" of finding the minimal set of $\mathcal A_2$-generators for $\mathscr P_d.$ It is equivalent to determining a $\mathbb Z/2$-basis for the space of "cohits"$$Q\mathscr P_d := \mathbb Z/2\otimes_{\mathcal A_2} \mathscr P_d \cong \mathscr P_d/\mathcal A_2^+\mathscr P_d.$$ This $Q\mathscr P_d$ is considered as a form modular representation of $GL_d$ over $\mathbb Z/2.$ The problem for $d= 5$ is not yet completely solved, and unknown in general. In this work, we give an explicit solution to the hit problem of five variables in the generic degree $n = r(2^t -1) + 2^ts$ with $r = d = 5,\ s =8$ and $t$ an arbitrary non-negative integer. An application of this study to the cases $t = 0$ and $t = 1$ shows that the Singer algebraic transfer is an isomorphism in the bidegrees $(5, 5+(13.2^{0} - 5))$ and $(5, 5+(13.2^{1} - 5)).$ Moreover, the result when $t\geq 2$ was also discussed. Here, the Singer transfer of rank $d$ is a $\mathbb Z/2$-algebra homomorphism from $GL_d$-coinvariants of certain subspaces of $Q\mathscr P_d$ to the cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A_2}^{d, d+*}(\mathbb Z/2, \mathbb Z/2).$ It is one of the useful tools for studying mysterious Ext groups and the Kervaire invariant one problem.

scholarly journals Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application to Singer's cohomological transfer

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Open Problem ◽  
General Linear Group ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Ext Groups ◽  
Hit Problem ◽  
Prime 2

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 problem is the content of "hit problem" of Frank Peterson. We study the $q$-th Singer algebraic transfer $Tr_q^{A}$, which is a homomorphism from the space of $GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the Adams $E_2$-term, ${\rm Ext}_{A}^{q, q+n}(k, k).$ 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 Singer transfer is one of the useful tools for describing mysterious Ext groups. In the present study, by using techniques of the hit problem of four variables, we explicitly determine the structure of the spaces $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and the representation of the fourth transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. These new results confirmed Singer's conjecture for the monomorphism of the rank $4$ transfer. Our approach is different from that of Singer.

The structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application, I

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Open Problem ◽  
General Linear Group ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Ext Groups ◽  
Hit Problem ◽  
Prime 2

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, can one write down a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This problem is the content of "hit problem" of Frank Peterson. We study the $q$-th Singer algebraic transfer $Tr_q^{A}$, which is a homomorphism from the space of $GL_q(k)$-coinvariants $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the Adams $E_2$-terms, ${\rm Ext}_{A}^{q, q+n}(k, k).$ 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 Singer transfer is one of the useful tools for describing mysterious Ext groups. In the present study, by using techniques of the hit problem of four variables, we explicitly determine the structure of the spaces $k\otimes _{GL_4(k)} P((P_4)_{n_j}^{*})$ in some generic degrees. Applying these results and the representation of the transfer $Tr_4^{A}$ over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. Our approach is different from that of Singer.

scholarly journals On Peterson's open problem and representations of the general linear groups

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
General Linear ◽  
Algebra Homomorphism ◽  
Linear Groups ◽  
Minimal Set ◽  
Prime Field ◽  
General Linear Groups ◽  
Ext Groups ◽  
Hit Problem

Fix $\mathbb Z/2$ is the prime field of two elements and write $\mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, \mathbb Z/2)$ the general linear group of rank $d$ over $\mathbb Z/2$ and by $\mathscr P_d$ the polynomial algebra $\mathbb Z/2[x_1, x_2, \ldots, x_d]$ as a connected unstable left $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson "hit problem" of finding the minimal set of $\mathcal A_2$-generators for $\mathscr P_d.$ Equivalently, we need to determine a basis for the $\mathbb Z/2$-vector space $$Q\mathscr P_d := \mathbb Z/2\otimes_{\mathcal A_2} \mathscr P_d \cong \mathscr P_d/\mathcal A_2^+\mathscr P_d$$ in each degree $n\geq 1.$ Note that this space is a representation of $GL_d$ over $\mathbb Z/2.$ The problem for $d= 5$ is not yet completely solved, and unknow in general.In this work, we give an explicit solution to the hit problem of five variables in the generic degree $n = r(2^t -1) + 2^ts$ with $r = d = 5,\ s =8$ and $t$ an arbitrary non-negative integer. An application of this study to the cases $t = 0$ and $t = 1$ shows that the Singer algebraic transfer of rank $5$ is an isomorphism in the bidegrees $(5, 5+(13.2^{0}-5))$ and $(5, 5+(13.2^{1}-5)).$ Moreover, the result when $t\geq 2$ was also discussed. Here, the Singer transfer of rank $d$ is a $\mathbb Z/2$-algebra homomorphism from $GL_d$-coinvariants of certain subspaces of $Q\mathscr P_d$ to the cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A_2}^{d, d+*}(\mathbb Z/2, \mathbb Z/2).$ It is one of the useful tools for studying these mysterious Ext groups.

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