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Dang Vo Phuc

Publications and source records attributed to Dang Vo Phuc.

At least 37 records · Page 2Linked to original sources

arXiv2025

On modular invariants of the truncated polynomial ring in rank four

We prove the rank-4 case of the conjecture of Ha-Hai-Nghia for the invariant subspace of the truncated polynomial ring $\mathcal{Q}_m(n)=\mathbb{F}_q[x_1,\dots,x_n]/(x_1^{q^m},\dots,x_n^{q^m}),$ under a new, explicit technical hypothesis. Our argument extends the determinant calculus for the delta operator by deriving crucial rank-4 identities governing its interaction with the Dickson algebra. We show that the proof of the conjecture reduces to a specific vanishing property, for which we introduce a sufficient condition, the "matching hypothesis" ($\mathrm{H_{match}}$), relating the degree structures of Dickson invariants. This condition is justified by theoretical arguments and verified computationally in many cases. Combining this approach with the normalized derivation approach from our prior work, we establish the conjecture. As a result, the Lewis-Reiner-Stanton Conjecture is also confirmed for rank four under the given hypothesis.

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arXiv2025

On the hit problem for the polynomial algebra and the algebraic transfer

This paper investigates Singer's conjecture by examining the cohit module $\mathbb F_2\otimes_{\mathcal A}P^{\otimes h}$ for specific degrees and values of $h$. Utilizing hit problem techniques, we extend previous work by Mothebe et al. and establish key dimensional results. Notably, for $h\geq 6$, we prove that the cohit module's dimension in certain degrees matches the order of a specific factor group. Our contributions include demonstrating that certain non-zero elements do not belong to the image of the Singer algebraic transfer. All results were verified using the OSCAR computer algebra system.

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arXiv2025

On the computation of $\mathrm{Ext}_{\mathcal A}^{k,k+*}(\mathbb{Z}/2,\mathbb{Z}/2)$ for $k \leq 5$

This Note presents a computational algorithm for determining a basis of the cohomology of the mod 2 Steenrod algebra, $\mathrm{Ext}_{\mathcal A}^{k, k+*}(\mathbb{Z}/2, \mathbb{Z}/2)$ for $k \leq 5$, based on the well-known generators and the Adams relations given in Chen's work [2]. The purpose of this algorithm is to verify the hand-computed results for $\mathrm{Ext}_{\mathcal A}^{4, 4+*}(\mathbb{Z}/2, \mathbb{Z}/2)$ presented in our corrected papers [8, 9, 10]. Combining our most recent works [11, 12], the verification of the domain and the codomain of the fourth algebraic transfer in specific degrees can now be completed.

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arXiv2025

On Singer's conjecture for the fourth algebraic transfer in certain generic degrees

Let $A$ be the Steenrod algebra over the finite field $k := \mathbb Z_2$ and $G(q)$ be the general linear group of rank $q$ over $k.$ A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, ${\rm Ext}^{q, *}_A(k, k),$ for all homological degrees $q \geq 0.$ The Singer algebraic transfer of rank $q,$ formulated by William Singer in 1989, serves as a valuable method for the description of such Ext groups. This transfer maps from the coinvariants of a certain representation of $G(q)$ to ${\rm Ext}^{q, *}_A(k, k).$ Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all $q\geq 4.$ This paper establishes Singer's conjecture for rank four in the generic degrees $n = 2^{s+t+1} +2^{s+1} - 3$ whenever $t\neq 3$ and $s\geq 1,$ and $n = 2^{s+t} + 2^{s} - 2$ whenever $t\neq 2,\, 3,\, 4$ and $s\geq 1.$ In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four. All the obtained results can be verified directly by using the program suite of our novel algorithms presented in [17, 18, 19, 20]. We note that although Singer's conjecture still holds for the case of rank 4, it no longer holds for rank 6, as announced in our most recent work [20].

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arXiv2025

Computing Invariant Spaces via Global Cluster Analysis and Representation Theory

The Singer algebraic transfer is a fundamental homomorphism in algebraic topology, providing a bridge between the homology of classifying spaces and the cohomology of the Steenrod algebra $\mathcal{A}$, which forms the $E_2$-term of the Adams spectral sequence. The domain of its dual is isomorphic to the space of $GL_k(\mathbb{F}_2)$-invariants in the quotient of the polynomial algebra, $(\mathcal{QP_k})^{GL_k(\mathbb{F}_2)}$, where $\mathcal{P}_k$ is regarded as a module over $\mathcal{A}$. A direct computation of this invariant space and its dual (i.e., the domain of the Singer transfer) remains a challenging problem. In this paper, we construct a new algorithm to compute $(\mathcal{QP_k})^{GL_k(\mathbb{F}_2)}$, which differs from the method presented in our recent work [15]. We refer to this new approach as the Global Cluster Analysis algorithm. It builds a \emph{weight interaction graph} to identify clusters of interacting weight spaces that form closed $Σ_k$-submodules (where $Σ_k \subset GL_k(\mathbb{F}_2)$). By performing invariance analysis on these larger clusters, our algorithm enables a complete and accurate computation of the global $Σ_k$-invariants, which are then used to determine the final $GL_k(\mathbb F_2)$-invariants. We also introduce an algorithm to directly compute the domain of the Singer transfer for ranks $k \leq 3$ in certain generic degrees, based entirely on Boardman's modular representation theory framework [2].

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arXiv2025

Computational Approaches to the Singer Transfer: Preimages in the Lambda Algebra and $G_k$-Invariant Theory

We present a systematic, algorithmic method to compute the preimage of elements under the Singer algebraic transfer. Using the lambda algebra and the invariant-theoretic formula of P.H. Chon and L.M. Ha [5], we formulate the preimage search as a solvable problem in linear algebra. This framework is applied to study key indecomposable elements in the Adams spectral sequence. As a consequence, we show that the proof of the known result that the indecomposable element $d_0 \in \mathrm{Ext}^{4,18}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ lies in the image of the fourth Singer transfer, as given by Nguyen Sum in [17], is false. Furthermore, we provide the explicit description of a preimage for the indecomposable element $p_0 \in \mathrm{Ext}^{4,37}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ This preimage had not been explicitly determined in the previous work of N.H.V. Hung and V.T.N. Quynh [8]. Finally, our most significant contribution is the construction of a complete \textsc{SageMath} algorithm that fully automates the computation of both the dimension and an explicit basis for the $G_k$-invariant space $[(Q\mathcal{P}_k)_d]^{G_k}$. This tool facilitates the verification of our results [11, 12, 13] that were previously computed manually in connection with Singer's conjecture for rank 4.

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arXiv2025

A matrix criterion and algorithmic approach for the Peterson hit problem: Part I

The Peterson hit problem in algebraic topology is to explicitly determine the dimension of the quotient space $Q\mathcal P_k = \mathbb F_2\otimes_{\mathcal A}\mathcal P_k$ in positive degrees, where $\mathcal{P}_k$ denotes the polynomial algebra in $k$ variables over the field $\mathbb{F}_2$, considered as an unstable module over the Steenrod algebra $\mathcal{A}$. Current approaches to this problem still rely heavily on manual computations, which are highly prone to errors due to the intricate nature of the underlying calculations. To date, no efficient algorithm implemented in any computer algebra system has been made publicly available to tackle this problem in a systematic manner. Motivated by the above, in this work, which is considered as Part I of our project, we first establish a criterion based entirely on linear algebra for determining whether a given homogeneous polynomial is "hit". Accordingly, we describe the dimensions of the hit spaces. This leads to a practical and reliable computational method for determining the dimension of $Q\mathcal{P}_k$ for arbitrary $k$ and any positive degrees, with the support of a computer algebra system. We then give a concrete implementation of the obtained results as novel algorithms in \textsc{SageMath}. As an application, our algorithm demonstrates that the manually computed result presented in the recent work of Sum and Tai [15] for the dimension of $Q\mathcal{P}_5$ in degree $2^{6}$ is not correct. Furthermore, our algorithm determines that $\dim(Q\mathcal{P}_5)_{2^{7}} = 1985,$ which falls within the range $1984 \leq \dim(Q\mathcal{P}_5)_{2^{7}} \leq 1990$ as estimated in [15].

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arXiv2025

On the algebraic transfers of ranks 4 and 6 at generic degrees

Let $\mathscr A$ denote the classical singly-graded Steenrod algebra over the binary field $\mathbb Z/2.$ We write $P_k:=\mathbb Z/2[t_1, t_2, \ldots, t_k]$ as the polynomial algebra on $k$ generators, each having a degree of one. Let $GL_k$ be the general linear group of rank $k$ over $\mathbb Z/2.$ Then, $P_k$ is an $ \mathscr A[GL_k]$-module. The structure of the cohomology groups, ${\rm Ext}_{ \mathscr A}^{k, k+\bullet}(\mathbb Z/2, \mathbb Z/2)$, of the Steenrod algebra has, thus far, resisted clear understanding and full description for all homological degrees $k$. In the study of these groups, the algebraic transfer -- constructed by W. Singer in [Math. Z. 202, 493--523 (1989)] -- plays an important role. The Singer transfer is represented by the following homomorphism: $$Tr_k: {\rm Hom}([(\mathbb Z/2\otimes_{ \mathscr A} P_k)_{\bullet}]^{GL_k}, \mathbb Z/2)\longrightarrow {\rm Ext}_{ \mathscr A}^{k, k+\bullet}(\mathbb Z/2, \mathbb Z/2).$$ Among Singer's contributions is an interesting open conjecture asserting the monomorphism of $Tr_k$ for all $k.$ For this reason, our main aim in this article is to ascertain the validity of the Singer conjecture for ranks 4 and 6 in certain families of internal degrees. We place particular emphasis on the rank 4 case. More precisely, we present a detailed proof for certain generic degree cases when verifying the conjecture of rank four, which were succinctly noted in our previous work [Proc. Roy. Soc. Edinburgh Sect. A 153, 1529--1542 (2023)].

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arXiv2025

The affirmative answer to Singer's conjecture on the algebraic transfer of rank four

In recent decades, the structure of the mod-2 cohomology of the Steenrod ring $\mathscr A$ has become a major subject of study in the field of Algebraic Topology. One of the earliest attempts to study this cohomology through the use of modular representations of the general linear groups was the groundbreaking work [Math. Z. 202 (1989), 493-523] by W.M. Singer. In that work, Singer introduced a homomorphism, commonly referred to as the "algebraic transfer," which maps from the coinvariants of a certain representation of the general linear group to the mod-2 cohomology group of the ring $\mathscr A.$ Singer's conjecture, in particular, which states that the algebraic transfer is a monomorphism for all homological degrees, remains a highly significant and unresolved problem in Algebraic Topology. In this research, we take a major stride toward resolving the Singer conjecture by establishing its truth for the homological degree four.

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arXiv2024

On the mod-2 cohomology of the product of the infinite lens space and the space of invariants in a generic degree

Let $\mathbb S^{\infty}/\mathbb Z_2$ be the infinite lens space and $\mathscr A$ be the Steenrod algebra over the binary field $\mathbb F_2.$ The cohomology $H^{*}((\mathbb S^{\infty}/\mathbb Z_2)^{\oplus s}; \mathbb F_2)$ is known to be isomorphic to the graded polynomial ring $\mathcal {P}_s:= \mathbb F_2[x_1, \ldots, x_s]$ on $s$ generators of degree 1, viewed as an unstable $\mathscr A$-module. The Kameko squaring operation $(\widetilde {Sq^0_*})_{(s; N)}: (\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s)_{2N+s} \longrightarrow (\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s)_{N}$ is rather useful in studying an open problem of determining the dimension of the indecomposables $(\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s)_N.$ As a continuation of our recent works, this paper deals with the kernel of the Kameko $(\widetilde {Sq^0_*})_{(s; N_d)}$ for the case where $s = 5$ and the "generic" degree $N_d$ is of the form $N_d = 5(2^{d} - 1) + 11.2^{d+1}$ for arbitrary $d > 0.$ We then rectify almost all of the main results that were inaccurate in an earlier publication [Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116:81 (2022)] by Nguyen Khac Tin. We have also constructed several advanced algorithms in SAGEMATH to validate our results. These new algorithms make an important contribution to tackling the intricate task of explicitly determining both the dimension and the basis for the indecomposables $\mathbb F_2 \otimes_{\mathscr A} \mathcal {P}_s$ at positive degrees, a problem concerning algorithmic approaches that had not previously been addressed by any author. Also, the present study encompasses an investigation of the behavior of the cohomological transfer in bidegrees $(5, 5+N_d)$, with the internal degree $N_d$ mentioned above.

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arXiv2022

On modules over the mod 2 Steenrod algebra and hit problems

Let us consider the prime field of two elements, $\mathbb F_2\equiv \mathbb Z_2.$ 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 $m\geq 5.$ Our intent in this work is of studying the hit problem of five variables. More precisely, we develop our previous work [Commun. Korean Math. Soc. 35 (2020), 371-399] 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. 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 to describe the modular representations of the general linear group of rank 5 over $\mathbb F_2.$ As a corollary, the cohomological "transfer", defined by William Singer [Math. Z. 202 (1989), 493-523], is an isomorphism in 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.

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arXiv2022

A note on the modular representation on the $\mathbb Z/2$-homology groups of the fourth power of real projective space and its application

One knows that, the connected graded ring $P^{\otimes h}= \mathbb Z/2[t_1, \ldots, t_h]= \{P_n^{\otimes h}\}_{n\geq 0},$ which is graded by the degree of the homogeneous terms $P^{\otimes h}_n$ of degree $n$ in $h$ generators with the degree of each $t_i$ being one, admits a left action of $\mathcal A$ as well as a right action of the general linear group $GL_h.$ A central problem of homotopy theory is to determine the structure of the space of $GL_h$-coinvariants, $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}.$ Solving this problem is very difficult and still open for $h\geq 4.$ In this Note, our intent is of studying the dimension of $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ for the case $h = 4$ and the "generic" degrees $n$ of the form $n_{k, r, s} = k(2^{s} - 1) + r.2^{s},$ where $k,\, r,\, s$ are positive integers. Applying the results, we investigate the behaviour of the Singer cohomological "transfer" of rank $4.$ Singer's transfer is a homomorphism from a certain subquotient of the divided power algebra $Γ(a_1^{(1)}, \ldots, a_h^{(1)})$ to mod-2 cohomology groups ${\rm Ext}_{\mathcal A}^{h, h+*}(\mathbb Z/2, \mathbb Z/2)$ of the algebra $\mathcal A.$ This homomorphism is useful for depicting the Ext groups. Additionally, in higher ranks, by using the results on $\mathcal A$-generators for $P^{\otimes 5}$ and $P^{\otimes 6},$ we show in Appendix that the transfer of rank 5 is an isomorphism in som certain degrees of the form $n_{k, r, s}$, and that the transfer of rank 6 does not detect the non-zero elements $h_2^{2}g_1 = h_4Ph_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_{6, 10, 1}}(\mathbb Z/2, \mathbb Z/2)$, and $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_{6, 10, 2}}(\mathbb Z/2, \mathbb Z/2).$ Besides, we also probe the behavior of the Singer transfer of ranks 7 and 8 in internal degrees $\leq 15.$

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arXiv2022

On the dimension of $H^{*}((\mathbb Z_2)^{\times t}, \mathbb Z_2)$ as a module over Steenrod ring

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $(\mathbb P^{\otimes t})^{G_t}$ as a module over Steenrod ring, $\mathscr A_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t, \mathbb Z_2).$ An equivalent problem is to find a monomial basis of the space of "unhit" elements, $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each $t$ and degree $n\geq 0.$ The structure of this tensor product is proved surprisingly difficult and has been not yet known for $t\geq 5,$ even for the trivial subgroup $G_t = \{e\}.$ In the present paper, we consider the subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators of $(\mathbb P^{\otimes t})^{G_t}$ in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [23, 25].

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arXiv2022

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

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.

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arXiv2021

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

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 the classical "hit problem" in literature [30]. Based on this problem, we are interested in the $q$-th cohomological transfer $Tr_q^{A}$ of Singer [39], 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 $A.$ Singer conjectured that $Tr_q^{A}$ is a monomorphism, but this remains unanswered for all $q\geq 4.$ The present paper is to devoted to the investigation of this conjecture for the rank 4 case. 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 $Tr_4^{A}$ over the lambda algebra, we notice that Singer's conjecture is true for the rank 4 transfer in those degrees $n$. 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 holds for $Tr_4^{A}.$ This study and our previous results have been provided a panorama of the behavior of the fourth cohomological transfer.

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arXiv2021

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

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 $\mathcal A_2$-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 also a 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 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 mysterious Ext groups and the Kervaire invariant one problem.

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arXiv2021

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

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 "cohits" 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 $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ 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 $n_s=5(2^{s}-1) + 42.2^{s}$ for every non-negative integer $s.$ As a result, it confirms Sum's conjecture [14] 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 $n_s$. As applications, 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 [11] is an isomorphism in bidegree $(5, 5+n_s)$.

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arXiv2021

On the lambda algebra and Singer's cohomological transfer

Writing $\mathbb A$ for the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of $\mathbb A$ is an important problem of Algebraic topology, because it is the initial page of the Adams spectral sequence converging to stable homotopy groups of the spheres. A relatively efficient tool to describe this cohomology is the Singer algebraic transfer of rank $n$ in [Math. Z. 202 (1989), 493-523], which passes from a certain subquotient of a divided power algebra to the cohomology of $\mathbb A.$ Singer predicted that this transfer is a monomorphism, but this remains open for $n\geq 4.$ This short note is to verify the conjecture in the ranks 4 and 5 and some generic degrees.

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