$\mathcal{A}(p)$ generators for $H^*V$ and Singer's homological transfer

Crossley, Martin D.

doi:10.1007/pl00004698

Cited by 14 publications

(17 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…al. [9], Chơn and Hà [11], Crossley [14], Hà [15], Hưng [19], Minami [26], Nam [31], the present author [41,42,43,44,46,47,48,49,50,51], Sum [61,62,63,65] and others). In [55], using the invariant theory, Singer claims that T r d is an isomorphism for d = 4 in a range of internal degrees, but T r 5 is not an epimorphism.…”

Section: A2mentioning

confidence: 64%

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

Phúc¹

2021

Preprint

103

View full text Add to dashboard Cite

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.

show abstract

Section: A2mentioning

confidence: 64%

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

Phúc¹

2021

Preprint

103

View full text Add to dashboard Cite

show abstract

“…Questions about excess and conjugation in A have been investigated notably by Don Davis [55] and others [20,21,70,103,116,191]. Recent advances in these areas can be traced through the work of Crabb, Crossley and Hubbuck [49,50,51,52,53], Ken Monks [139,138,142,143,140], and Bill Singer and Judith Silverman [174,175,178,182].…”

Section: Remarksmentioning

confidence: 99%

“…The proof uses the χ-trick and Theorem 4.6. In [42], an upper bound is proposed for the dimension of C d (n) for the prime 2 case in terms of a certain function of n. This has been generalised to the odd prime case by Crossley [51]. It remains a problem, however, to find the least upper bound.…”

Section: 3mentioning

confidence: 99%

Problems in the Steenrod Algebra

Wood

1998

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

show abstract

“…where P ((P q ) * n ) := {θ ∈ (P q ) * n : (θ)Sq i = 0, for all i > 0} = (Q ⊗q n ) * , the space of primitive homology classes as a representation of GL q (k) for all n and the coinvariant k⊗ GLq(k) P ((P q ) * n ) is isomorphic as an k-vector space to (Q ⊗q n ) GLq(k) , the subspace of GL q (k)-invariants of Q ⊗q . The Singer transfer has been studying for a long time: see Boardman [4], Chơn and Hà [9], Crossley [10], Hà [15], Hưng [18], Hưng-Quỳnh [19], Minami [26], the present writer [31,33,34,35,36,37], Sum [44,46], and others. By the works of Singer himself [39] and Boardman [4], T r A q is known to be an isomorphism for q ≤ 3.…”

Section: Introductionmentioning

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

$\mathcal{A}(p)$ generators for $H^*V$ and Singer's homological transfer

Cited by 14 publications

References 9 publications

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

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

Problems in the Steenrod Algebra

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