On the Peterson hit problem

Abstract

Abstract. We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra P k := F 2 [x 1 , x 2 , . . . , x k ] as a module over the mod-2 Steenrod algebra, A. In this paper, we study a minimal set of generators for A-module P k in some so-called generic degrees and apply these results to explicitly determine the hit problem for k = 4. Dedicated to Prof. N. H. V. Hưng on the occasion of his sixtieth birthday Introduction and statement of resultsLet V k be an elementary abelian 2-group of rank k. Denote by BV k the classifying space of V k . It may be thought of as the product of k copies of the real projective space RP ∞ . ThenHere the cohomology is taken with coefficients in the prime field F 2 of two elements.Being the cohomology of a space, P k is a module over the mod-2 Steenrod algebra A. The action of A on P k is explicitly given by the formulaand subject to the Cartan formulafor f, g ∈ P k (see Steenrod and Epstein [30]). A polynomial f in P k is called hit if it can be written as a finite sum f = i>0 Sq i (f i ) for some polynomials f i . That means f belongs to A + P k , where A + denotes the augmentation ideal in A. We are interested in the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra P k as a module over the Steenrod algebra. In other words, we want to find a basis of the F 2 -vector spaceThe hit problem was first studied by Peterson [22,23]

On the determination of the Singer transfer

“…This conjecture is true for k 3. It can be verified for k = 4 by using the results in [14,15]. The conjecture for k 5 is an open problem.…”

“…We also compute this space, however we use the admissible monomial basis for QP 3 that is different from the one of Boardman in [2]. Our approach can be apply for k = 4 by using the admissible monomial basis for QP 4 which is given in [14,15].…”

“…Proposition 4.1.1 (see [15,26]) Let n = 2 s+1 − 3 with s a positive integer. Then, the dimension of the F 2 -vector space (QP 4 ) n is determined by the following table:…”

A note on the Peterson hit problem for the Steenrod algebra

“…This problem has first been studied by Peterson [10], Wood [28], Singer [16], Priddy [14], who pointed out its relationship with some classical problems in homotopy theory such as the cobordism theory of manifolds, the modular representation theory of linear groups, Adams spectral sequences of stable homotopy of spheres, and stable homotopy type of the classifying space of finite groups. Then, this problem was investigated by Wood [28], Carlisle and Wood [1], Silverman [17], Nam [8,9], Mothebe [7], Sum [19,21], Cho'n and Hà [3], Kameko [5,6] and others. Recently, the hit problem and its applications to representations of general linear groups have been presented in the books of Walker and Wood [26,27].…”

“…For r ¼ k À 1 and m > 0, the problem was studied by Crabb and Hubbuck [2], Nam [8], Repka and Selick [15], Walker and Wood [25], Sum [21]. Now, the F 2 -vector space F 2 A P k was explicitly calculated by Peterson [10] for k ¼ 1; 2, by Kameko [4] for k ¼ 3 and by Sum [20,21] for k ¼ 4. However, for k > 4, it is still unsolved, even in the case of k ¼ 5 with the help of computers.…”