The hit problem forH^∗(BU(2); 𝔽_p)

Abstract

The hit problem for a module over the Steenrod algebra A seeks a minimal set of A-generators ("non-hit elements"). This problem has been studied for 25 years in a variety of contexts, and although complete results have been notoriously difficult to come by, partial results have been obtained in many cases.

Sparseness for the symmetric hit problem at all primes

“…Remark 2•2. This is the first sparseness result for A-generators of symmetric algebras that applies for all p and l. Its specialisation to p = 2 is best possible [7], but for p odd, there are degrees (at least for l = 1, 2, where a complete answer for the A-generators is known [12]) in which all elements are hit but the inequality is not satisfied. Thus the degrees of the A-generators are generally sparser yet than we prove here.…”

“…Applications include the Singer transfer to the cohomology of the Steenrod algebra, and the A-module structure of the cohomology of Thom spaces. Partial results have been obtained in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], among others, and we refer to [11,15] for further background.…”

“…We provide an analogous result for symmetric algebras for all primes (for odd primes our cohomology is concentrated in even degrees, so we use 'complex degree', half the topological degree). A Peterson-like sparseness conjecture analogous to p = 2 would be that the A-generators of H M (BU (l); F p ) can be concentrated in complex degrees τ such that α(τ + l) l. But this is obviously false for odd primes, even for l = 1 [12]. One might try using α(τ + l) l, where α is the number of nonzero digits, but this turns out to be false for l = 2 [12].…”

“…A Peterson-like sparseness conjecture analogous to p = 2 would be that the A-generators of H M (BU (l); F p ) can be concentrated in complex degrees τ such that α(τ + l) l. But this is obviously false for odd primes, even for l = 1 [12]. One might try using α(τ + l) l, where α is the number of nonzero digits, but this turns out to be false for l = 2 [12]. Instead, the inequality of our main theorem, while using α rather thanα, also involves p.…”

On modules over the mod 2 Steenrod algebra and hit problems

“…The classical "hit problem", which is concerned with seeking a minimal set of A -generators for P m , has been initiated in a variety of contexts by Peterson [24], Priddy [35], Singer [38], and Wood [48]. This problem is currently one of the central subjects in Algebraic topology and has a great deal of intensively studied by many authors like Ault-Singer [3], Ault [4], Crabb-Hubbuck [7], Inoue [9,10], Janfada-Wood [11], Kameko [12], Mothebe-Uys [20], Mothebe [21], Pengelley-William [23], the present author and Sum [25-31, 34, 42, 43], Walker-Wood [45,46], etc. As it is known, when F 2 is an A -module concentrated in degree 0, solving the hit problem is to determine an F 2 -basis for the space of indecomposables, or "unhit" elements, Q ⊗m := F 2 ⊗ A P m = P m /A P m where A is the positive degree part of A .…”