Sparseness for the symmetric hit problem at all primes

Abstract

AbstractThe hit problem for a module over the Steenrod algebra $\mathcal{A}$ seeks a minimal set of $\mathcal{A}$-generators (“non-hit elements,” those not in the image of positive degree elements of $\mathcal{A}$). This problem has been studied for 25 years in a variety of contexts and, although general or complete results have been difficult to obtain, partial results have been obtained in many cases.For any prime p ⩾ 2, consider the algebra of symmetric polynomials in l variables over $\mathbb{F}$p (the cohomology of BU(l) [BO(l) for p = 2]). We prove a general sparseness result: For any l, the $\mathcal{A}$-generators can be concentrated in complex [real for p = 2] degrees τ such that α((p - 1) (τ + l)) ⩽(p - 1)l, where α(m) denotes the sum of the digits in the p-ary expansion of m.The specialisation of this result to p = 2 was proved by Janfada and Wood using different methods. Key ingredients of our approach are an action of the Kudo–Araki–May algebra on the $\mathcal{A}$-primitives in homology, and a symmetric homology adaptation of the concept behind the χ-trick in cohomology.