The hit problem for $H^{*}(\mathrm{BU}(2);\mathbb{F}_{p})$

Abstract

The hit problem for a module over the Steenrod algebra $\mathcal{A}$ seeks a minimal set of $\mathcal{A}$–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.

For the cohomologies of classifying spaces, several such results possess two intriguing features: sparseness by degree, and uniform rank bounds independent of degree. In particular, it is known that sparseness holds for $H^{\ast }\left(BO\left(n\right);{\mathbb{F}}_2\right)$ for all $n$, and that there is a rank bound for $n\le 3$. Our results in this paper show that both these features continue at all odd primes for $BU\left(n\right)$ for $n\le 2$.

We solve the odd primary hit problem for $H^{\ast }\left(BU\left(2\right);{\mathbb{F}}_p\right)$ by determining an explicit basis for the $\mathcal{A}$–primitives in the dual $H_{\ast }\left(BU\left(2\right);{\mathbb{F}}_p\right)$, where we find considerably more elaborate structure than in the $2$–primary case. We obtain our results by structuring the $\mathcal{A}$–primitives in homology using an action of the Kudo–Araki–May algebra.