#### Volume 13, issue 4 (2013)

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

### David Pengelley and Frank Williams

Algebraic & Geometric Topology 13 (2013) 2061–2085
##### Abstract

The hit problem for a module over the Steenrod algebra $\mathsc{A}$ seeks a minimal set of $\mathsc{A}$–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 $\mathsc{A}$–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 $\mathsc{A}$–primitives in homology using an action of the Kudo–Araki–May algebra.

##### Keywords
Steenrod algebra, hit problem, primitive elements, Kudo–Araki–May algebra, symmetric invariants
##### Mathematical Subject Classification 2010
Primary: 16W22, 55R40, 55R45, 55S10
Secondary: 16W50, 55S05, 57T10, 57T25
##### Publication
Received: 7 October 2012
Revised: 23 January 2013
Accepted: 18 February 2013
Published: 29 May 2013
##### Authors
 David Pengelley Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003-8001 USA http://www.math.nmsu.edu/~davidp/ Frank Williams Department of Mathematics New Mexico State University Las Cruces, NM 88003-8001 USA