The hit problem for a module over the Steenrod algebra
seeks a minimal
set of –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
for all
, and that there is
a rank bound for .
Our results in this paper show that both these features continue at all odd primes for
for
.
We solve the odd primary hit problem for
by determining an explicit
basis for the –primitives
in the dual ,
where we find considerably more elaborate structure than in the
–primary
case. We obtain our results by structuring the
–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