Beyond the hit problem: Minimal presentations of odd-primary Steenrod modules, with application to CP($\infty) and BU

Abstract

We describe a minimal unstable module presentation over the Steenrod algebra for the odd-primary cohomology of infinite-dimensional complex projective space and apply it to obtain a minimal algebra presentation for the cohomology of the classifying space of the infinite unitary group. We also show that there is a unique Steenrod module structure on any unstable cyclic module that has dimension one in each complex degree (half the topological degree) with a fixed alpha-number (sum of 'digits') and is zero in other degrees.