Global structure of the mod two symmetric algebra, $H^*(BO;\mathbb{F}_{2})$, over the Steenrod algebra

Abstract

The algebra $\mathcal{S}$ of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra $\mathcal{A}$, and is isomorphic to the mod two cohomology of $BO$, the classifying space for vector bundles. We provide a minimal presentation for $\mathcal{S}$ in the category of unstable $\mathcal{A}$–algebras, ie, minimal generators and minimal relations.

From this we produce minimal presentations for various unstable $\mathcal{A}$–algebras associated with the cohomology of related spaces, such as the $BO\left(2^m-1\right)$ that classify finite dimensional vector bundles, and the connected covers of $BO$. The presentations then show that certain of these unstable $\mathcal{A}$–algebras coalesce to produce the Dickson algebras of general linear group invariants, and we speculate about possible related topological realizability.

Our methods also produce a related simple minimal $\mathcal{A}$–module presentation of the cohomology of infinite dimensional real projective space, with filtered quotients the unstable modules $\mathcal{ℱ}\left(2^p-1\right)∕\mathcal{A}{\overline{\mathcal{A}}}_{p-2}$, as described in an independent appendix.