#### Volume 3, issue 2 (2003)

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

### David J Pengelley and Frank Williams

Algebraic & Geometric Topology 3 (2003) 1119–1138
 arXiv: math.AT/0312220
##### Abstract

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

From this we produce minimal presentations for various unstable $\mathsc{A}$–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 $\mathsc{A}$–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 $\mathsc{A}$–module presentation of the cohomology of infinite dimensional real projective space, with filtered quotients the unstable modules $\mathsc{ℱ}\left({2}^{p}-1\right)∕\mathsc{A}{\overline{\mathsc{A}}}_{p-2}$, as described in an independent appendix.

##### Keywords
symmetric algebra, Steenrod algebra, unstable algebra, classifying space, Dickson algebra, $BO$, real projective space.
##### Mathematical Subject Classification 2000
Primary: 55R45
Secondary: 13A50, 16W22, 16W50, 55R40, 55S05, 55S10