## Algebraic & Geometric Topology

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

#### Abstract

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

From this we produce minimal presentations for various unstable $A$–algebras associated with the cohomology of related spaces, such as the $BO(2m−1)$ that classify finite dimensional vector bundles, and the connected covers of $BO$. The presentations then show that certain of these unstable $A$–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 $A$–module presentation of the cohomology of infinite dimensional real projective space, with filtered quotients the unstable modules $ℱ2p−1∕AA¯p−2$, as described in an independent appendix.

#### Article information

Source
Algebr. Geom. Topol., Volume 3, Number 2 (2003), 1119-1138.

Dates
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882431

Digital Object Identifier
doi:10.2140/agt.2003.3.1119

Mathematical Reviews number (MathSciNet)
MR2012968

Zentralblatt MATH identifier
1057.55004

#### Citation

Pengelley, David J; Williams, Frank. Global structure of the mod two symmetric algebra, $H^*(BO;\mathbb{F}_{2})$, over the Steenrod algebra. Algebr. Geom. Topol. 3 (2003), no. 2, 1119--1138. doi:10.2140/agt.2003.3.1119. https://projecteuclid.org/euclid.agt/1513882431

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