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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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


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 2p 1AA¯p2, as described in an independent appendix.

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
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Received: 24 October 2003
Published: 10 November 2003
David J Pengelley
New Mexico State University
Las Cruces, NM 88003
Frank Williams
New Mexico State University
Las Cruces, NM 88003