Global structure of the mod two symmetric algebra, H∗(BO; 𝔽2), over the Steenrod algebra

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 $B O$ , 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 $B O (2^{m} - 1)$ that classify finite dimensional vector bundles, and the connected covers of $B O$ . 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 $ℱ (2^{p} - 1) ∕ A {\bar{A}}_{p - 2}$ , 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

References

Forward citations

Publication

Received: 24 October 2003

Published: 10 November 2003

Authors

David J Pengelley
New Mexico State University Las Cruces, NM 88003 USA

Frank Williams
New Mexico State University Las Cruces, NM 88003 USA

Volume 3, issue 2 (2003)

David J Pengelley and Frank Williams