Algebraic & Geometric Topology

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

David J Pengelley and Frank Williams

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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(2m1) 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 2p1AA¯p2, as described in an independent appendix.

Article information

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

Received: 24 October 2003
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R45: Homology and homotopy of $B$O and $B$U; Bott periodicity
Secondary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 16W22: Actions of groups and semigroups; invariant theory 16W50: Graded rings and modules 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 55S05: Primary cohomology operations 55S10: Steenrod algebra

symmetric algebra Steenrod algebra unstable algebra classifying space Dickson algebra $BO$ real projective space.


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.

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