#### Algebraic and Geometric Topology 3 (2003),
paper no. 40, pages 1119-1138.

## Global structure of the mod two symmetric algebra, H^*(BO;F_2), over the Steenrod Algebra

### David J. Pengelley, Frank Williams

**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, i.e., 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(2^m-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 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.

**AMS subject classification**.
Primary: 55R45.
Secondary: 13A50, 16W22, 16W50, 55R40, 55S05, 55S10.

**DOI:** 10.2140/agt.2003.3.1119

**E-print:** `arXiv:math.AT/0312220`

Submitted: 24 October 2003.
Accepted: 5 November2003.
Published: 10 November 2003.

Notes on file formats
David J. Pengelley, Frank Williams

New Mexico State University

Las Cruces, NM 88003, USA

Email: davidp@nmsu.edu, frank@nmsu.edu

AGT home page

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