Global Structure of the
mod 2 Symmetric Algebra over the Steenrod algebra.
David J. Pengelley (davidp@nmsu.edu)
Frank Williams (frank@nmsu.edu)
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., a minimal set of generators and a minimal
set of relations.
From this we produce minimal presentations for various unstable
A-algebras associated with the cohomology of related spaces, such as
the BO(2^n - 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 mod 2 Dickson
algebras, and we speculate about possible related topological
realizability.
Our methods also produce a related simple A-module presentation of the
cohomology of infinite-dimensional real projective space, with a
filtration having well-known filtered quotients.