Global structure of the mod two symmetric algebra , H ∗ ( BO ; F 2 ) , 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 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 − 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 − 1) /AAp−2 , as described in an independent appendix. AMS Classification 55R45; 13A50, 16W22, 16W50, 55R40, 55S05, 55S10