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

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) /AA p−2 , as described in an independent appendix.

Beyond the hit problem: Minimal presentations of odd-primary Steenrod modules, with application to $CP(\infty)$ and BU

“…Finally, in roughly the same way that our minimal A-presentation of H * (RP (∞); F 2 ) led to a minimal unstable A-algebra presentation of the symmetric algebra H * (BO; F 2 ) [9], our minimal unstable A-module presentation for H * (CP (∞); F p ) will lead to a minimal unstable A-algebra presentation of the symmetric algebra H * (BU ; F p ). Many of our methods will be the same ones we used in [7,9,10] to determine minimal relations for unstable A-modules and A-algebras.…”

“…We accomplished this for H * (RP (∞); F 2 ) in [9], and will now do so for H * (CP (∞); F p ) (with p odd), where the answer has some fascinating extra twists but is still tractable. In so doing, we will analyze the cyclic subquotients of the Afiltration of H * (CP (∞); F p ) given by alpha-number of complex degree, and determine their minimal A-relations.…”

“…However, there are differences that are quite interesting. The filtered quotients in the mod two case are fairly well-known A-modules; i.e., they are isomorphic to the free unstable modules on single generators t 2 n−1 −1 in degree 2 n−1 − 1 subject to the A-relations Sq 2 k t 2 n−1 −1 = 0 for 0 k n − 3 [9]. In the odd-primary case, since the Steenrod algebra is concentrated in complex degrees divisible by p − 1, any A-module splits as a direct sum of p − 1 A-modules, each of these in degrees with fixed residue mod (p − 1).…”

“…Many of our methods will be the same ones we used in [7,9,10] to determine minimal relations for unstable A-modules and A-algebras. A fundamental element of our computations is the odd-primary even topological Kudo-Araki-May algebra, K, whose definition and properties we developed in [11], and which is well-suited to studying unstable A-modules.…”

Unstable module presentations for the cohomology of real projective spaces

“…First we describe minimal A-presentations for H * RP (∞) and its natural A-filtration pieces F s H * RP (∞). Our A-presentation for H * RP (∞) is in [6,Theorem 6.5], but the proofs there are circuitous and rely on methods that are ill-suited to our primary goal here of minimally presenting the cohomologies H * RP (m) of the finite projective spaces from the presentations of F s H * RP (∞)). We thus propose a new perspective, taking three distinctive and atypical points of view on unstable modules, and we provide proofs of all our results from this perspective.…”