Unstable Module Presentations for the Cohomology of Real Projective Spaces

There is much we still do not know about projective spaces. We describe here how the mod two cohomology of each real projective space is built as an unstable module over the Steenrod algebra A, or equivalently, over K, the algebra of inherently unstable mod two “lower operations” originally introduced by Steenrod. In particular, to produce the cohomology of projective space of each dimension we consider the well-known minimal set of unstable module generators and construct a minimal set of unstable relations. Three new perspectives we blend for this purpose are: • to focus solely on the two-power Steenrod squares that generate A to understand the A-action in a process we call “shoveling ones”, • to describe every element in a canonical way from a particular unstable generator by composing operations from the algebra K, • to shift attention when studying an unstable A-module to considering and analyzing it directly as an equivalent K-module.