Unstable module presentations for the cohomology of real projective
spaces
David J. Pengelley
New Mexico State University Las Cruces, NM 88003
davidp@nmsu.edu
Frank Williams
New Mexico State University
Las Cruces, NM 88003
frank@nmsu.edu
Primary 55R40; Secondary 55R45, 55S05, 55S10
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:
1. to focus solely on the two-power Steenrod squares that generate A to understand the A-action in a process we call ?shoveling ones?;
2. to describe every element in a canonical way from a particular
unstable generator by composing operations from the algebra K;
3. to shift attention when studying an unstable A-module to consid-
ering and analyzing it directly as an equivalent K-module.