The $RO(C_4)$ cohomology of the infinite real projective space

TL;DR

This paper computes the $C_4$-equivariant Bredon cohomology of an infinite real projective space, revealing its non-flatness and implications for Steenrod algebra computations, extending Hu-Kriz methods.

Contribution

It extends Hu-Kriz's $C_2$ methods to $C_4$, providing explicit cohomology calculations and showing the non-flatness of the module over homology of a point.

Findings

C_4$-equivariant cohomology computed explicitly.

Cohomology is not flat over the homology of a point.

Implications for Steenrod algebra generator methods.

Abstract

Following the Hu-Kriz method of computing the $C_2$ genuine dual Steenrod algebra $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$, we calculate the $C_4$ equivariant Bredon cohomology of the classifying space $\mathbf R P^{\infty \rho}=B_{C_4}\Sigma_{2}$ as an $RO(C_4)$ graded Green-functor. We prove that as a module over the homology of a point (which we also compute), this cohomology is not flat. As a result, it can't be used as a test module for obtaining generators in $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$ as Hu-Kriz do in the $C_2$ case.