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

Abstract

Following the Hu-Kriz method of computing the $C_2$ genuine dual Steenrod algebra $(H{\mathbf{F}}_2{)}_{★}(H{\mathbf{F}}_2)$, 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 $(H{\mathbf{F}}_2{)}_{★}(H{\mathbf{F}}_2)$ as Hu-Kriz do in the $C_2$ case.

Comments are most welcome! 41 pages, 5 figures. Code available on https://github.com/NickG-Math/Mackey

References in corpus (4)

• Equivariant local coefficients and the RO(G)-graded cohomology of classifying spaces
• $C_2$-equivariant Homology Operations: Results and Formulas
• The $RO(C_4)$ integral homology of a point
• On the $C_p$-equivariant dual Steenrod algebra