A structure theorem for $\mathit{RO}(C_2)$–graded Bredon cohomology

Abstract

Let $C_2$ be the cyclic group of order two. We present a structure theorem for the $\mathit{RO}\left(C_2\right)$–graded Bredon cohomology of $C_2$–spaces using coefficients in the constant Mackey functor $\underset{¯}{{\mathbb{𝔽}}_2}$. We show that, as a module over the cohomology of the point, the $\mathit{RO}\left(C_2\right)$–graded cohomology of a finite $C_2$–CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. The shifts are by elements of $\mathit{RO}\left(C_2\right)$ corresponding to actual (ie nonvirtual) $C_2$–representations. This decomposition lifts to a splitting of genuine $C_2$–spectra.