A structure theorem for $RO(C_2)$-graded Bredon cohomology

TL;DR

This paper establishes a decomposition theorem for $RO(C_2)$-graded Bredon cohomology of $C_2$-spaces, showing it splits into basic components related to the cohomology of a point and spheres with antipodal action.

Contribution

It provides a new structure theorem that describes the $RO(C_2)$-graded Bredon cohomology as a direct sum of fundamental modules, extending understanding of equivariant cohomology for $C_2$-spaces.

Findings

Cohomology decomposes into shifted copies of point and sphere cohomologies.

Decomposition applies to finite $C_2$-CW complexes.

Splitting lifts to genuine $C_2$-spectra.

Abstract

Let $C_2$ be the cyclic group of order two. We present a structure theorem for the $RO(C_2)$-graded Bredon cohomology of $C_2$-spaces using coefficients in the constant Mackey functor $\underline{\mathbb{F}_2}.$ We show that, as a module over the cohomology of the point, the $RO(C_2)$-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 $RO(C_2)$ corresponding to actual (i.e. non-virtual) $C_2$-representations. This decomposition lifts to a splitting of genuine $C_2$-spectra.