#### Volume 20, issue 4 (2020)

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

### Clover May

Algebraic & Geometric Topology 20 (2020) 1691–1728
##### 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.

##### Keywords
equivariant cohomology, equivariant homotopy, Toda bracket, Mackey functor, $\mathit{RO}(G)$–graded
Primary: 55N91