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

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.

arXiv comment29 pages, 13 figures. v2: strengthened result to spectrum level splitting and incorporated suggestions from referee

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• Equivariant fundamental classes in $RO(C_2)$-graded cohomology in $\underline{\mathbb{Z}/2}$-coefficients
• The $RO(C_4)$ cohomology of the infinite real projective space
• Equivariant Cohomology of Projective Spaces
• Equivariant homology decompositions for cyclic group actions on definite 4-manifolds