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

Clover May

Algebraic & Geometric Topology 20 (2020) 1691–1728

Let C2 be the cyclic group of order two. We present a structure theorem for the RO(C2)–graded Bredon cohomology of C2–spaces using coefficients in the constant Mackey functor 𝔽2 ¯. We show that, as a module over the cohomology of the point, the RO(C2)–graded cohomology of a finite C2–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(C2) corresponding to actual (ie nonvirtual) C2–representations. This decomposition lifts to a splitting of genuine C2–spectra.

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equivariant cohomology, equivariant homotopy, Toda bracket, Mackey functor, $\mathit{RO}(G)$–graded
Mathematical Subject Classification 2010
Primary: 55N91
Received: 12 June 2018
Revised: 15 August 2019
Accepted: 26 September 2019
Published: 20 July 2020
Clover May
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States