A structure theorem for RO(C2)–graded Bredon cohomology

May, Clover

doi:10.2140/agt.2020.20.1691

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

Clover May

Algebraic & Geometric Topology 20 (2020) 1691–1728

DOI: 10.2140/agt.2020.20.1691

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 $\underline{𝔽_{2}}$ . 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 (ie nonvirtual) $C_{2}$ –representations. This decomposition lifts to a splitting of genuine $C_{2}$ –spectra.

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Keywords

equivariant cohomology, equivariant homotopy, Toda bracket, Mackey functor, $\mathit{RO}(G)$–graded

Mathematical Subject Classification 2010

Primary: 55N91

References

Publication

Received: 12 June 2018

Revised: 15 August 2019

Accepted: 26 September 2019

Published: 20 July 2020

Authors

Clover May
Department of Mathematics University of California, Los Angeles Los Angeles, CA United States

Volume 20, issue 4 (2020)