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

May, Clover¹

¹Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, United States

Algebraic and Geometric Topology, Tome 20 (2020) no. 4, pp. 1691-1728

Résumé

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.

Détail

DOI : 10.2140/agt.2020.20.1691

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

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