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The $\mathrm{RO}(C_4)$ cohomology of the infinite real projective space

Nick Georgakopoulos

Algebraic & Geometric Topology 24 (2024) 277–323

Following the Hu–Kriz method of computing the C2 genuine dual Steenrod algebra π(H𝔽2 H𝔽2)C2, we calculate the C4–equivariant Bredon cohomology of the classifying space Pρ = BC4Σ2 as an RO (C4) graded Green-functor. We prove that as a module over the homology of a point (which we also compute), this cohomology is not flat. As a result, it can’t be used as a test module for obtaining generators in π(H𝔽2 H𝔽2)C4 as Hu and Kriz use it in the C2 case. Their argument for the Borel equivariant dual Steenrod algebra does generalize, however, and we give a complete description of π(H𝔽2 H𝔽2)hC2n for any n 2.

homotopy theory, equivariant, algebraic topology, Mackey functor, $\mathrm{RO}(G)$ graded homology, projective space, spectral sequences
Mathematical Subject Classification
Primary: 55N91, 55P91
Received: 20 July 2021
Revised: 8 July 2022
Accepted: 12 August 2022
Published: 18 March 2024
Nick Georgakopoulos
Department of Mathematics
University of Chicago
Chicago, IL
United States

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