The RO(C4) cohomology of the infinite real projective space

Georgakopoulos, Nick

doi:10.2140/agt.2024.24.277

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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

DOI: 10.2140/agt.2024.24.277

Abstract

Following the Hu–Kriz method of computing the $C_{2}$ genuine dual Steenrod algebra $π_{⋆} {(H 𝔽_{2} \land H 𝔽_{2})}^{C_{2}}$ , we calculate the $C_{4}$ –equivariant Bredon cohomology of the classifying space $ℝ P^{\infty ρ} = B_{C_{4}} Σ_{2}$ as an $RO (C_{4})$ 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 $π_{⋆} {(H 𝔽_{2} \land H 𝔽_{2})}^{C_{4}}$ as Hu and Kriz use it in the $C_{2}$ case. Their argument for the Borel equivariant dual Steenrod algebra does generalize, however, and we give a complete description of $π_{⋆} {(H 𝔽_{2} \land H 𝔽_{2})}^{h C_{2^{n}}}$ for any $n \geq 2$ .

Keywords

homotopy theory, equivariant, algebraic topology, Mackey functor, $\mathrm{RO}(G)$ graded homology, projective space, spectral sequences

Mathematical Subject Classification

Primary: 55N91, 55P91

References

Publication

Received: 20 July 2021

Revised: 8 July 2022

Accepted: 12 August 2022

Published: 18 March 2024

Authors

Nick Georgakopoulos
Department of Mathematics University of Chicago Chicago, IL United States

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Volume 24, issue 1 (2024)