The H𝔽2–homology of C2–equivariant Eilenberg–Mac Lane spaces

Petersen, Sarah

doi:10.2140/agt.2024.24.4487

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The $H \underline{\mathbb{F}}_2$–homology of $C_2$–equivariant Eilenberg–Mac Lane spaces

Sarah Petersen

Algebraic & Geometric Topology 24 (2024) 4487–4518

DOI: 10.2140/agt.2024.24.4487

Abstract

We extend Ravenel–Wilson Hopf ring techniques to $C_{2}$ –equivariant homotopy theory. Our main application and motivation is a computation of the $R O (C_{2})$ –graded homology of $C_{2}$ –equivariant Eilenberg–Mac Lane spaces. The result we obtain for $C_{2}$ –equivariant Eilenberg–Mac Lane spaces associated to the constant Mackey functor ${\underline{𝔽}}_{2}$ gives a $C_{2}$ –equivariant analogue of the classical computation due to Serre. We also investigate a twisted bar spectral sequence computing the homology of these equivariant Eilenberg–Mac Lane spaces and suggest the existence of another twisted bar spectral sequence with $E^{2}$ –page given in terms of a twisted Tor functor.

Keywords

Equivariant homotopy theory, $RO(C_2)$–graded homology, $C_2$–equivariant Eilenberg–Mac Lane spaces, Hopf rings

Mathematical Subject Classification

Primary: 55P91

Secondary: 55N91, 55P20

References

Publication

Received: 12 July 2022

Revised: 18 January 2023

Accepted: 6 February 2023

Published: 17 December 2024

Authors

Sarah Petersen
Department of Mathematics University of Colorado Boulder Boulder, CO United States

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Volume 24, issue 8 (2024)