The $H \underline{\mathbb{F}}_2$-homology of $C_2$-equivariant Eilenberg-MacLane spaces

TL;DR

This paper extends Hopf ring techniques to $C_2$-equivariant homotopy theory to compute the $RO(C_2)$-graded homology of equivariant Eilenberg-MacLane spaces, providing a $C_2$-analogue of Serre's classical result.

Contribution

It introduces a $C_2$-equivariant analogue of classical homology computations and explores spectral sequences for these spaces, advancing equivariant homotopy theory methods.

Findings

Computed $C_2$-equivariant homology of Eilenberg-MacLane spaces.

Established a $C_2$-analogue of Serre's classical homology result.

Investigated twisted bar spectral sequences for these computations.

Abstract

We extend Ravenel-Wilson Hopf ring techniques to $C_2$-equivariant homotopy theory. Our main application and motivation is a computation of the $RO(C_2)$-graded homology of $C_2$-equivariant Eilenberg-MacLane spaces. The result we obtain for $C_2$-equivariant Eilenberg-MacLane spaces associated to the constant Mackey functor $\underline{\mathbb{F}}_2$ 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-MacLane spaces and suggest the existence of another twisted bar spectral sequence with $E^2$-page given in terms of a twisted Tor functor.