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

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-MacLane spaces. The result we obtain for

C_{2}

-equivariant Eilenberg-MacLane spaces associated to the constant Mackey functor

u n d e r l i n e {m a t h b b 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.

Mathematics Subject Classification ID

Equivariant homology and cohomology in algebraic topology (55N91) Eilenberg-Mac Lane spaces (55P20) Equivariant homotopy theory in algebraic topology (55P91)

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