Abstract

We construct a $C_2$-equivariant spectral sequence computing $\mathrm{RO}<!--FUNCTION\; APPLICATION-->(C_2)$-graded homotopy groups by the $C_2$-equivariant Betti realization functor realizing the motivic effective slice filtration. We apply the spectral sequence to compute the $\mathrm{RO}<!--FUNCTION\; APPLICATION-->(C_2)$-graded homotopy groups of the completed $C_2$-equivariant connective real $K$-theory spectrum. The computation reproves the $C_2$-equivariant Adams spectral sequence results of Guillou, Hill, Isaksen and Ravenel. We also include the $2$-Bockstein spectral sequence computation to compute the $\mathrm{RO}<!--FUNCTION\; APPLICATION-->(C_2)$-graded homotopy ring of $H\underset{¯}{{\mathbb{Z}}_2}$ from that of $H\underset{¯}{{\mathbb{𝔽}}_2}$.

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