$C_2$-equivariant stable homotopy from real motivic stable homotopy

Abstract

We give a method for computing the $C_2$-equivariant homotopy groups of the Betti realization of a $p$-complete cellular motivic spectrum over $ℝ$ in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the $C_2$-equivariant $p$-complete stable homotopy category as a localization of the $p$-complete cellular real motivic stable homotopy category.