C_2-equivariant stable homotopy from real motivic stable homotopy

TL;DR

This paper introduces a method to compute C_2-equivariant homotopy groups of Betti realizations of p-complete cellular motivic spectra over R, linking motivic and equivariant stable homotopy theories.

Contribution

It establishes a framework for understanding the Betti realization as a localization of the real motivic stable homotopy category, advancing computational techniques in equivariant homotopy theory.

Findings

Provides a formula for C_2-equivariant homotopy groups in terms of motivic homotopy groups.

Shows Betti realization as a localization of the real motivic stable homotopy category.

Enables new computations in equivariant stable homotopy theory.

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.