Unstable motivic and real-étale homotopy theory
Aravind Asok, Tom Bachmann, Elden Elmanto, Michael J. Hopkins
TL;DR
This work analyzes unstable motivic homotopy theory over a base scheme with the real étale topology, linking it to unstable semialgebraic topology on real spectra. It develops destabilized analogues of prior stable results by introducing and exploiting the ρ-localization, showing that the real étale localization for connected motivic spaces is realized by smashing with the ρ-telescope, and establishing that locally constant real étale sheaves are $A^1$-invariant. The authors construct and leverage the James construction, prove a concrete equivalence between motivic spheres, and connect real realizations to $C_2$-equivariant topology, enabling computations for real-étale realizations of Eilenberg–Mac Lane spaces and nilpotent spaces. Collectively, the results provide an explicit, descent-friendly bridge between real-étale motivic homotopy theory and semialgebraic topology, with implications for understanding real realizations and EM-space behavior under real-étale localization.
Abstract
We prove that for any base scheme $S$, real étale motivic (unstable) homotopy theory over $S$ coincides with unstable semialgebraic topology over $S$ (that is, sheaves of spaces on the real spectrum of $S$). Moreover we show that for pointed connected motivic spaces over $S$, the real étale motivic localization is given by smashing with the telescope of the map $ρ: S^0 \to {\mathbb G}_m$.