Semitopologization in motivic homotopy theory and applications

Abstract

We study the semitopologization functor of Friedlander and Walker from the perspective of motivic homotopy theory. We construct a triangulated endofunctor on the stable motivic homotopy category $\mathcal{S}\mathcal{\mathscr{H}}\left(ℂ\right)$, which we call homotopy semitopologization. As applications, we discuss the representability of several semitopological cohomology theories in $\mathcal{S}\mathcal{\mathscr{H}}\left(ℂ\right)$, a construction of a semitopological analogue of algebraic cobordism and a construction of Atiyah–Hirzebruch type spectral sequences for this theory.