Semi-topologization in motivic homotopy theory and applications

TL;DR

This paper introduces a new functor in motivic homotopy theory called homotopy semi-topologization, which enables the development of semi-topological cohomology theories and analogues of algebraic cobordism with spectral sequences.

Contribution

It constructs a triangulated endo-functor on the stable motivic homotopy category and applies it to define semi-topological cohomology theories and cobordism analogues.

Findings

Construction of homotopy semi-topologization functor.

Representation of semi-topological cohomology theories.

Development of Atiyah-Hirzebruch type spectral sequences.

Abstract

We study the semi-topologization functor of Friedlander-Walker from the perspective of motivic homotopy theory. We construct a triangulated endo-functor on the stable motivic homotopy category $\mathcal{SH}(\C)$, which we call \emph{homotopy semi-topologization}. As applications, we discuss the representability of several semi-topological cohomology theories in $\mathcal{SH}(\C)$, a construction of a semi-topological analogue of algebraic cobordism, and a construction of Atiyah-Hirzebruch type spectral sequences for this theory.