O-minimal homotopy and generalized (co)homology

TL;DR

This paper extends semialgebraic homotopy theory to o-minimal structures, establishing equivalences between definable CW-complexes and classical topological CW-complexes, and linking generalized (co)homology theories across these frameworks.

Contribution

It develops an o-minimal homotopy theory over fields and shows the equivalence of associated homotopy categories with classical topological categories, connecting generalized (co)homology theories.

Findings

Homotopy category of definable CW-complexes is equivalent to that of topological CW-complexes.

In bounded o-minimal expansions, categories are equivalent to weakly definable spaces.

Generalized homology and cohomology theories correspond to classical theories on CW-complexes.

Abstract

This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspond (up to an isomorphism) to the known topological generalized homology and cohomology theories on pointed CW-complexes.