O-minimal cohomology: finiteness and invariance results

TL;DR

This paper establishes that o-minimal cohomology groups of definably compact sets are finitely generated and invariant under certain model-theoretic expansions, with additional results for structures expanding a field.

Contribution

It proves finiteness and invariance of o-minimal cohomology groups and explores their behavior under expansions and intersections of definably compact sets.

Findings

Cohomology groups are finitely generated.

Cohomology groups are invariant under elementary extensions.

Results extend to intersections of definably compact sets in structures expanding a field.

Abstract

We prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language. We also study the cohomology of the intersection of a definable decreas-ing family of definably compact sets, under the additional assumption that the o-minimal structure expands a field.