# O-minimal de Rham cohomology

## TL;DR

This paper develops an o-minimal de Rham cohomology theory for smooth manifolds within o-minimal structures, extending classical tools to a tame geometric setting with applications in arithmetic geometry.

## Contribution

It introduces a new o-minimal de Rham cohomology theory for definable smooth manifolds, establishing key properties like Mayer-Vietoris sequences and invariance under definable diffeomorphisms.

## Key findings

- Defined o-minimal cohomology groups for smooth manifolds.
- Proved Mayer-Vietoris sequence for the cohomology.
- Achieved invariance under definable diffeomorphisms.

## Abstract

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as Andr\'e-Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and \v Cech cohomology, which have been used for instance to prove Pillay's conjecture concerning definably compact groups. In the present paper we elaborate an o-minimal de Rham cohomology theory for abstract-definable $\mathcal{C}^\infty$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer-Vietoris sequence and the invariance under abstract-definable $\mathcal{C}^\infty$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must, working in a tame context that defines sufficiently many primitives, assume the validity of a statement related to Br\"ocker's question.