O-minimal De Rham cohomology

The aim of this dissertation lies in establishing an o-minimal de Rham cohomology theory for abstract-definable C ∞ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function, by following the classical de Rham cohomology.We can specify the o-minimal cohomology groups and attain some properties as the existence of Mayer-Vietoris sequence and the invariance under abstract-definable C ∞ diffeomorphisms.However, in order to obtain the invariance of our o-minimal cohomology under abstractdefinable homotopy we must, working in a tame context that defines sufficiently many primitives, assume the validity of a statement related to Bröcker's problem.