De Rham Cohomology In Diffeological Spaces

Abstract

Diffeological spaces are a generalization of smooth manifolds. They were introduced around 1970 by the French mathematician and physicist Jean-Marie Souriau (1922-2012), in an attempt to formalize quantum mechanics and geometric quantization. This approach proved to be useful in several settings which involve objects that rarely are finite dimensional manifolds, like the space of smooth maps between two manifolds or the space of leaves of a foliation. In this thesis we are mainly interested in the Cartan calculus on diffeological spaces. We clarify the properties of the De Rham cohomology groups, which are typical of the algebraic topology of manifolds but were less known in this setting. We introduce several new constructions like the Mayer-Vietoris sequence, the relative De Rham cohomology groups and the relative cup-product.