Differentiable Stacks and Equivariant Cohomology

In this thesis, we are interested in the study of cohomology of differentiable stacks and we want to provide a good notion of equivariant cohomology for differentiable stacks. For this we describe in detail some of the cohomology theories found in the literature and give some relations between them. As we want a notion of equivariant cohomology, we discuss the notion of an action on a stack by a Lie group G and how to define the quotient stack for this action. We find that this quotient stack is a differentiable stack and we describe its homotopy type. We provide a Cartan model for equivariant cohomology and we show that it coincides with the cohomology of the quotient stack previously defined. We prove that the Gequivariant cohomology can be expressed in terms of a T-equivariant cohomology for T a maximal torus of G and its Weyl group. Finally we construct some spectral sequences that relate the cohomology of the nerve associated to the Lie groupoid of the quotient stack with the previously described equivariant cohomology.