I am studying de Rham Cohomology of surfaces. I am following the book "An Introduction to Manifolds" by Loring Tu. I found that it uses the "Mayer Veitoris Sequence" and notions of Homology. But I want to study it more geometrically. My question is as follows. For an orientable two dimensional manifold (i.e. orientable surface), we can view it as a Riemann surface. Can we study de Rham Cohomology through Riemann surfaces? Please advise.
If yes, please advise me about some references and the learning road map.
Regards.