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The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
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Finiteness of De Rham cohomology of smooth quasi-projective varieties
I think there are proofs which are much easier.
For example, you can try to compute the cohomology using the Morse theory. For that you need existence of Morse functions having finitely many critical …