I'd like to find examples of theorems that express a link between the topology of, say, a manifold and some differential concept. So far I've thought about the de Rham theorem (isomorphism between de rham cohomology and singular or sheaf cohomology), poincare hopf theorem (relating the index of a vector field with the genus of a surface), Riemann Roch (relating the dimension of the space of meromorphic functions with the genus of a Riemann surface).
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