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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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Line bundle on $S^2$

How do you prove that a line bundle (vector bundle of rank 1) on $S^2$ is isomorphic to the trivial line bundle? Can you give a reference? Thanks.
Pedro Perez's user avatar