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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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Manifolds with polynomial transition maps

Let me develop Robert Bryant's answer : if $M$ is a simply-connected manifold of dimension $n$ with a polynomial atlas, then there exists a local isomorphism $M \rightarrow \mathbb{R}^n$. Conversely, …
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