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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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Existence of certain continuous curves

Here's a counter-example in the smooth case: take $p$ as the North pole, $f(\cdot,0)$ the Equator, and as $t$ grows, $f(\cdot,t)$ is a parallel of higher and higher latitude, until $f(\cdot,1)=p$. If …
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