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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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Isotopies of codimension-1 disks relative to boundary
I at first thought this should be an easy question, but then realized it might actually not even be known.
Let us work in the smooth category, though I am also wondering about PL and TOP (locally flat …