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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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uniqueness of regular/tubular neighborhood with equivariant boundary
Let's consider the smooth case. The basic idea is to solve the problem on the free quotient, and then to lift the solution by covering space theory. For the external reference below, I'm going to ass …