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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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Which manifolds admit free involutions?

The question of which manifolds admit a free involution seems natural enough, yet I couldn't find much about it online. It's not hard to see that such a manifold must be a boundary, but as pointed out …
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