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A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different.

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Embedded spheres in the K3 surface

As far as I know, this is an open question. It seems quite reasonable that such a surface should exist, but off the top of my head I'm not quite sure how you would construct something like that. In ge …
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