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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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Equivalence of knotted spheres in $S^4$

Say we have two smoothly embedded spheres $K, K' \subset S^4$ that are equivalent in the sense that there is a diffeomorphism of pairs $(S^4, K)$ and $(S^4, K')$. Does it follow that they are equivale …
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