Failure of The Whitney Trick in Dimension 4
According to Frank Quinn, "Two n-dimensional submanifolds of a manifold of dimension 2n will usually intersect themselves and each other in isolated points. The "Whitney trick" uses an isotopy across an embedded 2-disk to simplify these intersections. Roughly speaking this reduces the study of n-dimensional embeddings to embeddings of 2-disks. But this not a reduction when the embedding is 4: the 2 disks themselves are middle-dimensional, so trying to embed them encounters exactly the same problems they are supposed to solve. This is the phenomenon that separates dimension 4 from others."
Read more about this topic: 4-manifold
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