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The final open question can be restated as asking whether any 4-dimensional smooth manifold which is homeomorphic to the 4-sphere is necessarily diffeomorphic t
by kevinventullo 5y ago
The final open question can be restated as asking whether any 4-dimensional smooth manifold which is homeomorphic to the 4-sphere is necessarily diffeomorphic to the 4-sphere.
As a point of comparison, here is a surprising pair of facts about Euclidean space:
- For n != 4, any n-dimensional smooth manifold which is homeomorphic to n-dimensional Euclidean space is actually diffeomorphic to it.
- On the other hand, there are uncountably many pairwise non-diffeomorphic 4-dimensional smooth manifolds which are all homeomorphic to 4-dimensional Euclidean space.
- dpratt 5y agoIf I’m understanding this correctly, the statement is “There is a set of provably true assertions that are held in every n-dimensional geometry, except for n=4” If so, that’s fascinating. Has there been any proofs around why a dimensional ordinality (if that’s the right term) of 4 is different?
- kevinventullo 5y agoYes, and I agree it is fascinating! As for “why” 4-dimensional geometry is somehow special, I couldn’t tell you, but here’s a whole MO thread about it: https://mathoverflow.net/questions/47569/what-makes-four-dimensions-special https://mathoverflow.net/questions/47569/what-makes-four-dim...