In Topology When Are Two Shapes the Same?
quantamagazine.org
quantamagazine.org
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.
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?
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-dim...
They then go on to talk about 2D and 3D holes and Euler Characteristic etc.
The problem with their transformation was that they twisted the surface. You aren't allowed to do that. It works if you see the pants as a volume though, but they specifically talked about surfaces. Similarly a möbius strip is different from a regular loop as a surface, but not as a volume. The twist here is similar.
Rough quote: Technically, if these trousers were made from infinitely thin fabric, then this shape is not equal to this shape because of the twist problem. However, fabric is not infinitely thin [...] so I'm prepared to say a real pair of trousers are homeomorphic to the same pair with the ends of the legs stuck together.
From the article: "But Freedman’s [1981] proof left open the “smooth” four-dimensional Poincaré conjecture, which says that any four-dimensional smooth manifold that is homotopy equivalent to the four-dimensional sphere is also diffeomorphic to the four-dimensional sphere. This is an even stronger statement than the one Freedman proved — since a diffeomorphism is a stronger form of equivalence than a homeomorphism — and one that mathematicians today have no idea how to settle.
This leaves them in the strange position of being unable to perform one of the most basic classification tasks of all: recognizing when a smooth four-dimensional manifold is really a sphere."