In Topology When Are Two Shapes the Same?

https://www.quantamagazine.org/in-topology-when-are-two-shapes-the-same-20210928/

75 16 5 years ago

sarosh• 5 years ago

A nice summary of the current understanding (from a lay perspective) on manifold "sameness".

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."

donw• 5 years ago

This is one of the best lay-explanations of manifolds I’ve seen. Nice!

woopwoop• 5 years ago

When I took algebraic topology, the professor asked us if two topological spaces were the same. Then he looked at us for a moment, and said that actually, that was a "Bill Clinton question: it depends on what the meaning of the word 'is' is".

Koshkin• 5 years ago

A nice article, but it feels a bit weird that there was no mentioning of the Poincaré conjecture in its original form - the one that was named one of the seven Millennium Prize Problems.

kevinventullo2• 5 years 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• 5 years ago

If 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?

kevinventullo2• 5 years ago

Yes, 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-dim...

quickthrower22• 5 years ago

There is a good YouTube about this [1]: Pop a balloon and slowly let the air out. You can stretch the resulting fabric into a simple shape with no holes. The balloon therefore had -1 holes.

They then go on to talk about 2D and 3D holes and Euler Characteristic etc.

[1] https://www.youtube.com/watch?v=ymF1bp-qrjU

Jensson• 5 years ago

The pants example is wrong. Pants are equivalent to a disk with two holes, when you circle the two holes in a disk you get 3 separated areas you cannot move between. Topologically valid transformations of surfaces doesn't alter this property. The pants with connected legs however lets you circle two holes without separating any areas at all, you can move between any two points on the pants without crossing the lines just by circling the 8 they collapsed the connected pants into. So a punctured torus surface is not topologically equivalent to a pants surface.

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.

quickthrower22• 5 years ago

Thanks for that I was a bit bothered by the shorts transform but thought it was my not understanding something. All the animations require the twist to just kinda disappear.

Dylan168072• 5 years ago

He talks about that at the 29 minute mark.

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.

yazaddaruvala• 5 years ago

I also really like the Vsauce video essay about holes.

https://youtu.be/egEraZP9yXQ

gnu8• 5 years ago

Is the opening of the balloon no longer a hole because it is tied into a knot?

dukeofdoom• 5 years ago

3Blue1Brown has a nice intro video on Topology

https://www.youtube.com/watch?v=AmgkSdhK4K8

0xdeadb00f• 5 years ago

Thanks. This was really great.

In Topology When Are Two Shapes the Same?

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