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October 25th, 2015, 05:55 AM   #1
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Quest for the 4D Exotic Sphere

Hello everyone

here is the formal description of an Exotic Sphere

A manifold homeomorphic to a sphere but not diffeomorphic to the standard sphere.

In laymen terms does this mean to find a shape in 4D that has all the features of a sphere but not the smoothness?

As far as I am aware no one has yet found an exotic sphere in the 4th dimension but they have found this shape in higher dimensions.
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October 28th, 2015, 04:49 AM   #2
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I'm going to rephrase the question

Hello everyone

here is the formal description of an Exotic Sphere

A manifold homeomorphic to a sphere but not diffeomorphic to the standard sphere.

What does this mean in laymen terms?
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October 28th, 2015, 05:36 AM   #3
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Probably anyone who understands the term "diffeomorphism" is not what I'd call a layman. I'm not a topologist by any stretch, but here's my attempt:

Two shapes are homeomorphic if you could mold one into the shape of another (if they're made of, say, clay) without adding or removing holes. The classic example is that a coffee cup is homeomorphic to a doughnut:


Now this map between two homeomorphic objects is called, naturally enough, a homeomorphism. In particular, this mapping is continuous: if you pick a blob* on the second shape, even a very small one, then there's some (perhaps even smaller) blob on the first shape such that all of that second blob is mapped into the first blob. This means you can't do anything really crazy where some tiny bit of the first shape is smeared all over the second shape. You'e allowed to shrink (compress?) the clay, or to grow it, but only in this same continuous fashion.

With me so far?

Now some homeomorphisms will be a bit wilder than others. Maybe one little bit is being compressed and the piece right next to it is being expanded, where there's a sharp change between the two -- no matter how far you magnify the area, you'll always see this sharp change. It's still continuous, of course, but the rate of change itself is not continuous. Others are nicer: any of the changes from one area of the map to another is smooth, no sudden sharp changes anywhere. A map with this extra nice property is called a diffeomorphism.

Hmm, that's kind of hard to explain. Look at this picture

In the middle it's 'pinched', so that even though it's continuous it's not smooth. You can't use a diffeomorphism to map a smooth area of a shape into a crease like the one above, nor can you use a diffeomorphism to get rid of one of these creases. (Well, not directly at least -- but this is getting into surgery and other topics I'm not comfortable with.)

So an exotic sphere is a shape which is homoemorphic to a sphere -- you can continuously transform it to a sphere, just like the coffee cup can be transformed into a doughnut -- but which is not diffeomorphic to a sphere.

As complicated as this sounds, it's actually not even that simple! It's (relatively) easy to find a map from one shape to another which is a homeomorphism but not a diffeomorphism, but that's not what you need. To show that the two are homeomorphic 'all' you need is to find a single homeomorphism. To prove that they aren't diffeomorphic, it's not enough to find a single homeomorphism which is not a diffeomorphism. You need to prove that every single homeomorphism between the two shapes fails to be a diffeomorphism. This is where surgery theory comes in: very clever ways to make small changes to different places of the shape such that they map diffeomorphically in ways which are not at all obvious.

OK... did that help?

* Normally you'd pick a sphere, but that makes it sound like there's something special about that shape. Really, you could pick any reasonable shape for your blob. Crazy things like "only the points with rational coordinates" aren't OK though, they're not blobby enough.
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November 1st, 2015, 07:57 AM   #4
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A Cube in 4D is called a Tesseract

A Sphere in 4D is called a Hypersphere


So in 4D I guess an Exotic Sphere would be a Tesseract turning into a Hypersphere?

This question of course goes out to everyone.
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November 1st, 2015, 11:53 AM   #5
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Quote:
Originally Posted by HawkI View Post
So in 4D I guess an Exotic Sphere would be a Tesseract turning into a Hypersphere?
No, those are diffeomorphic I believe.
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November 28th, 2015, 10:05 AM   #6
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The manifold homeomorphic must comply with the operations of the diffeomorphic to the standard sphere.
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December 8th, 2015, 03:35 AM   #7
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Please help me understand. How do you shrink a sphere from infinity, to any lesser size.
How do you get to infinite, to start shrinking. also if you were able to start at infinity, how would you be able to shrink this sphere to a visible size, that could fit in a box I think this may take an infinite amount of time to accomplish. I do not understand this topic. This does not make logical sense to me. If you can help me understand this problem. I think it is funny that anything could reach any infinite. I feel if they did it would not be infinite? What are your views on this I do not understand. It seems physics and pure math does not fit together on this. Maybe I am wrong I do not understand this.
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January 26th, 2016, 04:27 AM   #8
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I've recently read the book 'Things to see and do in the 4th dimension' and the best thing I read was that spheres are naturally spiky.

Exotic Sphere description 'A manifold homeomorphic to a sphere but not diffeomorphic to the standard sphere.'

Diffeomorphic means smooth

Homeomorphic means 'looks locally like'

It would be homeomorphic to a sphere in 4D. A 4D Sphere is spiky.

Therefore how about this for a description of an Exotic Sphere.

It looks spiky and isn't smooth.

What do you think?

Last edited by HawkI; January 26th, 2016 at 04:33 AM. Reason: I changed A 4D Sphere looks spiky to, A 4D Sphere is spiky
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February 27th, 2016, 10:36 AM   #9
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Just a quick update on this quest of mine, I read recently that someone with klein in his name came up with a theory (which also had a name) that the 4th dimension is smaller than the 3rd.

This blew my mind!

On an unrelated note, I don't feel so strongly about my January 26th post.
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