Hopf Fibration, torus with fibers, do it 2 ways?

  • I
  • Thread starter Spinnor
  • Start date
  • Tags
    Torus
In summary: The two types of paths are distinguished (in the rational case) by the axis of the net rotation not being on the original 3-sphere. This leads to a contradiction, showing that the irrational case is pathological."That sounds like a juicy proof.In summary, the conversation discusses the projection of a set of fibers on a 3-sphere and whether it can be rotated to change its orientation. It also touches on the concept of smooth morphing and the potential existence of two inequivalent fibrations of a 3-sphere. The idea of helicity is also mentioned and a possible proof using rotations in 4-dimensional space is brought up.
  • #1
Spinnor
Gold Member
2,214
424
The following is a projection of a set of fibers of a 3-sphere?

1604148520047.png


Consider the mirror image of the above,

1604148503497.png


Can the 3-sphere be rotated so that (b) becomes (d)?

Is there an easy way to understand this?

Thank you.
 
Physics news on Phys.org
  • #2
Looks like a rotation around the y-axis by ##\pi##.
 
  • #3
fresh_42 said:
Looks like a rotation around the y-axis by ##\pi##.

That is a rotation of the 3-sphere around its y axis, 1 = x^2 +y^2 +z^2 +w^2 or of the y-axis of the image above? Are they the same?

Then one image should smoothly "morph" into the other?

Thanks.
 
  • #4
If one imagines a physical torus with the above lines of (b) drawn on it and one rotates it in front of you "end over end" (b) does not change into (d)? But if we rotate a 3-sphere can the projection (b) "morph" into the projection (d)?

Thanks.
 
  • #6
fresh_42 said:
very poor geometric phantasy

Could you put that in other words? Thanks.
 
  • #8
fresh_42 said:
imagination

You mathematical brute, you!

And if it not obvious, the above is a complement.
 
  • Haha
Likes pinball1970
  • #9
Maybe the answer is in this video below, the parts of the fibers closest to you seem to mostly go from "top left to bottom right". Just an observation not at all mathematical.

 
  • #10
So I think I have an answer to my original question which was the title of this thread,

Hopf Fibration, torus with fibers, do it 2 ways?

I think that the answer to my question is yes and there are 2 fibrations that cannot be rotated into each other? To understand this requires only some simple assumptions about Hopf Fibrations which I think to be true.

Imagine a very large 3-sphere and the set of fibers of its fibration. Pick a point of this 3-sphere and consider a fiber that goes through this point. Now with the point and the fiber we can define a 2-sphere that goes through the point, is perpendicular to the fiber, and divides the 3-sphere in two equal halves? Our fiber actually goes through this 2-sphere at 2 antipodal points and is perpendicular at both points? Now surround this fiber with with a small torus of radius r such that the fiber is perfectly centered in the center of the torus. This torus is in fact the union of 2 inequivalent sets of fibers, typical fibers shown in yellow and blue of the projection of our torus below? Ignore the green and light blue lines below.

1604493219095.png


If we were in the 3-sphere we would not see the image above but see a "straight" fibered torus. Keep that picture in your head, what you would actually see because under a rotation of the 3-sphere what you see does not change, the 3-sphere and its fibrations rotate "rigidly"? The two fibrations are inequivalent and can not be rotated into one another?

Please point out where I may have made questionable assumptions.

Thanks.

Image captured from,

 

Attachments

  • 1604492748815.png
    1604492748815.png
    54.1 KB · Views: 204
  • Like
Likes fresh_42
  • #11
I think you are right. One thing you mentioned sticks out to me as a solid negation of morphing one to the other.

Imagine that you begin simultaneously rotating all the fibers as if to "morph" to the other configuration. As they become parallel to the axis of the torus, they will trace circumferences of the torus. For this to be possible, the fibers must be unlinked.

I think the concept of a "smooth morph" requires that the fibers are linked at all time. Since there is a point halfway through the transformation where it is impossible for the fibers to be linked, then it is also impossible for them to make a smooth transition from the left hand configuration to the right hand one.

I think this primitive mental proof relies on a particular definition of "continuous". My calculus is strong, but the different types of continuity and analysis type underpinnings are a little arcane for me. Hopefully a mathematician will drop by and hand out some wisdom on that
 
  • Like
Likes Spinnor
  • #12
I think you msy want to look inho the fundamental groups of the sphere. I understand the Hopf Fibration is a non-trivial element of them ( I believe in this case both groups are equal/isomorphic). Edit: It, the Hopf map, generates ## \pi_3(S^2)= \mathbb Z ##, so the Hopf map is the only non-homotopically -trivial ( class of ) map(s) from ##S^3## to ## S^2 ##. I think @lavinia or @mathwonk may be able to verify this. It's been a while since I've seen this material.
 
Last edited:
  • #13
Had some time to think about the above. It may help to graph some of the fibers that make up a cylinder that surrounds a single fiber,

1645883943336.png


The graph above is meant to represent a cylinder that surrounds a single fiber. On this graph are drawn four fibers. As this cylinder sits in 3-dimensional space we can give it an orientation, let out of the computer screen be the outside of the cylinder. Thus defined the solid lines represent fibers of left-handed helicity and the dashed lines represent right-handed fibers? Each fiber links the central fiber once? In three space dimensions shape of a curve defines it's helicity? Physical rotation of a helix does not change it's helicity? Rotation of a three-sphere does not change a fibers helicity? There exist two inequivalent fibrations of a three-sphere of different helicity?

I think the statements with question marks are true but would appreciate corrections.
 

Attachments

  • 1645883836475.jpeg
    1645883836475.jpeg
    18.6 KB · Views: 133
  • #14
Spinnor said:
Maybe the answer is in this video below, the parts of the fibers closest to you seem to mostly go from "top left to bottom right". Just an observation not at all mathematical.


Love the rockin' torus.

Here is another nice graphic

https://samuelj.li/hopf-fibration/
 
  • #15
There may be another proof lurking in the group of rotations in 4 dimensions, SO(4)? The Wiki page on rotations in 4-dimensional Euclidean space has an image and comment,

1648560251500.png


"A 4D Clifford torus stereographically projected into 3D looks like a torus, and a double rotation can be seen as in helical path on that torus. For a rotation whose two rotation angles have a rational ratio, the paths will eventually reconnect; while for an irrational ratio they will not. An isoclinic rotation will form a Villarceau circle on the torus, while a simple rotation will form a circle parallel or perpendicular to the central axis."

https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space#Double_rotations

There are two families of "double" rotation that give the "flow" above and also its mirror inverse?

Thanks.

​

 

1. What is Hopf Fibration?

Hopf Fibration is a mathematical concept that describes the relationship between a torus (a doughnut-shaped object) and a sphere. It involves mapping each point on the surface of the torus to a point on the surface of the sphere in a specific way.

2. How does Hopf Fibration work?

Hopf Fibration works by taking a point on the surface of the torus and projecting it onto the surface of the sphere in a way that preserves certain geometric properties. This projection can be done in two different ways, resulting in two different types of Hopf Fibrations.

3. What is a torus with fibers in Hopf Fibration?

In Hopf Fibration, a torus with fibers refers to the torus being the base space and the sphere being the fiber space. This means that each point on the torus is connected to a point on the sphere through the Hopf Fibration mapping.

4. Can Hopf Fibration be done in more than 2 ways?

No, Hopf Fibration can only be done in 2 ways. This is because the mapping used in Hopf Fibration is based on the properties of a 3-dimensional sphere, which can only be projected onto a 2-dimensional surface (such as the torus) in 2 different ways.

5. What are the applications of Hopf Fibration?

Hopf Fibration has applications in various fields such as mathematics, physics, and computer science. It is used to study and understand the topology of higher-dimensional spaces, and has also been applied in the development of algorithms for computer graphics and in the study of quantum physics.

Similar threads

A Regarding fibrations between smooth manifolds
    • Topology and Analysis
Replies
1
Views
709
A The map from a complex torus to the projective algebraic curve
    • Topology and Analysis
Replies
27
Views
3K
I Hopf fibration, stereoprojected fibers look close, can be?
    • Differential Geometry
Replies
14
Views
2K
Heegard Splitting of the 2-Torus.
    • Topology and Analysis
Replies
2
Views
2K
I Living inside a Hopf fibration, seeing the fibers
    • Differential Geometry
Replies
26
Views
4K
A Proving that the real projective plane is not a boundary
    • Topology and Analysis
2
Replies
38
Views
4K
I What is an atlas for a torus manifold?
    • Topology and Analysis
Replies
4
Views
6K
A Torus with two Möbius Strips
    • Topology and Analysis
Replies
12
Views
3K
I Newton Galilean spacetime as fiber bundle
    • Special and General Relativity
Replies
26
Views
998
I Can you somehow make left handed helix into a right handed?
    • Topology and Analysis
Replies
8
Views
1K

Hot Threads

Recent Insights

Back
Top