Hopf fibration of 3-sphere

[Hornbein]

Can you explain why they can always be separated in the "ambient" $\mathbb R^4$ ?

Sure. Locally the two circles are one dimensional. When they approach closely enough they look like lines. Suppose one line W is [w, 0, 0, 0] while the other line X is [0,x,0,e]. (w and x are free variables, e is a constant.) If the X line tries to move directly to [0,x,0,-e] then the two lines will intersect at the origin. But if the X line uses the 4th dimension by slipping over to [0,x,e,e] then it can move to [0,x,e,-e] then slip back to [0,x,0,-e]. So it isn't possible to link two circles in R4. It is possible to link two 2-spheres. In N>=3 dimensions one may link (N-2)-spheres.

Great circles are always linked in S3 because they lack the freedom to maneuver like this. If either circle is not a great circle then they cannot be linked. Or at least so I believe.

 

[lavinia]

Sure. Locally the two circles are one dimensional. When they approach closely enough they look like lines. Suppose one line W is [w, 0, 0, 0] while the other line X is [0,x,0,e]. If the X lines tries to move to [0,x,0,-e] then the two lines will intersect at the origin. (w and x are free variables, e is a constant.) But if the X line uses the 4th dimension by slipping over to [0,x,e,e] then it can move to [0,x,e,-e] then slip back to [0,x,0,-e]. So it isn't possible to link two circles in R4. It is possible to link two 2-spheres. In N dimensions one may link (N-2)-spheres. In two dimensions one may link two 0-spheres.

Great circles are always linked in S3 because they lack the freedom to maneuver like this. If either circle is not a great circle then they cannot be linked. Or at least so I believe.

I think not all great circles. Some of them intersect with each other. The great circles in the Hopf fibration do not intersect with each other, this because they are orbits of the action of SO(2) on S^3. If two orbits intersected then where would the intersection point go under the action of SO(2)? Which orbit would it move on?

 

[Hornbein]

I think not all great circles. Some of them intersect with each other. The great circles in the Hopf fibration do not intersect with each other, this because they are orbits of the action of SO(2) on S^3. If two orbits intersected then where would the intersection go under the action of SO(2)?

That's true. If great circles intersect then they are not linked.

Two great circles chosen at random will be linked with probability one. They intersect with probability zero.

 

[lavinia]

That's true. If great circles intersect then they are not linked.

Two great circles chosen at random will be linked with probability one. They intersect with probability zero.

Can you make that probaility argument more precsise?

 

[Hornbein]

Can you make that probaility argument more precsise?

A great circle on a 3-sphere is the intersection of the sphere with a 2-plane that goes through the center of the sphere. Let's say that that center is the point [0,0,0,0] and you have two such 2-planes, each of which corresponds to a great circle.

For every vector in one plane one may get the projection of that vector in the other plane, then calculate the angle between these vectors. These angles will have a maximum and a minimum. These are the two principal angles. (Here in 3D only the maximum is of interest, as the minimum is always zero.)

Let's say plane A is [w,x,0,0]. Plane B is [cos(a)y, cos(b)z, sin(a)y, sin(b)z] with a and b the principal angles. If a and b are both zero then the planes are identical. If a is zero and b is not then the planes intersect in a line. If a and b are both non-zero then the only intersection of the planes is at the origin of [0,0,0,0]. Since the angles are on a continuum the probability of either angle being zero is zero. These are the only pairs of planes that correspond to intersecting great circles.

 

[lavinia]

A great circle on a 3-sphere is the intersection of the sphere with a 2-plane that goes through the center of the sphere. Let's say that that center is the point [0,0,0,0] and you have two such 2-planes, each of which corresponds to a great circle.

For every vector in one plane one may get the projection of that vector in the other plane, then calculate the angle between these vectors. These angles will have a maximum and a minimum. These are the two principal angles. (Here in 3D only the maximum is of interest, as the minimum is always zero.)

Let's say plane A is [w,x,0,0]. Plane B is [cos(a)y, cos(b)z, sin(a)y, sin(b)z] with a and b the principal angles. If a and b are both zero then the planes are identical. If a is zero and b is not then the planes intersect in a line. If a and b are both non-zero then the only intersection of the planes is at the origin of [0,0,0,0]. Since the angles are on a continuum the probability of either angle being zero is zero. These are the only pairs of planes that correspond to intersecting great circles.

Technically if sin(a) and sin(b) are both not zero. Without invoking a probability space, the set of pairs of angles with one or both zero has Lebesque measure zero. I think.

 

[Hornbein]

Technically if sin(a) and sin(b) are both not zero.

True. Principal angles are magnitudes always in the range [0,pi/2], just like in 3D.

 

[Hornbein]

The trouble I had with the Hopf linkage was I was using the wrong definition of linkage. I was looking for two distinct great circles in the 3-sphere for which one could say "you can't get there from here." Instead what they show is that you can't get from a circle to the same circle with the opposite sign. That is, you can't flip the circle over, just like linked circles here in 3D. The point is that there is a restriction on the movement of the circle, therefore the circles are linked. But in this context equality doesn't care about sign so you CAN move continuously without intersecting the forbidden circle between any two great circles that don't intersect the forbidden circle

.

 

[lavinia]

The trouble I had with the Hopf linkage was I was using the wrong definition of linkage. I was looking for two distinct great circles in the 3-sphere for which one could say "you can't get there from here." Instead what they show is that you can't get from a circle to the same circle with the opposite sign. That is, you can't flip the circle over, just like linked circles here in 3D. The point is that there is a restriction on the movement of the circle, therefore the circles are linked. But in this context equality doesn't care about sign so you CAN move continuously without intersecting the forbidden circle between any two great circles that don't intersect the forbidden circle

.

It took me a while to understand what you are saying. Yes. You can't rotate one circle in the link so that it is flipped without banging into the forbidden circle.

What I tried to do here was to show various ways to detect and define linkage. The definition which seems most intuitive, to me at least, and which generalizes to multiple linkings is the counting of oriented crossings of one loop through a disk that the other loop bounds. This oriented sum is called the linking number of the two loops.

In three space, there is the Gauss integral method which calculates the linking number of smooth closed loops. I'm still not sure why this works. Reference: Flanders, Differental Forms with Applications to the Physical Sciences

For smooth maps of the 3 sphere to the 2 sphere, such as the bundle projection map of the Hopf fibration, there is an integral that detects linking . This integral works for any mapping of the 3 sphere to the 2 sphere, not just the Hopf map. It is an integral of a 3 -form over the 3 sphere. It's value is always an integer and equals the linking number of pairs of closed loops in the 3 sphere that are the inverse images of pairs of regular values of the map. (I'm also not sure why this one works). This integral is called the Hopf invariant. It is what is called a homotopy invariant. That is: two maps of the S^3 into S^2 can be continuously deformed into each other over a finite time interval, if and only if the integral is the same. The integral for the Hopf fibration is 1 which is the linking number of any two fibers. The integral for a map that can be continuously deformed to the constant map is zero. So the Hopf map is not homotopic to the constant map. A reference for the the Hopf invariant is Bott and Tu Differentiable forms in Algebraic Topology

Hopf was interested in the mathematical question of whether there were homotopically non trivial (not homotopic to the constant map) mappings of S^3 into S^2. His discovery of the Hopf fibration was the first known non-trivial example. This led to a huge program to classify homotopy classes of maps of spheres into spheres of any dimension. In higher dimensions, inverse images of regular values of maps of S^(n+k) into S^n are submanifolds of dimension k. Pontryagin and Thom came up with a generalized notion of linking for these submanifolds. It generalizes the idea that two circles that bound a twisted cylinder are linked while two circles that bound a regular cylinder are not linked . And for that matter, two circles that bound an n times twisted cylinder are n-times linked. A refernce is Milnor's Topology from the Differentiable Viewpoint.