Hopf fibration of 3-sphere

[cianfa72]

TL;DR

About the Hopf fibration of the 3-sphere

As shown by this animation, the fibers of the Hopf fibration of the 3-sphere are circles (click on a point on the sphere to visualize the associated fiber). As far as I understand, they never intersect and their union is the 3-sphere itself.

I'd be sure whether the circles in the animation are given by stereographic projection of the 3-sphere from a point, say the "equivalent" of the $S^2$ north-pole. Assuming the viewpoint of 3-sphere defined by its embedding in $\mathbb C^2$ as "ambient" space, the 3-sphere is given as the locus of complex pairs $(z,w)$ such that $|z|^2 + |w|^2 = 1$. Then one can employ, for instance, the stereographic projection from the complex point $(0,i)$ that is $(0,0,0,1)$ using the natural identification $\mathbb C^2 \cong \mathbb R^4$. Such a stereographic projection maps the 3-sphere on the 3D space $v=0$ where $z = x +iy, w = u + iv$ including $\infty$.

Did I understand it correctly ? Thanks.

 

[cianfa72]

BTW, the fibers $S^1$ of the Hopf fibration are supposed to be what R. Penrose calls Clifford parallels in his book "The Road to Reality".

 

[Hornbein]

I don't really understand it but I think stereographic projection introduces much distortion that we don't see in the Hopf fibration. The Hopf fibration is a bijective map from points on the 2-sphere to a special class of great circles on the 3-sphere. The way I possibly understand it is that we fix a more or less arbitrary circle on the 4-sphere. Then any other great circle in that class has two angles relative to that circle. These two angles correspond to longitude and latitude on the 2-sphere. Note that the longitude angle is from -pi to pi, the latitude angle is from -pi/2 to pi/2. In general if we have two 2-planes that intersect at a point then their relative orientation can be characterized by two angles. The point is the center of the 3-sphere and the intersection of each plane containing that point with the 3-sphere is a great circle in that Hopf class.

I don't like their saying "the circles are the same distance apart." What they are trying to say is that if A and B are circles in the same Hopf fibration then each point in circle A has the same distance to the point nearest to it on circle B. This minimum distance varies from 0 to pi/2 depending on the choice of A and B. Relations are pretty much linear which is why I question whether stereographic is involved.

I also often see it said that the great circles are linked, but I don't understand what they mean by this.

I hope I'm not leading you astray.

 

[cianfa72]

I don't like their saying "the circles are the same distance apart." What they are trying to say is that if A and B are circles in the same Hopf fibration then each point in circle A has the same distance to the point nearest to it on circle B. This minimum distance varies from 0 to pi/2 depending on the choice of A and B.

Yes, this is the same for parallel straight lines in Euclidean space. The minimum distance between parallels A and B depends on our choice of which A and B are supposed to be.

 

[Hornbein]

From Wikipedia

Stereographic projection of the Hopf fibration induces a remarkable structure on R3, in which all of 3-dimensional space, except for the z-axis, is filled with nested tori made of linking Villarceau circles. Here each fiber projects to a circle in space (one of which is a line, thought of as a "circle through infinity"). Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product of two circles.) These tori are illustrated in the images at right. When R3 is compressed to the boundary of a ball, some geometric structure is lost although the topological structure is retained (see Topology and geometry). The loops are homeomorphic to circles, although they are not geometric circles.

So they are not saying that fibers are linked. They are saying that the images of the fibers in the stereographic projection are linked.

 

[cianfa72]

So they are not saying that fibers are linked. They are saying that the images of the fibers in the stereographic projection are llinked.

No, being pairwise linked (as in the image with keyrings in wikipedia link) is invariant under stereographic projection.

Happy 2026 !

 

[Hornbein]

No, being pairwise linked (as in the image with keyrings in wikipedia link) is invariant under stereographic projection.

Happy 2026 !

I say that circles (1-spheres) can't be linked in 4-space. In 4-space circles can always be separated, which to me means they aren't linked. 2-spheres can be linked in 4-space, (N-2)-spheres in N-space with N>2. An (N-2)-sphere is functionally a ring.

Hmm, can 0-spheres be link in 2-space? No, but they can be linked in 1-space.

 

[cianfa72]

I say that circles (1-spheres) can't be linked in 4-space. In 4-space circles can always be separated, which to me means they aren't linked.

True but not relevant. What matters is the (intrinsic) $\mathbb S^3$ topology. In this topology (homeomorphic to two 3-balls glued along their common spherical boundary) the fibers of the Hopf fibration can never be unlinked/separated.

 

[Hornbein]

True but not relevant. What matters is the (intrinsic) $\mathbb S^3$ topology. In this topology (homeomorphic to two 3-balls glued along their common spherical boundary) the fibers of the Hopf fibration can never be unlinked/separated.

I confess that this is completely over my head. I also know from experience that I am never able to find explanations of such things that I am able to understand.

When I was in math graduate school I came across a survey book on research on the Riemann hypothesis. I couldn't understand even the first page.

 

[cianfa72]

I confess that this is completely over my head. I also know from experience that I am never able to find explanations of such things that I am able to understand.

Try thinking of $\mathbb S^3$ in terms of the space obtained by gluing the spherical boundary of two 3-balls in 3D space. This space is homeomorphic to $\mathbb S^3$ and a fiber of the Hopf fibration is basically a circle passing through two antipodal points on the common/glued sphere.