Visualizing the Hopf Fibration

This is a guest post by Patrick Gillespie, who is currently a 2nd year Ph.D. student at the University of Tennessee Knoxville.

The Hopf map h:S^3\to S^2 is a classical example of a non-trivial fiber bundle. There are many great visualizations of the Hopf map which depict the fibers of various points in S^2. For example, Niles Johnson has a particularly good video which does this. These visualizations give a great deal of insight into the structure of the Hopf map, but I still felt as though I couldn’t see the Hopf map. So in this blog post, we will take an alternative approach to visualizing the Hopf map where we regard it as a loop of maps from the 2-sphere to itself. The identity map S^2\to S^2, whose homotopy class generates \pi_2(S^2) can be viewed as a loop of maps S^1\to S^2 and the result can be animated as shown below. We will do the same with the Hopf map viewed as a loop of maps S^2\to S^2. We’ll also take a look at another map, which is homotopic to the Hopf map, but visually simpler to understand. The post will conclude with a discussion of the J-homomorphism and how it can be used to visualize maps representing generators of \pi_{n+1}(S^n) for n\geq 2.

The identity map S^2\to S^2 expressed as a loop in \Omega S^2

First, let \Sigma X and \Omega X denote the reduced suspension and loop space of a pointed space X, let M_*(X,Y) be the space of pointed maps between two pointed spaces X and Y, and let M(X,A;Y,B) be the space of maps of pairs (X,A)\to (Y,B). By the loop-space suspension adjunction, recall that M_*(\Sigma X, Y) and M_*(X,\Omega Y) are naturally homeomorphic, and this homeomorphism induces an isomorphism of groups of pointed homotopy classes [\Sigma X,Y]_*\cong [X,\Omega Y]_*. Since S^3\cong\Sigma^2 S^1, we have that

M_*(S^3,S^2)\cong M_*(S^1,\Omega^2 S^2).

Thus we may identify any map S^3\to S^2 with a loop in \Omega^2 S^2, or equivalently, a loop in M(D^2,\partial D^2;S^2,*) where D^2 is the closed unit disk, D^2=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq1\}.

To express the Hopf map this way, we will first identify the Hopf map with a map h':D^2\times [0,1]\to S^2. So let C=D^2\times[0,1] and find a homeomorphism \psi between the interior of C and \mathbb{R}^3. By identifying \mathbb{R}^3 with S^3\setminus *, we may extend \psi to a map \Psi:(C,\partial C)\to (S^3,*). If h:S^3\to S^2 denotes the Hopf map, let h'=h\circ\Psi:(C,\partial C)\to (S^2,*). We may identify h' and h via the homeomorphism

M_*(S^3,S^2)\cong M(C,\partial C;S^2,*)

induced by \Psi. Now for t\in[0,1], let h'_t denote the restriction of h' to the disk D^2\times\{t\}\subset C. Then h'_t\in \Omega^2 S^2 for each t\in[0,1], and t\mapsto h'_t defines a loop [0,1]\to \Omega^2 S^2.

Below is the animation of the Hopf map represented this way. At time t, the left animation simply shows the domain of h'_t, regarded as a subspace of C. The right animation shows the image of h'_t:(D^2,\partial D)\to (S^2,*) at time t. The basepoint * of S^2 is (0,0,-1).

The Hopf map represented as a loop of maps (D^2,\partial D^2)\to (S^2,(0,0,-1)). The left animation depicts the domain of each map in the loop as a subspace of C=D^2\times [0,1]. The right animation depicts the image of each map in the loop.

It is a bit hard to keep track of what is going on in this animation since the maps S^2\to S^2 are not injective. To partially fix this, we can instead find a loop of maps S^2\to \mathbb{R}^3\setminus 0 which, after composing with the projection \mathbb{R}^3\setminus 0\to S^2, is homotopic to h'. Very briefly, if S^1_a=\{(x,y)\in\mathbb{R}^2: x^2+y^2=a^2\}, the idea will be to map circles of the form S^1_a\times \{t\} in C via the Hopf map to a sphere with radius r\in[1,2] and centered at (0,0,1-r)\in\mathbb{R}^3, where r is a continuous function of both a and t satisfying a couple key properties. Importantly, if t=0,1 or if a=1, then we want r=2 in order to guarantee that all of the maps S^2\to\mathbb{R}^3\setminus 0 share a consistent basepoint: (0,0,-3) in this case. The resulting loop of maps S^2\to\mathbb{R}^3\setminus 0 is animated below. We also include a point at the origin in the animation for reference.

A loop of maps (D^2,\partial D^2)\to (\mathbb{R}^2\setminus 0, (0,0,-3)) representing a map which is homotopic to the Hopf map after composing with the projection \mathbb{R}^2\setminus 0\to S^2.

This animation makes it a little easier to see how the Hopf map “loops” around the sphere. In particular, notice that the blue potion does twist around S^2 completely but that the red arcs only trace out disks. To simplify things even further, consider the following animation where both the red and blue arcs are “straightened out.”

Another loop of maps (D^2,\partial D^2)\to (\mathbb{R}^2\setminus 0, (0,0,-3)) which again represents a map that is homotopic to the Hopf map after composing with the projection \mathbb{R}^2\setminus 0\to S^2.

Let g:(C,\partial C)\to (S^2,*) be the map represented by the above animation. An explicit homotopy between g and h' was a by-product of constructing g and can be found here. The description of the homotopy is tedious so I’ll leave it in the attached pdf. For some intuition as to why the two maps should be homotopic, one can check that, for g, the fibers of a general pair of points form linked topological circles. The fibers of (0,0,-1), (0,0,1), (0,-1,0), and (0,1,0) are shown below in red, blue, green, and purple respectively (\partial C is not included in the picture of the fiber of (0,0,-1)).

The fibers of four select points in S^2 under the map g:(C,\partial C)\to (S^2,(0,0,-1))

Note that while the fiber of (0,0,1) is not a circle, it is a cylinder and thus homotopic to a circle. Also, while the fiber of (0,0,-1) is the union of the \partial C with red line shown, the image of this fiber under the identification (C,\partial C)\to (S^3,*) is homeomorphic to a circle.

Before we continue, let’s quickly establish some notation that will help us break down, not just loops, but some paths in \Omega^2 S^2. For paths \alpha,\beta:[0,1]\to X such that \alpha(1)=\beta(0), let \alpha\cdot\beta denote the concatenation of \alpha and \beta. Let \alpha^{-} denote the reverse of \alpha, i.e. \alpha^{-}(s)=\alpha(1-s). Finally, if \alpha is a loop, let \alpha^n be the n-fold concatenation of \alpha with itself, where \alpha^{-n}=(\alpha^{-})^n, and let \alpha^0 be the constant loop at \alpha(0).

Viewing g as a map [0,1]\to \Omega^2 S^2, we may write g=\alpha\cdot \gamma\cdot \alpha^- where \alpha and \gamma are simply g restricted to time intervals [0,1/3] and [1/3,2/3] respectively. Because g can be represented as a genuine path-conjugate, it follows that the n-th power of [g]\in\pi_1(\Omega^2 S^2)\cong \pi_3(S^2) is [\alpha\cdot \gamma^n\cdot\alpha^-]. If we were to animate \gamma^n, we would see the blue sphere rotate n times (clockwise or counterclockwise depending on the sign of n.) The takeaway here is that we have a visual correspondence between loops of rotations S^2\to S^2 which fix a point, which we can think of as representatives of elements of \pi_1(SO(2)), and powers of the Hopf map. We are seeing the J-homomorphism in action!

The J-homomorphism is really a collection of homomorphisms J_{k,n}:\pi_k(SO(n))\to \pi_{n+k}(S^n) originally defined by Whitehead as follows. An element f:\mathbb{R}^n\to\mathbb{R}^n of SO(n) restricts to an unbased map S^{n-1}\to S^{n-1} and this defines a map \theta:SO(n)\to M(S^{n-1},S^{n-1}). Then given \beta:S^k\to SO(n) representing an element of \pi_k(SO(n)), the composition \theta\circ \beta:S^k\to M(S^{n-1},S^{n-1}) is equivalent to a map S^k\times S^{n-1}\to S^{n-1} by the exponential law. Applying the Hopf construction to this, we obtain a map \gamma: S^k*S^{n-1}\to \Sigma S^{n-1}. Since S^k*S^{n-1}\cong S^{n+k} and \Sigma S^{n-1}\cong S^n, we define J_{k,n} by setting J_{k,n}([\beta])=[\gamma].

However, there is a equivalent definition of the J-homomorphism which will be more useful for our purposes. Consider the map \phi:SO(n)\to M_*(S^n,S^n)\cong \Omega^nS^n in which an element f:\mathbb{R}^n\to\mathbb{R}^n of SO(n) is sent to a based map \widehat f:S^n\to S^n by taking one-point compactifications (and where the basepoint is \infty). Then we can equivalently define the J-homomorphism as the map \phi_*:\pi_k(SO(n))\to\pi_k(\Omega^n S^n)\cong\pi_{n+k}(S^n) induced by \phi.

We glossed over some subtleties in the second definition. If we wish to work on the level of representatives, a map S^k\to SO(n) is sent to S^k\to \Omega^nS^n for which the basepoint of S^k is sent to the identity map in \Omega^nS^n rather than the constant map. Hence we cannot immediately apply the loop-space suspension adjunction to obtain a map S^{n+k}\to S^n as we might wish. Instead we should compose \phi with a homotopy equivalence \psi:\Omega^nS^n\to \Omega^n S^n which sends the path component of the identity map to the path component of the constant map (for example, we could take \psi to be the homotopy equivalence induced by multiplying by an element of \Omega^n S^n of degree -1). Then given \beta:S^k\to SO(n), the composition \psi\circ \phi\circ \beta:S^k\to \Omega^nS^n maps to the path component of the constant map. Finally, through a change of basepoints, we obtain S^k\to \Omega^nS^n in which the basepoint of S^k is sent to the constant map. We can then identify this with a map S^{n+k}\to S^n which represents the image of [\beta] under the J-homomorphism.

I’d like to draw our attention back to the map g=\alpha\cdot\gamma\cdot\alpha^-, where as before, \alpha and \gamma are g restricted to [0,1/3] and [1/3,2/3] respectively. At each time t, the regions shaded red and blue in the animation of g correspond to the northern and southern hemispheres of S^2 in the domain of g(t):S^2\to S^2, which we will denote S^2_+ and S^2_- respectively. Now define \gamma':[0,1]\to \Omega^2 S^2 by setting \gamma'(t)=\gamma(t)|_{S^2_-}, that is, we restrict \gamma(t) to the blue hemisphere at each time t. Strictly speaking, \gamma' is a map [0,1]\to M(S^2_-,\partial S^2_-;S^2,*), but this is of course equivalent to a map [0,1]\to\Omega^2 S^2. Then from the animation of g, we see that \gamma' factors as the composition \phi\circ\beta where \beta represents a generator of \pi_1(SO(2)) and \phi:SO(2)\to \Omega^2 S^2 is the map used in the definition of the J-homomorphism. Note that extending \gamma'(t) to \gamma(t) amounts to composing \gamma' with a homotopy equivalence \Omega^2 S^2\to \Omega^2 S^2 which maps the path component of the identity map to that of the constant map. Finally, g is the result of conjugating \gamma by the path \alpha. Not only does this show that [g] is the image of a generator of \pi_1(SO(2)) under the J-homomorphism, but it also shows how we could have arrived at the visualization of g through our second definition of the J-homomorphism.

With this in mind, we will now attempt to visualize the images of the homomorphisms J_{1,n}:\pi_1(SO(n))\to \pi_{n+1}(S^n) for all n\geq 2. It is classical that the J-homomorphism is an isomorphism in these cases, hence this will allow us to visually understand generators of \pi_{n+1}(S^n) for all n\geq 2. In order to do this, we first present an alternative way of visualizing the map g:[0,1]\to \Omega^2 S^2 which will be much easier to generalize. At each time t, we may identify the domain of g(t):S^2\to S^2 with the union of S^2_+ and S^2_- glued together along their boundaries in the obvious way. We may also regard the codomain of g(t):S^2\to S^2 as the quotient of D^2 in which the boundary \partial D^2 is collapsed to a single point. Then we may visualize g as shown below through two side-by-side animations where, at time t, the left animation shows the image of g(t) restricted to the northern hemisphere S^2_+, and the right animation shows the image of g(t) restricted to the southern hemisphere S^2_-. In the animation, the dotted circle represents the boundary \partial D^2 which we regard as a single point in S^2.

An alternative visual representation of g in which at each time t, the left and right animations depict the images of g(t):S^2\to S^2 when restricted to S^2_+ and S^2_- respectively and where the codomain S^2 is regarded as the quotient D^2/\partial D^2. The red and blue lines represent the images of the coordinate axes of S^2_+ and S^2_- viewed as unit disks D^2\subset\mathbb{R}^2 and simply serve to help see when a rotation is occuring.

What you’re seeing here is simply an alternative way to visualize the previous animation (of g). Technically, the above animation depicts a map \alpha'\cdot\gamma\cdot(\alpha')^-, homotopic to g, but for which the conjugating path \alpha' differs from \alpha very slightly. You have to look pretty closely to observe the difference between \alpha and \alpha '. As time progresses, both \alpha and \alpha ' pull the equator up toward the north pole. However, the way in which \alpha does this is not perfectly symmetric – at the start and end of the animation you can see a little more red than blue – whereas the expansion and shrinking of disks in this animation using \alpha ' is symmetric. This difference is certainly not homotopically significant.

We can now generalize this visualization to loops g_n:[0,1]\to \Omega^n S^n representing generators of \pi_{n+1}(S^n) for all n\geq 2. For example, below is an animation in the same style as above, but now representing a map g_3:[0,1]\to \Omega^3 S^3. If S^3_+ and S^3_- are two hemispheres of S^3 (each of which is homeomorphic to a closed 3-ball,) at each time t, the image of g_3(t) restricted to S^3_+ and S^3_- is shown on the left and right respectively.

A visual representation of g_3:[0,1]\to \Omega^3 S^3 in which at each time t, the left and right animations depict the images of g_3(t):S^3\to S^3 when restricted to S^3_+ and S^3_- respectively. The red, blue, and green lines represent the images of the coordinate axes of S^3_+ and S^3_- viewed as unit 3-disks D^3\subset\mathbb{R}^3.

Analogous to how we saw that [g] was the image of a generator [\beta]\in\pi_1(SO(2)) under the J-homomorphism, one can similarly check that the [g_3] is the image of a generator [\beta_3]\in\pi_1(SO(3)) under the J-homomorphism. Hence [g_3] indeed generates \pi_4(S^3).

Although we cannot animate the loops g_n:[0,1]\to \Omega^n S^n for n>3 as we have run out of spatial dimensions to work with, there is a clear pattern, which provides at least some visual understanding of the elements of \pi_{n+1}(S^n).

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