## How to “topologize” the fundamental group: a primer

The fundamental group, from algebraic topology, is one of the most widely used invariants in mathematics. Topological groups, such as pro-finite groups, Lie groups, ordered groups, etc, also arise in many different areas. So it’s natural to ask, can the fundamental group be given a topology in a meaningful or useful way? The answer, which I think is a resounding “yes,” is intertwined with my own personal story but some of my views have changed over the years. My choice to ask for “a topology” could easily be read the wrong way. I don’t mean to suggest that there could or should be just one such topology. What I really mean to ask is “do some useful topologies exist?” By the way, topologize is a verb I’ll be using that I’ll take to mean “the act of endowing a given set with a topology.”

It turns out that there are many interesting, useful, and functorial topologies that you can put on fundamental groups and I’m going to discuss some of them in a sequence of future posts. If I were to claim that one of these topologies is the “right one” or “best one,” a pragmatic mathematician would respond: right or best for what? This attitude of valuing constructions based on utility instead of aesthetic idealism is something I’ve grown into over the years. Of course, I invite everyone to choose their own favorite $\pi_1$-topology (I have one!). To my surprise, I needed my least favorite recently… there’s a certain topology on $\pi_1$, which I never considered very useful and so never bothered to study it very deeply. But it turned out to be exactly what I needed for characterizing the images of homomorphisms that characterize some previously unknown higher homotopy groups. It’s annoying to find out myself that I had such a wrong opinion about it but also pretty cool!

So in the end, if you want to topologize $\pi_1$, you want to ask yourself: what do you want the extra topological structure you’re adding to $\pi_1$ to remember about your space? Do you want it to remember the covering space lattice of your space? Maybe you want it to capture the shape theoretic properties of your space? Or maybe you want to remember a preferred metric structure rather than just topological information. There are many potentially interesting and applicable choices.

Just to be clear… if you’ve got a manifold $M$ and are hoping for a non-discrete topology on $\pi_1(M,x)$, you’re barking up the wrong tree. The fundamental group does a perfectly fine job on its own for locally contractible spaces. The idea here is to define a topology on $\pi_1(X,x)$ that makes $\pi_1$ a stronger invariant. Using an algebraically defined topology, e.g. by way of some kind of group completion, will not do this. Therefore, we want to define topologies, which remember non-trivial local structures of a space $X$, which can’t be “seen” by the algebra. When you have a locally-boring space $X$, all such topologies on $\pi_1(X,x)$ should be discrete.

I’m always surprised at how popular this topic is. My papers on topological $\pi_1$ are, by far, my most read and cited. Perhaps it is becuase it only takes having seen fundamental groups and topological groups separately to become curious about it. Regardless, I hope readers who find themselves here will enjoy this sequence of posts. Most importantly, I hope it will encourage young mathematicans to pursue theoretical curiosities connected to fundamental constructions. Often such investigations lead to surprising and useful advancements.

## An old and partially scandalous background

The idea of topologizing fundamental groups apparently goes way back to Hurewicz in 1935 [3]. In 1950, Dugundji [2] applied Hurewicz’s idea of using open covers of a space $X$ to topologize a group closely related and often equal to $\pi_1(X)$. In particular, Dugundji extended the classification of covering spaces for locally path-connected spaces: covering maps over a path-connected, locally path-connected space $X$ are classified up to equivalence by the open subgroups of $\pi_1(X)$ with Hurewicz’s topology. These results predate shape theory by several decades but, essentially, Hurewicz’s topology is what we’d now call the “Shape Topology.”

In the past 20 years, the literature about topologies on $\pi_1$ has grown enormously. It started to gain popularity following Daniel Biss’ paper The topological fundamental group and generalized covering spaces [1]. It’s important to note Biss’ paper is now retracted because almost none of the statements or proofs in it are correct. It doesn’t even get the earring group correct. I haven’t cited [1] in many years but according to Google, it has about 130 citations (as of 4/25/22), some of which are dated after the retraction. In my view, it’s a great “big idea” paper but that’s about it. With all respect to the author (who is now a politician), I do discourage people from reading [1] because it is so very wrong/misleading. That being said, it is still possible to cite a retracted paper in a responsible way. In particular, it should be cited as being “RETRACTED.” Being informed that a paper is retracted and continuing to cite it without the retraction is a rejection-worthy offense (this has happened!).

## Topological $\pi_1$ and my own story

When I was but a young Ph.D. student, Biss’ paper was still fairly hot news. Experts (at the time I was nowhere near one) knew some things were a bit sketchy but what parts of it were actually true remained unclear. An important claim in [1] was that the topology on $\pi_1(X,x)$ being studied made $\pi_1(X,x)$ into a genuine topological group (with continuous multiplication and inversion). However, the proof used the following tempting statement: If $q:X\to Y$ is a quotient map of topological spaces, then $q\times q:X\times X\to Y\times Y$ is a quotient map.

This tempting statement is extremely false. It’s failure is the cause of many a topologist’s headache since it’s basically equivalent to the disappointing fact that $\mathbf{Top}$ fails to be a Cartesian closed category.

Back in 2009, a friend/fellow grad student was working on a dissertation in math education but chose topology as their secondary field of expertise (a requirement at UNH). My advisor, Dr. Maria Basterra asked this student to give a talk on Biss’ paper. During the talk, Dr. Basterra realized that Biss’ proof of $\pi_1(X,x)$ being a topological group had used the above false statement about products of quotient maps. She printed a copy of the paper, handed it to me, and asked me to investigate if the intended topological group claim was true or if there was a counterexample. I learned later on that a few other mathematicians had independently caught Biss’ mistake but did not know of a counterexample.

Of course, now we know that there are lots of counterexamples. I’ll go into more details in my post on the quotient topology….but that, my friends, is how I wound up writing a dissertation on topologized homotopy invariants and finding a “fix” to Biss’ topology. And here I am now… still study wild topology and having lots of fun doing it.

On the practical/career side of things, I will say that choosing to work in a “niche” field has some ups and downs. The upside is that there are many tractible problems and wide open directions to consider within wild topology. It’s a smaller community, which makes it easier to get noticed. At the same time, the mathematics is really fundamental and is closely tied to so many different areas that it draws plenty enough interest to get published in good journals with a few stubborn exceptions…I won’t name names here. The downside to working in a smaller field is that job prospects do become more difficult. I had to develop my teaching ability first and continue to establish myself in research while holding a full-time teaching position for 5 years. By sticking with it, I was able to eventually get a tenure-track job that suits my professional and personal life very well.

## Where I’m taking this discussion

This post is a primer for a sequence of posts in which I plan to detail the most commonly studied topologies that we might want to put on $\pi_1$. Here are the topologies I plan to write about:

• The Whisker Topology
• The Quotient Topology
• The Tau Topology
• The Shape Topology
• The Uniform Metric Topology
• The Lasso Topology

I’ll insert links for these as I write the posts. This list is not meant to be exhaustive – these are the ones that have established uses. I have heard of some additional topologies, but their utility is not so clear just yet. There is a fundamental groupoid (or enriched groupoid) version for most of these but I don’t really plan to go down that rabbit hole.

## How I learned to stop worrying and love quasitopological groups

The issue around the continuity of operations in $\pi_1(X,x)$ is a subtle one.  One thing to be mentally prepared for, and which I hinted at earlier, is that for some $\pi_1$-topologies, the group operation $\pi_1(X,x)\times \pi_1(X,x)\to \pi_1(X,x)$ will not always be jointly continuous. In fact, for two of the topologies in the list above it will often happen that $\pi_1(X,x)$ can fail to be a topological group. When I was less experienced, I used to be upset about this. Now, I realize it’s just the way it is and what really matters is how we are able to use them. We’re not completely out of luck though. Topologized groups that are “almost” topological groups are a thing and have a substantial literature.

Definition: Let $G$ be a group with a topology and $a\in G$. Let $\lambda_{a}(b)=ab$ and $\rho_{a}(b)=ba$ be the left and right translation by $a$ respectively. The following terms have become fairly standard in topological algebra:

• If $\lambda_{g}$ is continuous for all $g\in G$, we call $G$ a left-topological group.
• If $\rho_{g}$ is continuous for all $g\in G$, we call $G$ a right-topological group.
• If $G$ is both a left- and right-topological group, then we call $G$ as semitopological group.
• If $G$ is a semitopological group and the inversion operation $g\mapsto g^{-1}$ is continuous, then we call $G$ a quasitopological group.
• If the group operation $G\times G\to G$ is continuous, we call $G$ a paratopological group.
• If $G$ is a paratopological group and the inversion operation $g\mapsto g^{-1}$ is continuous, then we call $G$topological group.

Of course, the last of these is probably the one you are the most familiar with. It turns out that many commonly used objects are these “weaker” structures. For example:

1. The space $Homeo(X)$ of self-homeomorphisms of a space $X$ with the compact-open topology is always a quasitopological group but is not always a topological group.
2. Any infinite group with the cofinite topology is a quasitopological group, which is not a topological group.

There are some classical theorems out there that allow you to move up in the list: Ellis’ Theorem states that every locally compact Hausdorff quasitopological group is actually a topological group! For more on these structures, I recommend the massive but helpful  book Topological Groups and Related Structures by

## References:

[1] D. Biss, The topological fundamental group and generalized covering spaces,
Topology and its Applications 124 (2002), 355–371. RETRACTED.

[2] J. Dugundji, A topologized fundamental group, Proc. Nat. Acad. Sci. 36
(1950), 141–143.

[3] W. Hurewicz, Homotopie, homologie und lokaler zusammenhang, Fundamenta
Mathematicae 25 (1935), 467–485.

## When is a local homeomorphism a semicovering map?

In a previous post “What is a Semicovering Map?,” I gave an introduction to semicovering maps. A semicovering is a slight generalization of covering map that becomes particularly relevant when you’re dealing with locally complicated spaces. In particular, a semicovering $p:E\to X$ is a local homeomrphism with unique lifting of all paths (for every path $\alpha:[0,1]\to X$ and point $e\in E$ with $p(e)=\alpha(0)$, there is a unique lift $\beta:[0,1]\to E$ with $\beta(0)=e$). For context, I recommend taking a look at the introductory post (or the original papers [1] and [3]) first to gain intuition for the subtle difference between covering maps and semicovering maps.

A natural question to ask is: how much can we weaken the definition of “semicovering map?” More precisely, is there a weaker condition that allows us to promote a local homeomorphism to a semicovering? I really enjoy the logical investigations that these kinds of questions lead to.

This question comes from the paper [4], which gives an answer similar to the one I’ll give in this post. My interest in this question was piqued by questions of some non-Archimedian geometers, who identified a geometric analogue of semicoverings. I shared some of my suspicions with these geometers and this post is just me writing down the details. Certainly, the authors of [4] deserve the credit for originally answering this question.

Definition: We say a map $p:E\to X$ has the endpoint-lifting property whenever $\alpha:[0,1]\to X$ is a path and $\beta:[0,1)\to E$ is a map satisfying $p\circ \beta=\alpha|_{[0,1)}$, $\beta$ extends (not necessarily uniquely) to a path $\beta:[0,1]\to E$ such that $p\circ\beta=\alpha$.

The endpoint-lifting property should be thought of as a completeness-type property of $E$ relative to $p$: if you have a map $[0,1)\to E$ and it agrees with the start of a true path in $X$, then you can complete it to a true path $[0,1]\to E$. There are some conditions on $E$ (without referring to $p$) that imply this, but these include very strong compactness conditions and do not improve known results.

Example: Let $T=\{0\}\times[-1,1]\cup\{(x,\sin(1/x)\mid 0 be the closed topologist sine curve and let $p:T\to [0,1]$ be the projection onto the x-axis. This map does not have the endpoint-lifting property because we can define continuous map $\beta:[0,1)\to T$ by $\beta(t)=(1-t,\sin(\frac{1}{1-t}))$. Now $\alpha:[0,1]\to [0,1]$, $\alpha(t)=1-t$ is a path and agrees with $p\circ\beta$ on $[0,1)$. However, we cannot assign a value to $\beta(0)$ so that $\beta$ is continuous at $0$. You can have both spaces path conecnted if you take the projection of the Warsaw circle onto the ordinary circle.

Throughout this post, we’ll assume the space $X$ is non-empty and path connected. With this assumption, a map $p:E\to X$ with the endpoint lifting property (or stronger property) will always be surjective.

Theorem: Suppose convergent sequences in $E$ have unique limits and $p:E\to X$ is a local homeomorphism. Then the following are equivalent:

1. $p$ is a semicovering map,
2. $p$ is a Hurewicz fibration,
3. $p$ has the endpoint-lifting property.

Note: the condition that convergent sequences in $E$ have unique limits is formally weaker than being Hausdorff. Sometimes a space with this property is called a “US-space.”

Proof of Theorem. The proof of 1. $\Leftrightarrow$ 2. is Theorem 7.5 in [2]. I’d like to focus on the equivalence of 1. and 3 because this part is, in my view, a little more fun. By definition, a semicovering has “unique lifts of all paths rel. basepoint.” Clearly, this implies the endpoint-lifting property. The direction 1. $\Rightarrow$ 3. follows. The rest of this post will be to prove 3. $\Rightarrow$ 1.

Suppose $p$ is a local homeomorphism with the endpoint-lifting property. We must show that $p$ has unique lifting of all paths. Let $\alpha:[0,1]\to X$ be a path starting at $\alpha(0)=x$ and $p(e)=x$. We must find a unique path $\beta:[0,1]\to E$ such that $\beta(0)=e$ and $p\circ \beta=\alpha$. We proceed similar to how you might prove that closed intervals are compact, that is, by defining a convenient set and analyzing the supremum. Let $$A=\{t\in (0,1]\mid \exists \beta_t:[0,t]\to E\text{ s.t. }\beta_t(0)=e\text{ and }p\circ\beta_t=\alpha|_{[0,t]}\}$$

Existence of path lifts: To show that lifts exist, we must show that $1\in A$. Since $p$ is a local homeomorphism, we can find an open neighborhood $U$ of $e$ that $p$ maps homeomorphically onto the neighborhood $p(U)$ of $x$. Find $s>0$ such that $\alpha([0,s])\subseteq p(U)$. The formula $\beta_s(t)=p|_{U}^{-1}(\alpha(t))$ gives a lift $\beta_s$ latex of $\alpha|_{[0,s]}$. Therefore $s\in A$ and we have $\sup(A)>0$. Moreover, the definition of $A$ also ensures that $A$ is an interval of the form $[0,t)$ or latex $[0,t]$ for $0. Now the assumption that $p$ has the endpoint-lifting property tells us that if if we knew $\alpha|_{[0,t)}$ could be lifted, then we can also lift $\alpha|_{[0,t]}$. Hence, $A$ must actually be a closed interval of the form $[0,v]$ where $v=\max(A)$.

But if $v<1$, we can only lift $\alpha|_{[0,v]}$ to a path $\beta:[0,v]\to E$ and no more. But $p$ is still a local homeomorphism. So we can find an open neighborhood $U'$ that $p$ maps homeomorphically onto the open set $p(U')$. Since $\alpha(v)\in p(U')$, we can find $v such that $\alpha([v,w])\subseteq p(U')$. Like before we define $\beta(t)=p|_{U'}^{-1}(\alpha(t))$ when $t\in [v,w]$. This extends $\beta$ to a map $\beta:[0,w]\to E$ that is a lift of $\alpha|_{[0,w]}$.

However $w>max(A)$; a contradiction. Since $v>0$ and $v<1$ is false, we must have $v=\max(A)=1$. This proves existence.

Uniqueness of path-lifts: To prove uniqueness, suppose $\beta_1,\beta_2:[0,1]\to E$ both satisfy $\beta_i(0)=e$ and $p\circ\beta_i=\alpha$. Let $B=\{t\in (0,1]\mid \beta_{1}|_{[0,t]}=\beta_{2}|_{[0,t]}\}.$

Using the neighborhood $U$ of $e$ (as above) that is mapped homeomorphically to $p(U)$, we can see that we must have $\beta_{1}|_{[0,t]}=\beta_{2}|_{[0,t]}$ for at least some small $t>0$. Hence, $B\neq \emptyset$. Just like $A$, the set $B$ must be an interval containing $0$. What if $B=[0,s)$ for some $0? Then $\beta_{1}|_{[0,s)}=\beta_{2}|_{[0,s)}$ but $\beta_1(s)\neq \beta_2(s)$. Picking a convergent increasing sequence $s_1 in $[0,s]$, we see that $\{\beta_1(s_n)\}$ and $\{\beta_2(s_n)\}$ are equal but converge to distinct limits points $\beta_1(s)$ and $\beta_2(s)$.

However, we have assumed that convergent sequences in $E$ have unique limits. Therefore, $B$ must be a closed interval.

Now we must show $v=\max(B)=1$. Suppose that $v<1$. Then $\beta_{1}|_{[0,v]}=\beta_{2}|_{[0,v]}$. Take a neighborhood $U'$ that maps homeomorphically on the neighborhood $p(U')$ of $\alpha(v)$. Because $p|_{U'}:U'\to p(U')$ is injective, $\beta_1([v,w])$ and $\beta_2([v,w])$ lie in $U'$ for some $v, and $p(\beta_1(t))=p(\beta_2(t))$ for all $v\leq t\leq w$, we must have $\beta_{1}|_{[v,w]}=\beta_{2}|_{[v,w]}$. Thus $w\in B$; a contradiction that $v$ is the maximum. We conclude that $v=1$, i.e. $\beta_1=\beta_2$. $\square$

## References:

[1] J. Brazas, Semicoverings: a generalization of covering space theory. Homology Homotopy Appl. 14, (2012) 33–63. Open Access.

[2] J. Brazas, A. Mitra, On maps with continuous path lifting. Preprint. 2020. https://arxiv.org/abs/2006.03667

[3] H. Fischer, A. Zastrow, A core-free semicovering of the Hawaiian Earring. Topology Appl. 160, (2013) 1957–1967. Open Access.

[4] M. Kowkabi, B. Mashayekhy, H. Torabi, When is a local homeomorphism a semicovering map? Acta Mathematica Vietnamica, 42, (2017) 653-663. https://arxiv.org/abs/1602.07260

## 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 $n\geq 2$.

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 $(0,0,-1)$.

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.

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.”

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 $(0,0,-1)$).

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 $S^2$.

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.

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 $\pi_{n+1}(S^n)$.