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

What is a semicovering map?

I’ve heard twice in the past year from folks who study non-Archimedian geometry and have found connections to “semicoverings,” which are a generalization of covering maps used in wild topology. The questions I received had me revisiting the basics and motivated this post.

There is a massive history and literature of attempts to generalize covering space theory. The goal of such generalizations is usually to expand the incredibly useful symbiotic relationship between topology (covering spaces) and algebra (the fundamental group) that covering space theory provides. One can distinguish spaces using fundamental groups and, on the flip-side, one can also prove algebraic results by realizing algebraic objects as invariants of a space, e.g. a group realized as the fundamental group of some space that topologically “encodes” the structure of the group. Generalizations of covering space theory often result in an even richer symbiotic relationship between more complicated spaces (maybe failing to be locally path-connected or semilocally simply connected) or space-like objects and more intricate/enriched algebraic objects (like pro-groups, topological groups, etc.) In my experience, the tricky part of this business is starting with an intended application and then finding just the right generalized notion of “covering map” that does exactly what you want it to do. This can require a lot of fussing around with the hierarchy of properties that covering maps enjoy and seeking out the appropriate combination.

Background

“Semicovering maps” and their classification are something that I worked on at the end of grad school in 2011. I had spent a lot of time studying topological versions of homotopy groups for my thesis, including the “tau topology,” and I had my eye on using it to fill in some long-stand gaps in topological group theory. If only I had the right generalization of covering maps to do the job! It all feels “obvious” in hindsight but back then it was not so easy to do and it took a lot of wrong definitions to sort out one that ended up working. Even then, the original definition of semicoverings in [1] was not the simplest possible one. Honestly, I can’t remember if I made any attempt to simplify it back in 2011. I just wanted a working classification and toward that end I found a combination of properties that made it work. Not too much later, several folks realized that the definition could be simplified.

In this post, I’ll introduce semicoverings using the refined definition. Perhaps in a future post, I’ll discuss their classification and applications. Spoiler: semicoverings over a space $X$ are classified by open subgroups of a natural topologized version of the fundamental group $\pi_1(X,x_0)$. Since “nice” spaces have discrete fundamental groups, this contains the usual classification of covering spaces as a special case.

Much of the original work in [1] is done in categorical language using topologically enriched groupoids. I love groupoids but some of my colleagues don’t and this groupoid-heavy approach was partly the product of a referee’s preferences. I was but a baby-child of a mathematician and wanted this published so I didn’t push back…what can I say?. Anyway, the fundamental group version of the classification appears at the very end of the paper anyway.

Getting to the point…What is a semicovering?

We’ll say that a map $p:E\to X$ has unique lifting of all paths rel. basepoint if whenever $\alpha:[0,1]\to X$ is a path and $e\in p^{-1}(\alpha(e))$, there exists a unique path $\beta:[0,1]\to E$ such that $\beta(0)=e$ and $p\circ\beta=\alpha$.

Definition: A map $p:E\to X$ is a semicovering map if $p$ is a local homeomorphism and has unique lifting of all paths rel. basepoint. We refer to $E$ as a semicovering space of $X$.

Notice that $p$ is NOT defined to be locally trivial like a covering map is. So although a semicovering must have discrete fibers (because it is a local homeomorphism) it will not typically be a fiber bundle.

Two-of-three: in general, the composition of a two coverings maps is not always a covering map. However, compositions of local homeomorphisms are local homeomorphisms. Also, maps with unique lifting of all paths rel. basepoint are closed under composition. Therefore, semicoverings are closed under composition. In fact, semicoverings satisfy the two-of-three property that if $f\circ g=h$ where two of the maps $f,g,h$ are semicoverings, then the third one must also be a semicovering (provided the spaces involved are path-connected).

Every covering map is a semicovering map: this is true essentially by choice of our definition. Every covering map is a local homeomorphism and the usual theory shows that covering maps have unique lifting of paths rel. basepoint. It’s not really obvious at this point but if you have a space $X$ with the usual conditions (path-connected, locally path-connected, and semilocally simply connected), then any semicovering $p:E\to X$ where $E$ is connected, will be a true covering map. So, the situations where semicoverings are intended to be useful in non-trivial ways are related to locally complicated spaces like the earring space, other locally path-connected spaces, and even many non-locally path-connected spaces.

Example: The two-of-three property immediately tells you how to find examples of semicoverings that are not covering maps: take two covering maps whose composition is not a covering map. The composition will be a semicovering but not a covering map. There are more extreme examples in [1] that can’t be realized this way, but this is a good start.

A composition of two covering maps, which is a semicovering map but not a covering map. The lower map “unwinds” the outermost circle of the earring. The upper map is a 2-fold covering where as you look both to the left and right you see more and more circles based at fiber points becoming arcs connecting the two fiber points.

What is the difference between a covering and a semicovering over the earring space? Let $\mathbb{E}$ be the earring space with wild point $b_0$ and $n$-th circle $C_n$. If you have a true covering map $p:E\to \mathbb{E}$, you can find some neighborhood $U$ of $b_0$ which is evenly covered by $p$. Now all but finitely many of the circles of $\mathbb{E}$, will be contained in $U$. In particular we have a smaller copy of $\mathbb{E}_{\geq N}=\bigcup_{n\geq N}C_n$ in $U$. Since $U$ is evenly covered by $p$, we have $p^{-1}(U)$ decomposing as a disjoint collection of open neighborhoods $\coprod_{k\in K}V_k$ where $p$ maps $V_k$ homeomorphically onto $U$. In particular, we have a copy of $\mathbb{E}_{\geq N}$ in each $V_k$ that gets mapped homeomorphically onto $\mathbb{E}_{\geq N}$. In this way covering spaces of $\mathbb{E}$ are “uniformly wild” by which I mean there is a copy of $\mathbb{E}_{\geq N}$ hanging around each point in the wild fiber. Said another way, there must be some uniform upper bound $N$ on the number of circles $C_n$ can lift to an arc in $E$.

Now, a semicovering map $p:E\to \mathbb{E}$ is still a local homeomorphism but there may not be a neighborhood of $b_0$ that is evenly covered by $p$. Being a local homeomorphism means that at each wild fiber point $e\in p^{-1}(b_0)$ there will be exist natural number $m_e$ and a copy of $\mathbb{E}_{\geq m_e}$ attached at $e$. However, like in the illustrated example, $F=\{m_e\in\mathbb{N}\mid e\in p^{-1}(b_0)\}$ may not be bounded above (see the image below). In fact $p$ will be a true covering if and only if $F$ is bounded above.

A non-trivial semicovering over the earring space, which is not a traditional covering map. Each step that moves you one edge away from the given basepoint, unwraps the next circle of the earring.

This informs how we should think about semicovering spaces in general. We should still think of a semicovering space $E$ over $X$ as looking like $X$ locally but as we “unwind” the path-homotopy classes of $X$ to obtain $E$, we are allowed to unwind smaller and smaller paths as we move further away from a fixed basepoint $e\in E$.

Continuous Lifting

What’s the most interesting property of semicoverings? I think it’s that even without local triviality, paths lift not only uniquely but also continuously. This is what makes the connection to topologized fundamental groups possible.

Let $P(X,x_0)$ be the space of paths $\alpha:[0,1]\to X$ with $\alpha(0)=x_0$ equipped with the compact-open topology. Every based map $f:(X,x)\to (Y,y)$ induces a continuous function $P(f):P(X,x)\to P(Y,y)$, $P(f)(\alpha)=f\circ\alpha$.

Using path spaces provides a nice way to understand lifting: A map $p:E\to X$ has unique lifting of paths rel. basepoint if and only if for every $e\in E$, the map $P(p):P(E,e)\to P(X,p(e))$ is a bijection.

Lemma (Continuous Lifting): If $p:E\to X$ is a semicovering, then for every $e\in E$, the induced map $P(p):P(E,e)\to P(X,p(e))$ is a homeomorphism.

Proof sketch. Set $x=p(e)$. A subbasic set in $P(E,e)$ is of the form $\langle K,U\rangle=\{\alpha\in P(E,e)\mid \alpha(K)\subseteq U\}$ where $K\subseteq [0,1]$ is compact and $U$ is open in $E$. Let $K_{n}^{j}=\left[\frac{j-1}{n},\frac{j}{n}\right]$. With some basic general topology arguments, you can show that there is a basis for the topology of $P(E,e)$ consisting of open sets of the form $\mathcal{U}=\bigcap_{j=1}^{n}\langle K_{n}^{j},U_j\rangle$ for open sets $U_1,U_2,\dots, U_n\subseteq E$. One should think of $\mathcal{U}$ as a finite set of instructions determining how the paths it contains can proceed through $E$.

It’s tempting to think that $P(p)$ will map $\mathcal{U}$ onto the open set $\bigcap_{j=1}^{n}\langle K_{n}^{j},p(U_j)\rangle$ in $P(X,x)$. But this doesn’t exactly work out because of how the intersections might overlap. Therefore, we need to define

$\mathcal{V}=\bigcap_{j=1}^{n}\langle K_{n}^{j},p(U_j)\rangle\cap \bigcap_{j=1}^{n-1}\langle \{\frac{j}{n}\},p(U_{j}\cap U_{j+1})\rangle.$

which is still a basic open set in $P(X,x)$. With this set defined, showing that $P(p)(\mathcal{U})=\mathcal{V}$ requires a direct set-inclusion argument.

Corollary: If $p:E\to X$ is a semicovering map and $H:(D^2,(1,0))\to (X,x)$ is a map from the closed unit disk, then for every $e\in p^{-1}(x)$, there is a unique map $\widetilde{H}:(D^2,(1,0))\to (E,e)$ such that $p\circ\widetilde{H}=H$.

Proof. Recall that a map $f:(D^2,(1,0))\to (Y,y)$ may be identified uniquely with a loop $F:[0,1]\to P(Y,y)$ using exponential properties of spaces. Specifically $F(t)$, $0 is the path given by restricting $f$ to the line from $(1,0)$ to $(\cos(2\pi t),\sin(2\pi t))$. Therefore, may view $H$ is a loop $h:[0,1]\to P(X,x)$. Recall that $P(p):P(E,e)\to P(X,x)$ is a homeomorphism. Therefore, $P(p)^{-1}\circ h:[0,1]\to P(E,e)$ is a loop. Going backward in the adjunction, have a map $\widetilde{H}:(D^2,(1,0))\to (E,e)$ that satisfies $p\circ \widetilde{H}=H$. $\square$

With unique lifting of paths and path-homotopies established, we can conclude that nearly all of the lifting properties that covering maps enjoy follow as well. A little less obvious is that every semicovering is a Hurewicz fibration (see Theorem 7.5 in [2]). Thus, semicoverings fit snuggly between the two:

Covering map $\Rightarrow$ Semicovering map $\Rightarrow$ Hurewicz fibration with discrete fibers.

I don’t know of a Hurewicz fibration with discrete fibers that is not a semicovering map but I expect that one exists. Think you can find one?

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

The Griffiths twin cone and the harmonic archipelago have isomorphic fundamental group (Part 3)

We saw in the previous post that the problem of producing our isomorphism is solved provided we can produce a sufficiently large coherent collection of coi triples. But how is this to be accomplished? For example, given a (perhaps quite complicated) word $W \in \textbf{Red}_T$, is there a way to find some $U \in \textbf{Red}_H$ and coi $\iota$ from $W$ to $U$ so that the one-element collection $\{\text{coi}(W, \iota, U)\}$ is coherent? More challengingly, if we have already defined a coherent collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ of coi triples and we are given a word $W \in \textbf{Red}_T$ then can we find $\iota$ and $U \in \textbf{Red}_H$ so that the slightly larger collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is again coherent? And even if we can surmount this challenge for a reasonable coherent collection, might we still fail to produce a sufficiently large coherent collection on account of the fact that

$\beth_T(\text{P-fine}(\{U_x\}_{x \in X})) = \textbf{Red}_T/\langle\langle \textbf{Pure}_T \rangle\rangle$

but

$\beth_H(\text{P-fine}(\{W_x\}_{x \in X})) \neq \textbf{Red}_H/\langle\langle\textbf{Pure}_H\rangle\rangle$.

In other words, we may have exhausted the codomain but have failed to fully extend the homomorphism to have the appropriate domain. The reverse problem could also occur: we could exhaust the codomain before producing the isomorphism.

The last two potential problems are solved by alternately considering the elements of $\textbf{Red}_H$ and $\textbf{Red}_T$, ensuring that no $[[\cdot]]$-classes of words are left out of the homomorphism by a transfinite induction. The addition of “just one more coi” can require a great deal of technical care, and we will attempt to give the big ideas behind the ability to do so. We let $\|W\| = \frac{1}{n +1}$ where $n$ is the smallest subscript on a letter in $W \in \textbf{Red}_H$ (and $\|E\| = 0$) and similarly $\|U\| = \frac{1}{n + 1}$ where $n$ is the smallest second subscript of a letter in the word $U \in \textbf{Red}_T$.

To begin our collection of coi we notice that $\{\text{coi}(W, \iota_W, E)\}_{W \in \textbf{Pure}_H}$ is coherent (each $\iota_W$ is obviously the empty function). So far our collection is countable (since $|\textbf{Pure}_H| = \aleph_0$) and more particularly of cardinality less than $2^{\aleph_0}$. Next one can prove the following (we’ll number lemmas within this post).

Lemma 1. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent and that $\epsilon > 0$.

(1) If $W \in \text{P-fine}(\{W_x\}_{x \in X})$ then we can find $U \in \textbf{Red}_T$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)$ is coherent, and $\|U\| \leq \epsilon$, and $U \not\equiv E$ provided $W \not\equiv E$.

(2) If $U \in \text{P-fine}(\{U_x\}_{x \in X})$ then we can find $W \in \textbf{Red}_H$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)$ is coherent, and $\|W\| \leq \epsilon$, and $W \not\equiv E$ provided $U \not\equiv E$.

The proof of this not-very-surprising lemma uses the fact that changing finitely many pure p-chunks of a word does not change the $[[\cdot]]$ equivalence class. Next we tackle infinitary concatenations of order type $\mathbb{N}$ (and we will need to use the crucial fact that the coi collection is not very large).

Lemma 2. Suppose that $\{ \text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine}(\{W_x\}_{x \in X})$ and we can write $\text{p-index}(W) \equiv \prod_{n \in \mathbb{N}} I_n$ with each $I_n \neq \emptyset$ and $W\upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$, then we can find $U \in \textbf{Red}_{T}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X})$ and we can write $\text{p-index}(U) \equiv \prod_{n \in \mathbb{N}} I_n$ with each $I_n \neq \emptyset$ and $U \upharpoonright_p I_n \in \text{P-fine}(\{U_x\}_{x \in X})$, then we can find $W \in \textbf{Red}_H$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

To prove part (1) we inductively use Lemma 1 (1) to produce a coherent collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W\upharpoonright_p I_n, \iota_n, U_n)\}_{n \in \mathbb{N}}$ so that $U_n \neq E$ and $\|U_{n+1}\| \leq \frac{\|U_n\|}{2}$. Now an obvious candidate for $U$ would be $\prod_{n \in \mathbb{N}}U_n$, and this infinitary concatenation is indeed a word by the requirement $\|U_{n+1}\| \leq \frac{\|U_n\|}{2}$, but it may not be reduced. Therefore we instead will introduce a sequence of words $\{V_n\}_{n \in \omega} \subseteq \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$ with $\|V_n\| = \|U_n\|$ and $|\text{p-index}(V_n)| \in \{1, 2\}$ and so that each concatenation $U_nV_nU_{n + 1}$ is reduced. The ability to make such a selection is guaranteed be the fact that the number of pure elements in $\text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$ is at most $(|X| + |\mathbb{N}|) \cdot \aleph_0 < 2^{\aleph_0}$. The fact that

$U \equiv U_0V_0U_1V_1\cdots$

is reduced uses the fact that each subword $U_nV_nU_{n + 1}$ was reduced (and we allowed $\text{p-index}(V_n)$ to have cardinality either $1$ or $2$ depending on how the word $U_n$ ends and how the word $U_{n + 1}$ begins). The function $\iota$ will be given in the obvious way: $\iota = \bigcup_{n \in \mathbb{N}} \iota_n$ and the tedious check that

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W\upharpoonright_p I_n, \iota_n, U_n)\}_{n \in \omega} \cup \{\text{coi}(W, \iota, U)\}$

is coherent (and therefore so is $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$) uses the fact that $\{V_n\}_{n \in \omega} \subseteq \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$.

The proof for part (2) is somewhat similar: one inductively extends to a larger coherent collection

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W_n, \iota_n, U \upharpoonright_p I_n )\}_{n \in \omega}$

using Lemma 1 (2), but “buffer” words $V_n \in \textbf{Red}_H$ are selected during the induction to be of form $V_n \equiv h_{k_n}^{m_n}$. The sequences $\{k_n\}_{n \in \mathbb{N}}$ and $\{m_n\}_{n \in \mathbb{N}}$ are selected so that for each $n$ we have

$W_nV_nW_{n + 1}V_{n + 1}\cdots \notin \text{P-fine}(\{W_x\}_{x\in X} \cup \{W_n\}_{n \in \mathbb{N}})$

(this selection makes use of the fact that $|X| < 2^{\aleph_0}$).

Another difficult situation arises with concatenations which are of order type $\mathbb{Q}$.

Lemma 3. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine}(\{W_x\}_{x \in X})$ and we can write $\text{p-index}(W) \equiv \prod_{q \in \mathbb{Q}} I_q$ with each $I_q \neq \emptyset$ and $W \upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$ and $I_q$ is a maximal such interval, then we can find $U \in \textbf{Red}_{T}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x \in X})$ and we can write $\text{p-index}(U) \equiv \prod_{q \in \mathbb{Q}} I_q$ with each $I_q \neq \emptyset$ and $U \upharpoonright_p I_n \in \text{P-fine}(\{U_x\}_{x \in X})$ and $I_q$ is a maximal such interval, then we can find $W \in \textbf{Red}_{H}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

For (1) we make a list $\{W_n\}_{n\in \mathbb{N}}$ so that for each $q\in \mathbb{Q}$ exactly one of $W \upharpoonright_p I_q$ or $(W \upharpoonright_p I_q)^{-1}$ appears in the enumeration. As in Lemma 2 we produce a coherent collection

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W_n, \iota_n, U_n )\}_{n \in \omega}$

by inductively using Lemma 1 and the sequence $\{U_n\}_{n \in \mathbb{N}}$ is again selected to satisfy nice properties; for example the values $\|U_n\|$ shrink to $0$ quite rapidly. Now we select two buffer words $V_{n, a}, V_{n, b}$, this time for both the front and tail of the word $U_n$, so that $V_{n, a}U_nV_{n, b}$ is reduced and some other technical properties hold. Now define the word $U \equiv \prod_{q\in \mathbb{Q}} (V_{f(q), a}U_f(q)V_{f(q), b})^{\delta_q}$ where $W \upharpoonright_p I_q \in \{W_{f(q)}^{\pm 1}\}$ and $\delta_q \in \{\pm 1\}$ with $\delta_q = 1$ if and only if $W \upharpoonright_p I_q \equiv W_{f(q)}$. From how cleverly the buffer words were selected, one argues that $U$ is reduced, and a coi $\iota$ is produced from the collection $\{\iota_n\}_{n \in \mathbb{N}}$ in the natural way. Part (2) requires similar modifications as those used in Lemma 2 (2). In both (1) and (2) the ability to select suitably nice buffer words makes essential use of the fact that $|X| < 2^{\aleph_0}$.

Lemma 4. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine(}\{W_x\}_{x \in X})$ then there exists $U \in \textbf{Red}_T$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x \in X})$ then there exists $W \in \textbf{Red}_{H}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

The proof of part (2) is essentially that of part (1), with obvious modifications. For (1) we ask whether there exists a sequence of intervals $\{I_n\}_{n \in \mathbb{N}}$ in $\text{p-index}(W)$ where all $I_n$ have the same minimum or all have the same maximum, $I_n$ is properly included into $I_{n + 1}$, $W \upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$ for all $n$, and $W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n \notin \text{P-fine}(\{W_x\}_{x \in X})$. If such an interval does not exist then we proceed to the next paragraph. If it does exist, then we extend the coi collection so as to include $W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n$ using Lemma 2 (1) (applied to $(W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n)^{-1}$ in case all the $I_n$ have a common maximum) and we once again ask whether such a sequence exists for the new collection. We do this over and over again, taking unions of the previously defined coherent collections at limit ordinals. Using certain parameters to keep track of how many times this process iterates, we deduce that it can only be executed countably many times. Thus we move on to the next step.

If $W$ is in $\text{P-fine}(\{W_y\}_{y\in Y})$, where $\{\text{coi}(W_y, \iota_y, U_y)\}_{y \in Y}$ is the slightly enlarged coi collection, then we produce $U$ and $\iota$ using Lemma 1 (1). Else we can write $\text{p-index}(W) \equiv \prod_{\lambda \in \Lambda} I_{\lambda}$ where $\Lambda$ is infinite dense-in-itself and each interval $I_{\lambda}$ is nonempty and maximal such that $W \upharpoonright_p I_{\lambda} \in \text{P-fine}(\{W_y\}_{y\in Y})$. The set $\Lambda$ may have a maximum and/or minimum, so we let $\Lambda' \subseteq \Lambda$ be the subset excluding such elements. Then $\Lambda' \equiv \mathbb{Q}$ and we use Lemma 4 (1) to extend to a collection, say, indexed by $Y'$, so that $W\upharpoonright_p \bigcup_{\lambda \in \Lambda'} I_{\lambda} \in \text{P-fine}(\{W_y\}_{y \in Y'})$ and by applying Lemma 1 (1) perhaps once or twice (in case we have a maximum and/or minimum in $\Lambda$) we then obtain the $U$ and $\iota$ so that the collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

Now that we are armed with Lemma 4 we can define a suitable collection by induction over $2^{\aleph_0}$. Let $\prec_H$ (respectively $\prec_T$) be a well-ordering of $\textbf{Red}_H$ (resp. $\textbf{Red}_T$) such that each element has fewer than $2^{\aleph_0}$ predecessors. We already have $\{\text{coi}(W_n, \iota_n, E)\}_{m \in \mathbb{N}}$ in our collection, where $\textbf{Pure}_H = \{W_m\}_{m \in \mathbb{N}}$ is an enumeration. Recall that each ordinal $\alpha$ can be expressed uniquely as $\alpha = \gamma + n$ where $\gamma$ is either zero or a limit ordinal and $n \in \mathbb{N}$; in particular each ordinal can be considered either even or odd depending on the number $n \in \mathbb{N}$.

Suppose that we have already defined a coherent collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \zeta}$ for all $\zeta < \alpha$ where $\mathbb{N} \leq \alpha$ is an ordinal below $2^{\aleph_0}$. Then the collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \alpha}$ is coherent (this is easy to check). If $\alpha$ is even then we select $W \in \textbf{Red}_H$ such that $[[W]] \notin \beth_H(\text{Pfine}(\{W_{\beta}\}_{\beta < \alpha}))$ (such a $W$ exists using a cardinality argument) which is minimal under $\prec_H$ and by Lemma 4 (1) we choose suitable $U$ and $\iota$ to coherently extend and write $W_{\alpha} \equiv W$, $\iota_{\alpha} = \iota$, and $U_{\alpha} \equiv U$. In case $\alpha$ is odd we instead select $U \in \textbf{Red}_T$ with $[[U]] \notin \beth_T(\text{P-fine}(\{U_{\beta}\}_{\beta < \alpha}))$ which is minimal under $\prec_T$, use Lemma 4 (2) and extend accordingly. Thus we obtain a larger coherent collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \alpha + 1}$.

Perform the process on all $\alpha < 2^{\aleph_0}$ and it is clear that $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < 2^{\aleph_0}}$ is coherent and

$\beth_H(\text{P-fine}(\{W_{\beta}\}_{\beta < 2^{\aleph_0}})) = \textbf{Red}_H/\langle\langle\textbf{Pure}_H\rangle\rangle$

and similarly

$\beth_T(\text{P-fine}(\{U_{\beta}\}_{\beta < 2^{\aleph_0}})) = \textbf{Red}_T/\langle\langle\textbf{Pure}_T\rangle\rangle$.

The argument is finished.