## Topologized Fundamental Groups: The Whisker Topology, Part 2

This post is the second on the “whisker topology” on fundamental groups, in a larger sequence of posts on topologized fundamental groups. If you haven’t seen it, you’ll probably want to start with the first post to get the basic definitions and terminology. The topology on $\pi_{1}^{wh}(X,x_0)$ is generated by the sets $B([\alpha],U)=\{[\alpha\cdot\epsilon]\in \pi_1(X,x_0)\mid \epsilon([0,1])\subseteq U\}$ where $U$ is an open neighborhood of $x_0$.

One thing I kind of glossed over earlier is the fact that changing the basepoint can change the whisker topology. To convince you of this, let’s take a look at the following question, which I left as an exercise in the previous post.

## When is the Whisker Topology discrete?

In the first post, we saw that $\pi_{1}^{wh}(X,x_0)$ need not be a topological group but that it is still a homogeneous space. Consequently, if any one-point set is open, including the trivial subgroup, then the space will be discrete. Based on this observation, we know that $\pi_{1}^{wh}(X,x_0)$ will be discrete if and only if some basic neighborhood $B(1,U)$ of the identity contains only the identity element, i.e. $B(1,U)=\{1\}$. This will happen if and only if $[\alpha]=1$ for any loop $\alpha$ in $U$, that is, if any loop in $U$ based at $x_0$ is null-homotopic in $X$. This is precisely the definition of being semilocally simply connected at $x_0$!

Here, we’re using the “based” version of this property: $X$ is semilocally simply connected at $x\in X$ if there exists an open neighborhood of $U$ such that the homomorphism $\pi_1(U,x)\to \pi_1(X,x)$ induced by inclusion is trivial.

Proposition: $\pi_{1}^{wh}(X,x_0)$ is discrete if and only if $X$ is semilocally simply connected at $x_0$.

So the whisker topology only detects wildness only the basepoint. Algebraically, if $x_1\in X$, then there is a basepoint-change (group) isomorphism $\pi_1(X,x_0)\cong \pi_1(X,x_1)$. However, the topologized groups $\pi_{1}^{wh}(X,x_0)$ and $\pi_{1}^{wh}(X,x_0)$ will not always be homeomorphic!

Example: If $y_0$ is a point in the earring space $\mathbb{E}$ other than the wild point, then $\pi_{1}^{wh}(\mathbb{E},y_0)$ will be an uncountable discrete group. But if $x_0$ is the wild point, we saw in the last post that $\pi_{1}^{wh}(\mathbb{E},y_0)$ is not discrete. So these two fundamental groups (with the same space but different basepoints) are isomorphic as groups but not homeomorphic as spaces.

That changing the basepoint can change the whisker topology emphasizes, once again, that this topology really only tells us about the topology at the basepoint.

## How Connected is the Whisker Topology?

One of the interesting things about the whisker topology is the following lemma, which tells us that for a fixed neighborhood $U$ of $x_0$, the basic neighborhoods $B([\alpha],U)$ partition $\pi_{1}^{wh}(X,x_0)$. Recall that $\widetilde{X}$ is the set of homotopy classes of paths starting at $x_0$ and $N([\alpha],U)$ denotes the corresponding basic neighborhoods in $\widetilde{X}$.

Neighborhood Lemma: In $\widetilde{X}$, if $[\beta]\in N([\alpha],U)$, then $N([\alpha],U)=N([\beta],U)$.

Proof. If $[\beta]\in N([\alpha],U)$, write $[\beta]=[\alpha\cdot\epsilon]$ for $\epsilon$ with image in $U$. Then $[\alpha]=[\beta\cdot\epsilon^{-}]$. With $\epsilon$ fixed, we prove both subset inclusions. Given $[\gamma]\in N([\alpha],U)$, write $[\gamma]=[\alpha\cdot\delta]$ for $\delta$ with image in $U$. Thus $[\gamma]=[\beta\cdot(\epsilon^{-}\cdot\delta)]$ where $\epsilon^{-}\cdot\delta$ has image in $U$ (see the illustration below). Thus $[\gamma]\in N([\beta],U)$. This proves $N([\alpha],U)\subseteq N([\beta],U)$. The other inclusion is similar. $\square$

Proof of the Neighborhood Lemma in $\widetilde{X}$.

Corollary: Given an open neighborhood $U$ of $x_0$ in $X$, two sets $B([\alpha],U)$ and $B([\beta],U)$ in $\pi_{1}^{wh}(X,x_0)$ are either equal or disjoint.

Proof. If $B([\alpha],U)$ and $B([\beta],U)$ are not disjoint for loops $\alpha,\beta$ based at $x_0$, then we have $[\alpha\cdot\delta]=[\beta\cdot\epsilon]$ for loops $\delta,\epsilon$ with image in $U$. Then $[\alpha]=[\beta\cdot(\epsilon\cdot\delta^{-})]$ where $epsilon\cdot\delta^{-}$ has image in $U$ and so $[\alpha]\in B([\beta],U)$. Recalling that $B([\alpha],U)=\pi_1(X,x_0)\cap N([\alpha],U)$, it follows from the Neighborhood Lemma that $B([\alpha],U)=B([\beta],U)$. $\square$

This corollary implies that all basic open sets in $\pi_{1}^{wh}(X,x_0)$ are always clopen, that is, the small inductive dimension of $\pi_{1}^{wh}(X,x_0)$ is zero.

Theorem: Whenever $\pi_{1}^{wh}(X,x_0)$ is Hausdorff, it is zero-dimensional (in the sense of small inductive dimension).

Another way to see what’s going on here is to notice that the neighborhoods $B(1,U)$ of the identity element $1$ are all clopen. Therefore, if $\mathscr{U}(x_0)$ is the set of open neighborhoods of $x_0$, then $\bigcap_{U\in\mathscr{U}(x_0)}B(1,U)$ is equal to the closure of the trivial subgroup $\overline{\{1\}}$.

## When is the Whisker Topology Hausdorff?

At this point, we’ve seen that whenever $\pi_{1}^{wh}(X,x_0)$ is Hausdorff, is pretty highly structured – zero dimensional homogeneous spaces are only so complicated. So what would it take for $\pi_{1}^{wh}(X,x_0)$ to be Hausdorff?

Topological group theory tells us that a topological group is completely regular (Tychonoff) if and only if the trivial subgroup is closed. In fact, we have close to the same thing for the whisker topology even though it’s often far from being a topological group.

Theorem: The following are equivalent.

1. The trivial subgroup is closed in $\pi_{1}^{wh}(X,x_0)$,
2. $\pi_{1}^{wh}(X,x_0)$ is Hausdorff,
3. $\pi_{1}^{wh}(X,x_0)$ is $T_3$,

Proof. 3. $\Rightarrow$ 2. $\Rightarrow$ 1. is clear. Also, every zero-dimensional Hausdorff space is $T_3$. So it suffices to show 1. $\Rightarrow$ 2. Suppose the trivial subgroup $\{1\}$ is closed and $[\alpha]\neq [\beta]$ in $\pi_{1}^{wh}(X,x_0)$. Then $[\beta^{-}\cdot\alpha]\neq 1$ and so there exists an open neighborhood $U$ of $x_0$ such that $1\notin B([\beta^{-}\cdot\alpha],U)$. Suppose, to obtain a contradiction, that $B([\alpha],U)$ and $B([\beta],U)$ are disjoint. The Neighborhood Lemma gives $[\alpha]\in B([\beta],U)$. Thus $[\alpha]=[\beta\cdot\delta]$ for loop $\delta$ in $U$. Then $[\beta^{-}\cdot\alpha]=[\delta]$, which means $[\beta^{-}\cdot\alpha]\in B(1,U)$. The Neighborhood Lemma then implies $1\in B([\beta^{-}\cdot\alpha],U)$; a contradiction. $\square$

While this theorem tells us that we gain a lot of ground by just knowing the trivial subgroup is closed, understanding when this happens is a different story. It really depends on the space $X$, particularly how homotopy classes interact with the local topology at $x_0$.

There’s an idea that I wrote about previously called the “Homotopically Hausdorff” property. I’ll remind you of that here.

Definition: A space $X$ is Homotopically Hausdorff at $x\in X$ if for every non-trivial element $1\neq [\alpha]\in \pi_1(X,x)$, there exists an open neighborhood such that no loop in $U$ based at $x$ is path-homotopic to $\alpha$.

This definition is stated a little differently than in my earlier post but, as noted in the remark afterward, the two are equivalent. I’d say this definition is conceptually simpler but the other one is easier to apply. I would imagine that the following theorem is apparently the original justification for the name “homotopically Hausdorff” but the origin of the definition is surprisingly difficult to track down. I first encountered this proof in [2]. It’s a pretty straightforward proof that would make a nice exercise. It’s utility is just that it gives a different characterization of the homotopically Hausdorff property, namely, one in terms of a functorial topology on $\pi_1$.

Theorem: The group $\pi_{1}^{wh}(X,x_0)$ is Hausdorff if and only if $X$ is homotopically Hausdorff at $x_0$.

Proof. The previous theorem tells us that it suffices to prove that the trivial subgroup is not closed if and only if $X$ is not homotopically Hausdorff at $x_0$. Suppose $\{1\}$ is not closed. Then there exists $1\neq [\alpha]$ in the closure of $\{1\}$. Now let $U$ be an open neighborhood of $x_0$. Since $B([\alpha],U)$ is an open neighborhood of $[\beta]$, we have $1\in B([\alpha],U)$. Thus $1=[\alpha][\epsilon]$ for some loop $\epsilon$ in $U$. It follows that $[\alpha]=[\epsilon^{-}]$, i.e. $\alpha$ is path-homotopic to a loop in $U$. This proves $X$ is not homotopically Hausdorff at $x_0$.

For the converse, suppose $X$ is not homotopically Hausdorff at $x_0$. Then there is a non-null-homotopic $\alpha$ based at $x_0$, which is path-homotopic to a loop in every neighborhood of $x_0$. We claim that $[\alpha]$ lies in the closure of $1$. Let $U$ be a neighborhood of $x_0$. Then we have $[\alpha]=[\epsilon]$ for some loop $\epsilon$ in $U$. Equivalently, $1=[\alpha\cdot \epsilon^{-}]$. Since $\epsilon^{-}$ has image in $U$, this means $1\in B([\alpha],U)$. Since $1$ lies in every neighborhood of $[\alpha]$, we conclude that $[\alpha]$ lies in the closure of the trivial subgroup. $\square$

Example: There are lots of homotopically Hausdorff spaces. In my bestiary, I almost always list whether or not this property holds – given an example space, it’s usually easy to decicde whether this property is present or not. Generally, all one-dimensional Hausdorff spaces and subsets of surfaces are homotopically Hausdorff at all of their points and so whenever $X$ is one of these spaces $\pi_{1}^{wh}(X,x_0)$ will be Hausdorff. A fairly non-trivial example is the harmonic pants space. Of course, if you’ve got a locally “nice” space like a manifold or CW-complex, then $\pi_{1}^{wh}(X,x_0)$ is trivially Hausdorff because it is discrete. Examples of spaces, which are not homotopically Hausdorff include the harmonic archipelago and Griffiths twin cone. Actually, for both of these spaces, $\pi_{1}^{wh}(X,x_0)$ is an uncountable indiscrete group.

Example: For an example that is neither Hausdorff nor indiscrete, you can take $\mathbb{G}$ to be the Griffiths twin cone and set $X=\mathbb{G}\vee S^1$ where $x_0$ is the wedgepoint (see the figure below). Let $r: X\to S^1$ be the based retraction that collapses $\mathbb{G}$ to $x_0$. We know $\pi_{1}^{wh}(S^1,x_0)$ is isomorphic to the discrete group $\mathbb{Z}$ and functorality tells us that $r_{\#}:\pi_{1}^{wh}(X,x_0)\to \pi_{1}^{wh}(S^1,x_0)$ is continuous. Therefore $N=\ker (r_{\#})$ is a proper, non-trivial open subgroup of $\pi_{1}^{wh}(X,x_0)$. So $\pi_{1}^{wh}(X,x_0)$ is not discrete. However, we may view $\pi_1(\mathbb{G},x_0)$ naturally as a subgroup of $\pi_{1}^{wh}(X,x_0)$. It’s not too hard to see that $\pi_1(\mathbb{G},x_0)$ is actually equal to the closure of the trivial subgroup in $\pi_{1}^{wh}(X,x_0)$ (but it’s not equal to $N$!). In particular, $\pi_{1}^{wh}(X,x_0)$ is not Hausdorff.

The Griffiths twin cone with an extra circle attached.

Remark: At the start of Part 1, I mentioned that I was particularly interested in compact subsets of $\pi_{1}^{wh}(X,x_0)$. At this point, we can say that if $X$ is homotopically Hausdorf,f then a compact subset $K\subseteq \pi_{1}^{wh}(X,x_0)$ will be zero-dimensional. This is helpful but it would be nice to know when $K$ is metrizable too. So in the third and final post on the whisker topology, I’ll dive into the metrizability of the group $\pi_{1}^{wh}(X,x_0)$.

## References

[1] M. Abdullahi Rashid, S.Z. Pashaei, B. Mashayekhy, H.Torabi, On the Whisker Topology on Fundamental Group. Conference Paper from 46th Annual Iranian Mathematics Conference 46 (2015). pdf available here.

[2] H. Fischer, A. Zastrow, Generalized universal covering spaces and the shape
group, Fund. Math. 197 (2007) 167-196.

## Topologized Fundamental Groups: The Whisker Topology, Part 1

This post follows the primer: How to “topologize” the fundamental group. I’m starting with the “whisker topology” because it comes from a very familiar construction. Also, with such a simple definition, a lot of the questions one could ask about it are pretty straightforward to answer. I found writing this to be a string of fun little exercises.

Despite the apparent simplicity, compact sets in the whisker topology are not always so easy to understand. To my own bewilderment, I’ve been needing to understand exactly such things is some recent projects on wild higher homotopy groups. So I’m also hoping these posts will help me clarify some of my own thoughts about it.

Let’s jump right into it. If you have seen covering space theory, you are likely to recognize the following construction.

## A construction from covering space theory

Let $X$ be a path-connected space with basepoint $x_0\in X$. If we were trying to construct a universal covering space for $X$ (whether it actually exists or not), we might try the following “standard” construction: Let $\widetilde{X}$ be the set of path-homotopy classes $[\alpha]$ of paths $\alpha:[0,1]\to X$ starting at $x_0$. We define a topology on $\widetilde{X}$ by determining a basis. A basic open neighborhood of $[\alpha]$ is of the form $N([\alpha],U)=\{[\alpha\cdot\delta]\mid \delta([0,1])\subseteq U\}$ where $U$ is an open neighborhood of $\alpha(1)$ in $X$. Here, $\alpha\cdot\delta$ denotes path concatenation. For later on, $\alpha^{-}$ will denote the reverse path of $\alpha$ and $c_{x_0}$ will be denote the constant path at $x_0$ so that $1=[c_{x_0}]$ is both the identity element of $\pi_1(X,x_0)$ and the basepoint of $\widetilde{X}$.

Visualizing a basic neighborhood $N([\alpha],U)$ of $[\alpha]$ in the whisker topology

Given a path $\alpha:([0,1],0)\to (X,x_0)$ and a neighborhood $U$ of $\alpha(1)$ in $X$, the neighborhood $N([\alpha],U)$ of $[\alpha]$ in $\widetilde{X}$ consists of only homotopy classes that differ from $[\alpha]$ by a “small” change or “whisker” at it’s end. This is why this topology is often referred to as the whisker topology. The directionality in the definition of the whisker topology is important! Nearby homotopy classes only differ by small differences on the right side.

This definition can be found in most textbooks that include covering space theory but it’s studied in detail in [5] and [6]. I’m not sure who first named it the “whisker” topology. In many settings, it does not need a name but since we are studying and comparing several different topologies, a name is appropriate for this discussion.

We define a function $p:\widetilde{X}\to X$ as the endpoint projection $p([\alpha])=\alpha(1)$. This function is certainly continuous since $p(N([\alpha],U))\subseteq U$ for any neighborhood $U$ and path $\alpha$ with $\alpha(1)\in U$.

If you recognize this construction, you might also recall the following theorem, which is often used to cover the existence part of the classifciation of covering maps: If $X$ is locally path connected and semilocally simply connected, then $p:\widetilde{X}\to X$ is a universal covering map.

The space $\widetilde{X}$ can be constructed for an arbitrary based space $(X,x_0)$. Even when $p:\widetilde{X}\to X$ does not end up being a universal covering map, this construction is often useful.

Since $\widetilde{X}$ is often a covering space, our minds quickly want to identify $\widetilde{X}$ with some familiar space. For example, if $X$ is a torus, we’d identify $\widetilde{X}$ with $\mathbb{R}^2$. But formally, $\widetilde{X}$ is still a set of homotopy classes that contains the fundamental group $\pi_1(X,x_0)$ as a genuine subset. Specifcially, the fundamental group is the basepoint fiber $p^{-1}(x_0)=\pi_1(X,x_0)$.

Therefore, we may give $\pi_1(X,x_0)$ the subspace topology inherited from $\widetilde{X}$. When we intersect the basic neighborhoods of $\widetilde{X}$ with $\pi_1(X,x_0)$, all of the small “whiskers” end up being loops.

Definition: The whisker topology on $\pi_1(X,x_0)$ is generated by basic open sets

$B([\alpha],U)=N([\alpha],U)\cap \pi_1(X,x_0)$

where $[\alpha]\in \pi_1(X,x_0)$ and $U$ is an open neighborhood of $x_0$ in $X$. We will write $\pi_{1}^{wh}(X,x_0)$ to denote the fundamental group equipped with the whisker topology.

A basic neighborhood $B([\alpha],U)$ of an element $[\alpha]$ in the fundamental group with the whisker topology consists of homotopy classes of the form $[\alpha\cdot\epsilon]$.

## What kind of thing is $\pi_{1}^{wh}(X,x_0)$?

It’s tempting to think that because the whisker topology comes from such a standard and simple construction that $\pi_{1}^{wh}(X,x_0)$ will be a topological group. However, it turns out that this very often not true. This is the functorial topology on $\pi_1$ that is, in a sense, the furthest from being a group topology.

Definition: Let $G$ be a group with a topology. Let $\lambda_{g}:G\to G$, $\lambda_{g}(a)=ga$ denote left translation by $g$. If $\lambda_{g}$ is continuous for all $g\in G$, then $G$ is a left topological group. Similarly, if the right translations $\rho_{g}(a)=ag$ are continuous for all $g$, then $G$ is a right topological group. If $G$ is both a left and right topological group and inversion is continuous, then $G$ is a quasitopological group.

If $G$ is a left topological group, then all of the left-translation maps $\lambda_{g}$ will be homeomorphisms since $\lambda_{g^{-1}}$ is the inverse of $\lambda_{g}$.

Proposition: $\pi_{1}^{wh}(X,x_0)$ is a left topological group.

Proof. Fix $[\beta]\in\pi_{1}^{wh}(X,x_0)$ and consider the left translation map $\lambda_{[\beta]}:\pi_{1}^{wh}(X,x_0)\to \pi_{1}^{wh}(X,x_0)$, $\lambda_{[\beta]}([\gamma])=[\beta][\gamma]$ is continuous. A straightforward check gives that $\lambda_{[\beta]}(B([\alpha],U))=B([\beta\cdot\alpha],U)$. This makes it clear that $\lambda_{[\beta]}$ is a homeomorphism. $\square$

Consequently, $\pi_{1}^{wh}(X,x_0)$ is a homogeneous space. But what about the other operations? For this, let’s explore an example.

Example: Let $\mathbb{E}=\bigcup_{n\geq 1}C_n$ be the usual earring space where $C_n\subseteq \mathbb{R}^2$ is the circle of radius $1/n$ centered at $(1/n,0)$. We take $x_0=(0,0)$ to be the wild point and and let $\ell_n:[0,1]\to \mathbb{E}$ denote a loop that traverses the $n$-th circle once counterclockwise. Let $U_n$ be a neighborhood of $x_0$ that contains $\bigcup_{k\geq n}C_k$ and meets $C_k$, $k in an open arc.

Then notice that $\{[\ell_n]\}_{n\geq 1}\to 1$ where $1$ denotes the identity element. Also, $\{[\ell_1\cdot\ell_n]\}_{n\geq 1}$ converges to $[\ell_1]$. However, $\{[\ell_n\cdot\ell_1]\}_{n\geq 1}$ does NOT converge to $[\ell_1]$ because $N([\ell_1],U_2)$ doesn’t contain any of the elements $[\ell_n\cdot\ell_1]$; it can only contain elements of the form $[\ell_1][\beta]$ where $\beta$ has image in $\bigcup_{k\geq 2}C_k$.

As a consequence, we see that $\pi_{1}^{wh}(\mathbb{E},x_0)$ is NOT a right topological group and certainly is not a topological group. Moreover, inversion is NOT always continuous. If inversion was continuous, $\{[\ell_1\cdot\ell_n]\}_{n\geq 1}\to [\ell_1]$ would give $\{[\ell_{n}]^{-1}[\ell_{1}]^{-1}\}_{n\geq 1}\to [\ell_{1}]^{-1}$ but this can’t occur because the sequence $\{[\ell_{n}]^{-1}[\ell_{1}]^{-1}\}_{n\geq 1}$ never enters the open set $B([\ell_{1}]^{-1},U_2)$.

## Functorality

I won’t say to much about the details here but $\pi_{1}^{wh}$ is a functor. Let $\mathbf{LTopGrp}$ be the category of left topological groups and continuous group homomorphisms.

Exercise: If $f:(X,x_0)\to (Y,y_0)$ is a based map, prove that the induced homomorphism $f_{\#}:\pi_{1}^{wh}(X,x_0)\to \pi_{1}^{wh}(Y,y_0)$, $f_{\#}([\alpha]=[f\circ\alpha]$ is continuous.

Once the exercise is done, the fact that $\pi_{1}$ is a a functor, direclty implies that $\pi_{1}^{wh}:\mathbf{Top_{\ast}}\to \mathbf{LTopGrp}$ is a functor.

In fact, since $f_{\#}=g_{\#}$ whenever $f,g:(X,x_0)\to (Y,y_0)$ are homotopic rel. basepoint, $\pi_{1}^{wh}$ induces a functor $\pi_{1}^{wh}:\mathbf{HTop_{\ast}}\to \mathbf{LTopGrp}$ on the pointed-homotopy category (objects are based spaces and morphisms are basepoint-preserving homotopy classes of based maps).

## When CAN we expect continuous operations?

Some of us might be a little unhappy that so many operations in $\pi_{1}^{wh}(X,x_0)$ are going to be discontinuous so often. You might think, well…what if we allowed for whiskers to appear on both ends instead of just on the right. Here, I mean define $B([\alpha],U)=\{[\epsilon\cdot\alpha\cdot\delta]\mid \epsilon([0,1])\cup \delta([0,1])\subseteq U\}$ instead. It’s not a bad idea to consider this for a moment but remember that $\pi_{1}^{wh}(X,x_0)$ is a left topological group. Right multiplication is often discontinuous precisely because we have the whiskers on the right. So if we allow for whiskers on the left and right, then left multiplication won’t be continuous either.

It may seem like this could not possibly be useful until we recall that this topology came from $\widetilde{X}$. So the whisker topology of $\pi_{1}^{wh}(X,x_0)$ is really meant to be more like the vertex set some kind of graph or tree that branches off in infinite and topologically non-trivial ways (maybe think of something that you could consider a non-discrete vertex set in an $\mathbb{R}$-tree if that idea is familiar to you).

However, if you’re really hung up on the whole “fails to be a topological group” thing, we can work to understand this failure better through the following theorem. I’ve never seen this theorem in the literature before but it wasn’t that hard to figure out or prove. If anyone know of a reference, please let me know so I can provide some attribution.

Theorem: Let $G=\pi_{1}^{wh}(X,x_0)$. Then the following are equivalent:

1. Group inversion $in:G\to G$, $in(g)=g^{-1}$ is continuous,
2. $G$ is a topological group,
3. For every $g\in G$, conjugation $c_{g}:G\to G$, $c_g(h)=ghg^{-1}$ is continuous.

Proof. The hardest direction is 1. $\Rightarrow$ 2 so we’ll prove that first. Suppose group inversion $in:G\to G$, $in(g)=g^{-1}$ is continuous. To show that $G$ is a topological group, it’s enough to show that multiplication $([\alpha],[\beta])\mapsto [\alpha\cdot\beta]$ is continuous. Let $U$ be a neighborhood of $x_0$ in $X$ so that $B([\alpha\cdot\beta],U)$ is a basic open neighborhood of $[\alpha\cdot\beta]$. Since inversion is continuous, specifically at $[\beta^{-}]\in G$, we may find an open neighborhood $V$ of $x_0$ such that $V\subseteq U$ and $in(B([\beta^{-}],V))\subseteq B([\beta],U)$. The second inclusion means that whenever $\delta$ is a loop in $V$ based at $x_0$, we have $[\delta^{-}][\beta]=[\beta][\gamma]$ for some loop $\gamma$ in $U$.

We will show that group multiplication maps $B([\alpha],V)\times B([\beta],V)$ into $B([\alpha\cdot\beta],U)$. Let $[\alpha\cdot\delta]\in B([\alpha],V)$ and $[\beta\cdot\epsilon]\in B([\beta],V)$ for loops $\delta,\epsilon$ in $V$. Since $\delta^{-}$ has image in $V$, we have $[\delta][\beta]=[\beta][\gamma]$ for some loop $\gamma$ in $U$. Thus

$[\alpha\cdot\delta][\beta\cdot\epsilon]=[\alpha][\delta][\beta][\epsilon]=[\alpha][\beta][\gamma][\epsilon]=[\alpha\cdot\beta][\gamma\cdot\epsilon]$

where $\gamma\cdot\epsilon$ has image in $U$. Thus the product $[\alpha\cdot\delta][\beta\cdot\epsilon]$ lies in $B([\alpha\cdot\beta],U)$, completing the proof of the first direction.

The implication 2. $\Rightarrow$ 3. is clear so it is suffices to prove 3. $\Rightarrow$ 1. Suppose for every $g\in G$, that conjugation by $g$ is continuous. Let $[\beta]\in G$. We will check that inversion is continuous at $[\beta]$. Let $U$ be an open neighborhood of $x_0$ in $X$ so that $B([\beta^{-}],U)$ is an open neighborhood of $in([\beta])=[\beta^{-}]$. Since conjugation $c_{[\beta]}$ is continuous at the identity element, we may find a neighborhood $V$ of $x_0$ such that $[\beta]B(1,V)[\beta^{-}]\subseteq B(1,U)$. In other words, if $\delta$ is any loop in $V$ based at $x_0$, then there is a loop $\epsilon$ in $U$ such that $[\beta][\delta][\beta^{-}]=[\epsilon]$. Now it suffices to show that $in(B([\beta],V))\subseteq B([\beta^{-}],U)$. Let $\delta$ be a loop in $V$ based at $x_0$. Since $\delta^{-}$ is also in $V$, we may find a loop $\epsilon$ in $U$ such that $[\beta][\delta^{-}][\beta^{-}]=[\epsilon]$. Then $in([\beta\cdot\delta])=[\delta^{-}\cdot\beta^{-}]=[\beta^{-}][\epsilon]\in B([\beta^{-}],U)$. $\square$

The whisker topology becomes a group topology if and only if for every $\beta$ and neighborhood $U$, there exists a neighborhood $V$ such that for any loop $\epsilon$ in $V$, the conjugate $\beta\cdot\epsilon\cdot\beta^{-}$ is homotopic to some loop in $U$.

What I like about this theorem is that the proof really uses the whisker topology (showing how it depends on the topology of $X$ at $x_0$) and can’t just be proven by analyzing compositions of operations. I made sure to include Condition 3. in the theorem because even if we know $\pi_{1}^{wh}(X,x_0)$ is abelian, it’s not immediately clear that inversion is continuous. However, we do know that these conjugation maps in abelian groups are always the identity!

Corollary: If $\pi_1(X,x_0)$ is abelian, then $\pi_{1}^{wh}(X,x_0)$ is a topological group.

But just being non-commutative doesn’t guarantee that $\pi_{1}^{wh}(X,x_0)$ will fail to be a topological group.

Exercise: Show that $\pi_{1}^{wh}(X,x_0)$ is discrete if and only if $X$ is semilocally simply connected at $x_0$.

Example: If $X=\bigvee_{a\in A}S^1$ is an ordinary wedge of circles with the weak topology, then $\pi_{1}(X,x_0)$ is a free group but $\pi_{1}^{wh}(X,x_0)$ will be discrete and all discrete groups are topological groups.

So, being a topological group here is not just about purely algebraic things like commutativity or conjugation. What’s really going on? If you read the proof closely you can see that for Condition 3. you really only need to use that the conjugation maps are continuous at the identity element. This idea is exactly what is described in the above illustration. So for $\pi_{1}^{wh}(X,x_0)$ to be a topological group, all elements $g$, no matter how “large,” have to conjugate small loops to small loops. Or maybe think about the negation this way…If there is some $g\neq 1$ and also arbitrarily small $a\neq 1$ with $gag^{-1}$ remaining “large,” then you’re in trouble.

Example: The harmonic archipelago gives an example where $\pi_{1}^{wh}(X,x_0)$ is non-commutative but is a topological group because it is indiscrete

So now here’s a question. I haven’t thought too deeply about it so I don’t exactly know how hard it is. If you know of an answer, feel free to share.

Question: Is there a space $X$ where $\pi_{1}^{wh}(X,x_0)$ is a non-discrete, Hausdorff, and non-commutative topological group?

In the next post, I’ll discuss more about separation axioms where we’ll run into the familiar term “homotopically Hausdorff” and see that when this happens the group $\pi_{1}^{wh}(X,x_0)$ is always zero dimensional!

## References

The references here include several papers that involve the whisker topology from a group of Iranian researchers who have done a lot of research in this area.

[1] N. Jamali, B. Mashayekhy, H. Torabi, S.Z. Pashaei, M. Abdullahi Rashid, On
topologized fundamental groups with small loop transfer viewpoints, Acta Math. Vietnamica, 44 (2019) 711–722.

[2] M. Abdullahi Rashid, N. Jamali, B. Mashayekhy, S.Z. Pashaei, H. Torabi, On subgroup topologies on the fundamental group. Hacettepe Journal of Mathematics & Statistics 49 (2020), no. 3, 935 – 949.

[3] M. Abdullahi Rashid, B. Mashayekhy, H. Torabi, S.Z. Pashaei, On subgroups of
topologized fundamental groups and generalized coverings, Bull. Iranian Math. Soc.
43 (2017), no. 7, 2349–2370.

[4] M. Abdullahi Rashid, S.Z. Pashaei, B. Mashayekhy, H.Torabi, On the Whisker Topology on Fundamental Group. Conference Paper from 46th Annual Iranian Mathematics Conference 46 (2015). Note: easily found through a google search but I can’t get a link to work.

[5] H. Fischer, A. Zastrow, Generalized universal covering spaces and the shape
group, Fund. Math. 197 (2007) 167-196.

[6] Z. Virk, A. Zastrow, The comparison of topologies related to various concepts of
generalized covering spaces, Topology Appl. 170 (2014) 52–62.

## 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.