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.

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2 Responses to How to “topologize” the fundamental group: a primer

1. Paul Fabel says:

Great post! These are important stories, many still yet to be worked out and told well, hopefully ultimately resonating at low frequency or higher, with the liquid tensor experiment.

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