Topologized Fundamental Groups: The Quotient Topology Part 2 (Discreteness)

Posted on February 13, 2023 by Jeremy Brazas

In Part 1, I described the construction of \pi_{1}^{qtop}(X,x_0), the fundamental group equipped with the quotient topology and some of the drama around \pi_{1}^{qtop}(X,x_0) failing to always be a topological group. In this second post, I plan to connect \pi_{1}^{qtop}(X,x_0) back to spaces with “nicer” local structure by discussing when \pi_{1}^{qtop}(X,x_0) is discrete. The theorem we’ll prove is the main result from [1].

If you’ve got a CW-complex, manifold, simplicial complex, or some other locally contractible space, you should very much hope that \pi_{1}^{qtop}(X,x_0) is a discrete group. If \pi_{1}^{qtop}(X,x_0) is NOT discrete, then it’s because the topological part of \pi_{1}^{qtop} is detecting some non-trivial local structures in X. In this post, we’ll explore what non-discreteness is really telling you.

Reminder: \Omega(X,x_0) is the loop space with the compact-open topology and \pi_{1}^{qtop}(X,x_0) is the fundamental group with the quotient topology inherited from the map q:\Omega(X,x_0)\to\pi_{1}(X,x_0), q(\alpha)=[\alpha] identifying homotopy classes of loops. We’ll need to use the description of a basis for the compact-open topology from Part 1.

First, let’s identify a clear case where \pi_{1}^{qtop}(X,x_0) is not discrete. Recall that a space X is semilocally simply connected at x\in X if there exists an open neighborhood U of x such that the homomorphism pi_1(U,x)\to \pi_1(X,x) induced by the inclusion map U\to X is trivial, that is, if every loop in U based at a x contracts by a null-homotopy in X. We say X is semilocally simply connected if it has this property at all of its points.

Lemma 1: If X is path connected and \pi_{1}^{qtop}(X,x_0) is discrete, then X is semilocally simply connected.

Proof. Suppose X is not semilocally simply connected at some point x\in X. Fix a path \alpha:[0,1]\to X from x_0 to x and let \mathscr{N} be the set of open sets in X containing x. Since X is not semilocally simply connected at, for every U\in \mathscr{N}, there exists a loop \beta_{U}:[0,1]\to U based at x such that \beta_U is not null-homotopic in X, that is, [\beta_{U}]\neq 1 in \pi_1(X,x). Thus [\alpha\cdot\beta_{U}\cdot\alpha^{-}]\neq 1 in \pi_1(X,x_0). However, \mathscr{N} is a directed set (by subset inclusion) and so \{\alpha\cdot\beta_{U}\cdot\alpha^{-}\}_{U\in\mathscr{N}} is a net in \Omega(X,x_0). Moreover, \{\alpha\cdot\beta_{U}\cdot\alpha^{-}\}_{U\in\mathscr{N}} converges to \alpha\cdot c_x\cdot \alpha^{-} in the compact-open topology where c_x denotes the constant loop at x. Since q:\Omega(X,x_0)\to \pi_{1}^{qtop}(X,x_0) is continuous, the net of non-trivial homotopy classes \{[\alpha\cdot\beta_{U}\cdot\alpha^{-}]\}_{U\in\mathscr{N}} converges to the identity element 1=[\alpha\cdot c_x\cdot \alpha^{-}] in \pi_{1}^{qtop}(X,x_0). Since a net of non-identity elements converges to the identity element, there is no way the trivial subgroup \{1\} can be open. Thus \pi_{1}^{qtop}(X,x_0) is not discrete. \square

So immediately, all of the regular suspects on this blog, e.g. the earring space, harmonic archipelago, Menger cube, etc. have non-discrete fundamental group.

Now, the above result doesn’t say anything about local path connectivity. There a plenty of simply connected spaces that are not locally path connected, e.g. the Warsaw circle. But to prove a converse to Lemma 1, we need to add a local path connectivity condition and, in the, end we’ll see this actually is necessary if we’re looking for fully classify when \pi_{1}^{qtop}(X,x_0) is discrete.

Lemma 2: If X is locally path connected and semilocally simply connected, then \pi_{1}^{qtop}(X,x_0) is a discrete group.

Proof. To show that \pi_{1}^{qtop}(X,x_0) is discrete, we must show that for any loop \alpha:[0,1]\to X, the 1-point set \{[\alpha]\} is open. Since q:\Omega(X,x_0)\to\pi_{1}^{qtop}(X,x_0) is a quotient map, \{[\alpha]\} is open in \pi_{1}^{qtop}(X,x_0) if and only if q^{-1}(\{[\alpha]\})=[\alpha] is open in \Omega(X,x_0). Remember that [\alpha] is the homotopy class of \alpha, that is, the set of all loops path-homotopic to \alpha. In order to do this we must show that all loops “nearby” \alpha are homotopic to \alpha where “nearby” really means “in some compact-open neighborhood.”

For each t\in [0,1], let U_t be an open neighborhood of \alpha(t) such that every loop in U_t is null-homotopic in X (here, we are using the semilocally simply connected property). Using the Lebesgue Number Lemma, we may find an integer n\geq 1 such that if K_{n}^{j}=\left[\frac{j-1}{n},\frac{j}{n}\right] (as in Part 1), then for each j\in \{1,2,\dots,n\}, we have \alpha(K_{n}^{j})\subseteq U_{s_j} for some s_j. To simplify notation, we’ll write U_j for U_{s_j}.

A sequence of open sets covering the path alpha.

Now we have \alpha(t_j)\in U_{j}\cap U_{j+1}. However, it may not be the case that the intersection U_{j}\cap U_{j+1} is path connected. We can address this issue in the following way. Since X is locally path connected, for each j\in \{1,2,\dots ,n-1\}, find a path-connected neighborhood latex V_j of \alpha(\frac{j}{n}) such that V_j\subseteq U_{j}\cap U_{j+1}.

Now we are ready to define the neighborhood 

\mathscr{U}=\bigcap_{j=1}^{n}\langle K_{n}^{j},U_j\rangle\cap \bigcap_{j=1}^{n-1}\langle \{\frac{j}{n}\},V_j\rangle

Recall that the notation \langle C,U\rangle denotes the set consisting of all loops that map the compact set C into the open set U. So we can think of \mathscr{U} above as the set of all loops that follow an ordered list of instructions. If \beta\in \mathscr{U}, then \beta must first proceed through U_1 and end somehwere in V_1. It must then proceed through U_2 and end in V_2, etc.

Our desired neighborhood of alpha in the compact-open topology

The neighborhood \mathscr{U} of \alpha.

Our remaining job is to show that [\alpha] is open and this will be done if we can show that \mathscr{U}\subseteq [\alpha], that is every loop in \mathscr{U} is homotopic to \alpha.

Let \beta\in \mathscr{U}. We’ll construct a homotopy \alpha\simeq\beta. For j\in\{1,2\dots,n-1\}, both \alpha(t_j) and \beta(t_j) lies in V_j and so we may find a path \gamma_j:[0,1]\to V_j from \alpha(t_j) to \beta(t_j).

The path beta and connected to alpha by small paths gamma_j.

The path \beta and the connecting paths \gamma_j.

Notice that

  • \alpha|_{K_{n}^{1}}\cdot \gamma_1\cdot \beta|_{K_{n}^{1}}^{-} is a loop in U_1,
  • \gamma_{j-1}^{-}\cdot \alpha|_{K_{n}^{j}}\cdot \gamma_{j}\cdot \beta|_{K_{n}^{j}}^{-} is a loop in U_j when 2\leq j\leq n-1,
  • \gamma_{n-1}^{-}\cdot \alpha|_{K_{n}^{n}}\cdot  \beta|_{K_{n}^{n}}^{-} is a loop in U_n.

A close-up view of the general case where we have create a loop in U_j with corresponding portions of \alpha and \beta. Since this loop lies in U_j, it is null-homotopic in X.

By our choice of the sets U_j, all of these loops are null-homotopic in X. In particular, this means that 

  • \alpha|_{K_{n}^{1}}\simeq \beta|_{K_{n}^{1}}\cdot \gamma_{1}^{-},
  • \alpha|_{K_{n}^{j}}\simeq\gamma_{j-1}^{-}\cdot \beta|_{K_{n}^{j}}\cdot \gamma_{j}^{-} when 2\leq j\leq n-1,
  • \alpha|_{K_{n}^{n}}\simeq \gamma_{n-1}\cdot\beta|_{K_{n}^{n}}.

Composing these homotopies “horizontally” gives that \alpha=\prod_{j=1}^{n}\alpha|_{K_{n}^{j}} is homotopic to

(\beta|_{K_{n}^{1}}\cdot \gamma_{1}^{-})\cdot \left(\prod_{j=2}^{n-1}\gamma_{j-1}^{-}\cdot \beta|_{K_{n}^{j}}\cdot \gamma_{j}^{-}\right)\cdot (\gamma_{n-1}\cdot\beta|_{K_{n}^{n}})

cancelling the inverse pairs \gamma_{j}^{-}\cdot\gamma_{j} gives a homotopy with \prod_{j=1}^{n}\beta|_{K_{n}^{j}}=\beta. Thus \alpha\simeq\beta. We conclude that \mathscr{U}\subseteq [\alpha]. \square

Lemmas 1 and 2 tell us that for locally path-connected spaces, being semilocally simply connected is equivalent to \pi_{1}^{qtop}(X,x_0) being a discrete group.

Theorem: Suppose X is locally path connected. Then \pi_{1}^{qtop}(X,x_0) is discrete if and only if X is semilocally simply connected.

This tells us that, for locally path-connected spaces, non-discreteness of \pi_{1}^{qtop}(X,x_0) as a topological invariant, really is detecting the existence of local 1-dimensional wildness in a space. When our space in question is not locally path connected, things get a bit trickier.

Example: The following “hoop earring” space Y is semilocally simply connected but is not locally path-connected. This space includes the sequence of circles that all meet at a point y_0 and includes the limit circle too. If \ell_n:[0,1]\to Y is the loop going once around the n-th circle (parameterized in a standard way) and \ell_{\infty}:[0,1]\to Y goes once around the limit circle, then \{\ell_n\}\to\ell_{\infty} in the compact-open topology and so \{[\ell_n]\}\to [\ell_{\infty}] in \pi_{1}^{qtop}(Y,y_0) even though none of these homotopy classes are the same. Since \pi_{1}^{qtop}(X,x_0) contains a non-trivial convergent sequence, it can’t be discrete. In fact, this group is isomorphic to something called a free topological group (but that’s much harder to show)!

The space Y: a wedge of converging circles has non-discrete fundamental group.

 

References.

[1] J. Calcut, J. McCarthy, Discreteness and homogeneity of the topological fundamental group, Topology Proc.  34 (2009) 339-349.

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Topologized Fundamental Groups: The Quotient Topology Part 1

Posted on August 25, 2022 by Jeremy Brazas

Next up for topologies on the fundamental group is what I’d consider the most “natural” one. It’s almost certainly the topology you’d most often get if you asked random topologists on the street to construct one for you. This is the natural quotient topology. I’m excited about this one. I studied this one a lot as a grad student so it’s near and dear to my heart.

Now, even though it’s easy to construct and the added topology often contains way more information than the ordinary non-topologized fundamental group, it is actually quite tricky to work with. I also mentioned in my primer post that this topology comes with historical baggage and I’ll get to that later in this post.

Let’s go ahead and define it. Throughout, X will be a path-connected space with basepoint x_0\in X. Let \Omega(X,x_0) denote the space of loops in X based at x_0, that is, maps \alpha:[0,1]\to X with \alpha(0)=\alpha(1)=x_0. We give \Omega(X,x_0) the usual compact-open topology, which is generated subbasis sets \langle K,U\rangle=\{\alpha\mid \alpha(K)\subseteq U\} for compact K\subseteq [0,1] and open U\subseteq X.

Every based loop \alpha\in\Omega(X,x_0) has a corresponding path-homotopy class [\alpha]\in \pi_1(X,x_0). This defines a surjection q:\Omega(X,x_0)\to\pi_1(X,x_0), q(\alpha)=[\alpha] which identifies homotopy classes.

Definition: Let \pi_{1}^{qtop}(X,x_0) denote \pi_1(X,x_0) equipped with the quotient topology with respect to the map q:\Omega(X,x_0)\to\pi_1(X,x_0). We refer to this topology as the natural quotient topology on \pi_{1}(X,x_0).

What this means: Remember that a surjective function q:X\to Y is a quotient map if U is open in Y if and only if it’s a preimage q^{-1}(U) is open in X. Hence, a set A\subseteq \pi_{1}^{qtop}(X,x_0) is open (closed) if and only if the set q^{-1}(A) of all loops representing the homotopy classes in A is open (closed) in \Omega(X,x_0).

The “qtop” superscript is actually going to do double duty here. Yes, it stands for “quoteint topology” but it also invokes the term “quasitopological” as in quasitopological group. The early literature on the quotient topology is messy but has corrected itself and has seen a lot of growth in the past 10 years. A summary appears in the following paper.

J. Brazas, P. Fabel, On fundamental groups with the quotient topology, J. Homotopy and Related Structures 10 (2015) 71-91

Terminology: Occasionally, authors will call the quotient topology on \pi_1 the “compact-open topology.” Tisk tisk. Look…I get it. The quotient topology does descend from the compact-open topology. Fine. But it’s not the compact-open topology itself so why call it the compact-open topology. “c.o.-quotient” topology would be better. I get a little hot about this because other topologies on \pi_1 like the tau-topology also depend very closely on the compact-open topology on loop spaces so when authors do this, I see it as an example of choosing to confuse terminology that avoids readily available descriptive terms. It’s a quotient topology… just call it what it is. Sometimes I’ve called it the quasitopological fundamental group, which is not ideal for more subtle reasons. At this point just “the natural quotient topology” is probably best. Sorry* for the rant.

In this sequence of posts, I’m going to detail some of the history, properties, and uses of the natural quotient topology on the fundamental group.

Understanding the compact-open topology

Since we’re form a quotient space from the compact-open topology, let’s briefly unpack what the compact-open topology is really about. A subbasic set of the form \langle K,U\rangle in \Omega(X,x_0) can be thought of as a single instruction. It contains all of the paths that do something similar, namely map K into U. A basis set is an intersection of subbasic sets, say \bigcap_{i=1}^{n}\langle K_i,U_i\rangle, which can be thought of as a finite set of instructions. If we have \alpha\in \bigcap_{j=1}^{n}\langle K_j,U_j\rangle, then \alpha must map K_1 into U_1, and K_2 into U_2, and so on. Provided the compact sets K_i cover [0,1], we have some kind of restriction on every point in the domain. For example, let K_{n}^{j}=\left[\frac{j-1}{n},\frac{j}{n}\right]. Then \mathscr{U}=\bigcap_{j=1}^{n}\langle K_{n}^{j},U_j\rangle is a basic open set and every element of \mathscr{U} must be a path which proceeds sequentially through the sets U_1,U_2,\dots,U_n in X at a certain rate.

A basic open set in the compact open topology

A basic open set in the compact open topology

It’s a folklore lemma that’s a bit tedious to prove that the sets of the form \mathscr{U} actually form a basis for the compact-open topology on \Omega(X,x_0).

The compact-open topology is the one most often used on loop spaces because

  1. it usually has nice categorical properties,
  2. it generalizes the topology of uniform convergence. In particular, if X is a metric space, then \{\alpha_n\}\to \alpha in \Omega(X,x_0) with the compact-open topology if and only if \{\alpha_n\}\to \alpha uniformly in X,
  3. standard operations on loops are continuous.

Let’s expand upon 3. The usual concatenation of loops gives us an operation c:\Omega(X,x_0)\times\Omega(X,x_0)\to \Omega(X,x_0), c(\alpha,\beta)=\alpha\cdot\beta where \alpha\cdot\beta is the loop that does \alpha on [0,1/2] and \beta on [1/2,1]. We also have a reverse path operation rev:\Omega(X,x_0)\to \Omega(X,x_0), rev(\alpha)=\alpha^{-}. Here, \alpha^{-}(t)=\alpha(1-t) simply does \alpha in the opposite orientation. It’s a nice exercise to check that c and rev are continuous. One should beware that c is not stricly unital or associative.

We can also restrict the concatenation map c to continuous right- and left-concatenation maps:

  • \rho_{\beta}:\Omega(X,x_0)\to\Omega(X,x_0), \rho_{\beta}(\alpha)=\alpha\cdot\beta;
  • \lambda_{\beta}:\Omega(X,x_0)\to\Omega(X,x_0), \lambda_{\beta}(\alpha)=\beta\cdot\alpha.

Both of these are continuous as they can be identified with restrictions of c.

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

The definition of \pi_{1}^{qtop}(X,x_0) is so natural and simple that this topologized group should to be a pretty nice thing, right? Let’s see what we can do. We really only need one tool from general topology.

University property of quotient maps: Consider spaces A,B,C and functions q,f,g that make the following diagram commute (so f\circ q=g).

If q is a quotient map and g is continunous, then f is continuous too.

The universal property is a powerful tool for showing that functions are continuous. Note that the universal property might appear in slightly different ways. For instance, we might instead have the following commutative square.

In a situation like this, the universal property becomes useful if we’re not sure if h is continuous or not. If the left map q is a quotient map and the upper composition g\circ f is continuous, then h will also be continuous. Of course, this is just a special case of the triangle above, but for the sake of applications it’s good to be ready to use it in diagrams of various shapes.

Let’s try our best to prove that \pi_{1}^{qtop}(X,x_0) is a topological group. First, we’ll prove that inversion is continuous.

Proposition: Group inversion inv:\pi_{1}^{qtop}(X,x_0)\to \pi_{1}^{qtop}(X,x_0), inv([\alpha])=[\alpha]^{-1} is continuous.

Proof. Consider the following commutative square.

Here, rev(\alpha)=\alpha^{-} is the reverse map. Since inv(q(\alpha))=[\alpha]^{-1}=[\alpha^{-}]=q\circ rev(\alpha), the diagram does indeed commute. Now both vertical maps are the quotient map q and as noted above, rev is continuous. Since the left map is quotient and the upper composition q\circ rev is continuous, the bottom maps inv is continuous by the universal property of quotient maps. \square.

Ok, with the universal property in hand, that wasn’t so bad! Let’s see if we can do the same thing for group multiplication \mu:\pi_{1}^{qtop}(X,x_0)\times \pi_{1}^{qtop}(X,x_0)\to \pi_{1}^{qtop}(X,x_0). WARNING: I’m about to propose an incorrect proof. It’s going to have an error in it. See if you can find it.

Proposed Proof. Consider the following commutative diagram where c is the concatenation map and \mu is the group operation in the fundamental group.

Note that \mu(q\times q(\alpha,\beta))=[\alpha][\beta]=[\alpha\cdot\beta]=q(c(\alpha,\beta) so the diagram does indeed commute. By definition q is a quotient map and as noted above c is continuous. Since the left vertical map q\times q is quotient and the upper composition q\circ c is continuous, the bottom map \mu is continuous by the universal property of quotient maps. (QED?)

Seem ok? See any problems? The diagram commutes just fine. The upper composition is continuous. The error is in the phrase: “Since the left vertical map q\times q is quotient.” Here’s a valuable lesson friends. If we’re using the product topology (which is what you use for a topological group G to have continuous multiplication G\times G\to G), the direct product of two quotient maps is not always a quotient map. There are counterexamples in most introductory general topology books.

Just because a proof is wrong doesn’t always mean the claim is false. However…our wish here is false. Spoiler Alert: \pi_{1}^{qtop}(X,x_0) is NOT always a topological group. In fact, I’d say it’s rarely a topology group when it’s Hausdorff and non-discrete. Paul Fabel gave the first counterexample [4] and shortly after I published one [2] that connects to other structures from topological group theory.

Oops

The above logical error is an easy one to make and miss. The most notable place this mistake was made is in the 2002 paper:

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

This paper had some really cool global ideas. However, there are a ton of independent errors in it…. I mean a ton. Very few results in it are correct. As you can see it is now retracted – and it even made retraction watch. According to Google Scolar, as of 8/25/2022, this paper has 90 citations! The errors within created something of a mess for a while because some papers called upon Biss’ false claims. For one, we no longer called it the “topological fundamental group” because we now know it’s not always “topological” in the sense of being a “topological” group. Unfortunately, several papers have cited [1] without mentioning the many errors or retraction.

The same error was used in [6] to claim the higher homotopy groups with the natural quotient topology are always topological groups. Paul Fabel constructed counterexamples to this higher-dimensional claim in [5].

Actually, the same mistake was made 12 years earlier in an appendix [7] written by my mathematical grandfather J.P. May. This short note involves a groupoid version of \pi_{1}^{qtop}(X,x_0) but it includes essentially the same error. However, unlike in Biss’ paper, the overall results are not damaged by this error because of the kinds of spaces being used.

I think most professional mathematicians, incluidng myself, have published some false arguments or statements. Sometimes our intuition guides our writing more than formal logic does. This can lead to a correct result but a proof with an easily fixable logical gap. Occasionally, like in this case, a published error might be unfixable.

Nobody wants to have their published mathematics end up being wrong. The point of publicly mentioning these specific papers is not to be hard on the people making the mistakes. I mean…. J.P. May, who asked me about this situation back in 2010, is one of the most influential and prolific algebraic topologists in the history of the subject. Also, Biss’ paper did end up being very influential even if it’s not in the intended way. That’s something to consider. Rather, the point of this part of the post is to:

  1. Encourage awareness about published errors so others don’t fall into the trap of doing mathematics that depends on false claims.
  2. Emphasize to young mathematicians that incorrect math can lead to new (and correct) ideas and areas of research and that even great mathematicians make mistakes here and there. It’s possible to own up to a mistake without getting defensive or letting it defeat you personally.
  3. Show that it is possible to point out mistakes of others at the research level in a respectful and kind manner.

How I learned to love horrifyingly complicated things

Now, look…sometimes the direct product of two quotient maps is a quotient map. And it’s true that for some spaces X, this works and \pi_{1}^{qtop}(X,x_0) does end up being a topological group. But this is really only guaranteed when the domain and codomain satisfy some (local) compactness criteria. This apparent failure of the topological category \mathbf{Top} is closely related to the fact that it is not Cartesian closed. Are there ways to cheat the system? Sure. This apparent deficiency goes away if you replace the usual category of spaces with a coreflective Cartesian closed category like compactly generated spaces, sequential spaces, delta-generated spaces, etc. However, if you work internal to one of these categories, the group-objects in those categories may not be true topological groups.

Before, you demand that we switch to one of these categories consider where this is going. Maybe consider the question “what is the fundamental group good for?” I’d say its utility is that it is an invariant, which creates a symbiotic relationship between topology and algebra. You can study and classify spaces using the algebra and functorality of fundamental groups and, on the flip side, if you want to prove things about a collection of groups it’s often a good idea to realize them as fundamental groups of spaces with some common features and then use the topology to prove things about the groups. For example, fundamental groups can be used to classify surfaces and, in the other direction, covering space theory provides nice proofs of results like the Nielsen-Schreier and Kurosh Theorems.

What can a topologized fundamental group like \pi_{1}^{qtop}(X,x_0) be good for? Potentially, it could create an extended symbiotic relationship between spaces with complicated local structure and topologized groups. I’d say this has been carried out in a successful way and the progress is ongoing. For example, long-standing gaps in free topological group theory have been filled using topologized fundamental groups. The only known proofs rely on working with \pi_{1}^{qtop}(X,x_0). Such things could not have been found if everyone had just decided to shove everything into another category.

Also, from a pragmatic viewoint…there are international communities of general topologists studying the topology of actual topological groups. There are not large communities who just study groups internal to, say, compactly generated spaces. Not that there shouldn’t be…but there just aren’t.

Personally, the fact that \pi_{1}^{qtop} ends up being really complicated taught me a valuable lesson early on in my research career. If I became unwilling to struggle to work with complicated structures that some mathematicians might find ugly or terrifying, my mathematical world would remain small and leave me with less potential to reveal fascinating and beautiful possibilities.

\pi_{1}^{qtop}(X,x_0) is a quasitopological group

If \pi_{1}^{qtop}(X,x_0) not being a topological group is a castle in ruins, let’s pick up some of rubble and build a tiny house out of it.

Definition: a group G with topology is a quasitopological group if inversion g\mapsto g^{-1} is continuous and if multiplication G\times G\to G is continuous in each variable, that is, if all left and right translation maps G\to G given by h\mapsto gh and h\mapsto hg are continuous for all g\in G.

For [\beta]\in\pi_{1}^{qtop}(X,x_0), let \rho_{[\beta]}:\pi_{1}^{qtop}(X,x_0)\to \pi_{1}^{qtop}(X,x_0), \rho_{[\beta]}([\alpha])=[\alpha][\beta] be right multiplication by [\beta], and \lambda_{[\beta]}:\pi_{1}^{qtop}(X,x_0)\to \pi_{1}^{qtop}(X,x_0), \lambda_{[\beta]}([\alpha])=[\beta][\alpha] be left multiplication by [\beta].

Lemma: For any based space (X,x_0), \pi_{1}^{qtop}(X,x_0) is a quasitopological group.

Proof. We have already seen above that inversion in \pi_{1}^{qtop}(X,x_0) is continuous. Fix [\beta]\in\pi_{1}^{qtop}(X,x_0)  and consider the following commutative diagrams involving left and right concatenation/multiplication.

In both squares, the left map is quotient and the upper composition is continuous. Therefore the bottom maps are continuous by the universal property of quotient maps. \square.

Theorem: \pi_{1}^{qtop}:\mathbf{Top_{\ast}}\to \mathbf{qTopGrp} is a functor from the category of based topological spaces to the category of quasitopological groups and continuous homomorphisms.

Proof. The previous lemma, tells us that \pi_{1}^{qtop} is well-defined on objects. The underlying algebraic structure of \pi_{1}^{qtop} is the usual functor \pi_{1} so we only need to check that if f:(X,x_0)\to (Y,y_0) is a based map, then the induced homomorphism f_{\#}: \pi_{1}^{qtop}(X,x_0)\to \pi_{1}^{qtop}(Y,y_0) is continuous. First, note that there is an induced loop-space function \Omega(f):\Omega(X,x_0)\to \Omega(Y,y_0), \Omega(f)(\alpha)=f\circ \alpha. If \langle K,U\rangle is a subbasic open set in \Omega(Y,y_0), then \Omega(f)^{-1}(\langle K,U\rangle)=\langle K,f^{-1}(U)\rangle is a subbasic open subset of \Omega(X,x_0). Thus \Omega(f) is continuous. The map f also induces the homomorphism f_{\#}:\pi_1(X,x_0)\to\pi_1(Y,y_0), f_{\#}([\alpha])=[f\circ \alpha]. Now consisder the following diagram.

Since f_{\#}(q(\alpha))=f_{\#}([\alpha])=[f\circ \alpha]=q\circ \Omega(f)(\alpha), the diagram commutes. Moreover, the left vertical map is quotient and the upper composition is continuous. Therefore, the bottom function is continuous by the universal property of quotient maps. \square.

If f:(X,x_0)\to (Y,y_0) and g:(Y,y_0)\to (X,x_0) are based homotopy inverses, then the homomorphisms f_{\#} and g_{\#} they induce are continuous inverses and therefore are homeomorphisms. This means that \pi_{1}^{qtop} is an invariant of based homotopy type.

Exercise: Use the universal property of quotient maps to prove that if \beta:[0,1]\to X is a path from x_0 to x_1, then the basepoint-change isomorphism \varphi_{\beta}:\pi_{1}^{qtop}(X,x_0)\to\pi_{1}^{qtop}(X,x_1), \varphi([\alpha])=[\beta^{-}\cdot\alpha\cdot\beta] is a homeomorphism.

From this exercise, the same argument that \pi_1 is an invariant of unbased homotopy type can be used to show that \pi_{1}^{qtop} is an invariant of unbased homotopy type.

Corollary: If X and Y are path connected and X\simeq Y, then \pi_{1}^{qtop}(X,x_0)\cong \pi_{1}^{qtop}(Y,y_0) as quasitopological groups for any choice of x_0\in X and y_0\in Y.

There’s a lot more to say about \pi_{1}^{qtop} and I’ll get to some of it in future posts. For now we can take away the fact that even though \pi_{1}^{qtop}(X,x_0) is not always a topological group, it is still pretty close to being a topological group. Moreover, the natural quotient topology gives us a homotopy invariant, which is much stronger invariant than the usual fundamental group, particularly when it comes to spaces with complicated local structures.

References

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

[2] J. Brazas, The topological fundamental group and free topological groups, Topol. Appl. 158 (2011) 779–802.

[3] J. Brazas, P. Fabel, On fundamental groups with the quotient topology, J. Homotopy Relat. Struct. 10 (2015) 71–91.

[4] P. Fabel, Multiplication is discontinuous in the Hawaiian earring group, Bull. Pol. Acad. Sci., Math. 59 (2011) 77–83.

[5] P. Fabel, Compactly generated quasitopological homotopy groups with discontinuous multiplication, Topol. Proc. 40 (2012) 303–309.

[6] H. Ghane, Z. Hamed, B. Mashayekhy, and H. Mirebrahimi, Topological
homotopy groups, Bull. Belgian Math. Soc. 15 (2008), 455–464.

[7] J.P. May, G-spaces and fundamental groupoids, appendix, K-Theory 4
(1990), 50–53.

 

*I’m not really sorry. It does bug me.

Posted in Algebraic Topology, compact-open topology, Fundamental group, Quasitopological groups, quotient topology, Topological groups, Uncategorized | Tagged , , , , , | 2 Comments

Topologized Fundamental Groups: The Whisker Topology, Part 3

Posted on June 30, 2022 by Jeremy Brazas

In Part 1 and Part 2, I gave detailed introductory exposition about the whisker topology on the fundamental group. In general, this topologized fundamental group \pi_{1}^{wh}(X,x_0) is a left topological group and therefore a homogeneous space. Moreover, whenever this group is T_1, it’s also zero-dimensional. This bit about zero-dimensionality is a huge restriction which tells us the whisker topology is a very fine topology. Most often, we’re interested in metrizable spaces so it’s natural to ask what we can acheive in this situation.

The proof I’m going to give isn’t too hard but it does require working through several individual steps. If nothing else, this proof is basically the same as a certain proof that a universal covering space of a locally path-connected, metrizable space is metrizable. 

(Pseudo)metrizability of the whisker topology

Supposing that X is metrizable, pick a metric d that induces the topology of X. Let’s define a distance function \rho on \pi_{1}^{wh}(X,x_0). For loops \alpha,\beta based at x_0, set

\rho([\alpha],[\beta])=\inf\{diam(\epsilon)\mid [\epsilon]=[\alpha^{-}\cdot\beta]\}

Now, it’s possible that \rho([\alpha],[\beta])=0 even when [\alpha]\neq [\beta] so we should only expect that \rho will be a pseudometric in general. Symmetry is pretty clear from the definition. The triangle inequality is maybe a little less clear but working it out is a nice exercise. Let’s do it. There are some different ways to write proofs like this involving infimums and supremums that use a variety of established/known facts but these are pretty efficient (though they are by contradiction).

Lemma: If \alpha,\beta are loops in X based at x_0 and \gamma=\alpha\cdot\beta, then diam(\gamma)\leq diam(\alpha)+diam(\beta).

Proof. Recall the supremum definition of the diameter of a loop. We’ll use the general fact that \sup(S)>a if and only if there exists s\in S with s>a. If diam(\gamma)> diam(\alpha)+diam(\beta), then there exists a,b\in [0,1] with d(\gamma(a),\gamma(b))>diam(\alpha)+diam(\beta). If a,b\in [0,1/2], then d(\gamma(a),\gamma(b))=d(\alpha(a/2),\alpha(b/2))>diam(\alpha), which is a contradiction. If a,b\in [1/2,1], then d(\gamma(a),\gamma(b))=d(\beta(2a-1),\alpha(2b-1))>diam(\beta). Without loss of generality, suppose a\in[0,1/2] and b\in [1/2,1]. Then

d(\alpha(2a),\beta(2b-1))= d(\gamma(a),\gamma(b))>diam(\alpha)+diam(\beta)\geq d(\alpha(2a),x_0)+d(x_0,\beta(2b-1))

but this is a violation of the triangle inequality in X. \square

Lemma: Given [\alpha],[\beta],[\gamma]\in\pi_{1}^{wh}(X,x_0), \rho([\alpha],[\gamma])\leq \rho([\alpha],[\beta])+\rho([\beta],[\gamma]).

Proof. We’ll play a similar game in this lemma using the analagous elementary fact for infimums:  a>\inf(S) if and only if there exists s\in S with a>s,

Suppose \rho([\alpha],[\gamma])>\rho([\alpha],[\beta])+\rho([\beta],[\gamma]). Since \rho([\alpha],[\gamma])-\rho([\beta],[\gamma])>\rho([\alpha],[\beta]), there exists a loop \epsilon with [\epsilon]=[\alpha^{-}\cdot\beta] and \rho([\alpha],[\gamma])-\rho([\beta],[\gamma])>diam(\epsilon). Since \rho([\alpha],[\gamma])-diam(\epsilon)>\rho([\beta],[\gamma]), there exists a loop \delta with [\delta]=[\beta^{-}\cdot\gamma] and \rho([\alpha],[\gamma])-diam(\epsilon)>diam(\delta). Now [\epsilon\cdot\delta]=[\alpha^{-}\cdot \beta][\beta^{-}\cdot\gamma]=[\alpha^{-}\cdot\gamma] and \rho([\alpha],[\gamma])>diam(\epsilon)+diam(\delta). We have diam(\epsilon)+diam(\delta)\geq diam(\epsilon\cdot\delta) from the previous lemma so \rho([\alpha],[\gamma])>diam(\epsilon\cdot\delta). Since [\epsilon\cdot\delta]=[\alpha^{-}\cdot\gamma], this is a contradiction. \square.

With the triangle inequality in hand, we have a pseudometric! Now we should check that it induces the topology \pi_{1}^{wh}(X,x_0). In general, if d is a (pseudo)metric on a set S, O_d(s,r)=\{t\in S\mid d(t,s)<r\} will denote the open ball of radius r about s.

Lemma: The pseudometric induces the topology of \pi_{1}^{wh}(X,x_0).

Proof. Let B([\alpha],U) be a basic neighborhood of [\alpha] in \pi_{1}^{wh}(X,x_0). Find r>0 such that O_{d}(x_0,r)\subseteq U where O_{d}(x_0,r) is the open r-ball about x_0. Suppose [\beta]\in O_{\rho}([\alpha],r). Since \rho([\alpha],[\beta])<r, there exists a loop \gamma based at x_0 with [\gamma]=[\alpha^{-}\cdot\beta] and diam(\gamma)<r. Since \gamma has image in O_{d}(x_0,r), it has image in U. This gives [\beta]=[\alpha\cdot\epsilon]\in B([\alpha],U).

For the other direction, suppose r>0 and consider the neighborhood O_{\rho}([\alpha],r). We’ll show that B([\alpha],O_d(x_0,r/3))\subseteq O_{\rho}([\alpha],r). Suppose [\beta]\in B([\alpha],O_d(x_0,r/3)). Write [\beta]=[\alpha\cdot\epsilon] for loop \epsilon in O_d(x_0,r/3)). Then [\epsilon]=[\alpha^{-}\cdot\beta] and diam(\epsilon)\leq \frac{2r}{3}. Thus \rho([\alpha],[\beta])\leq \frac{2r}{3}<r, showing [\beta]\in O_{\rho}([\alpha],r). \square

Theorem: If X is metrizable, then \pi_{1}^{wh}(X,x_0) is pseudometrizable.

Remark: This proof also goes through for the space \widetilde{X} of all path-homotopy classes of paths starting at x_0 with the whisker topology of which \pi_{1}^{wh}(X,x_0) is a subspace. Indeed, whenever X is metrizable, \widetilde{X} is pseudometrizable (See [5,Lemma 2.12]). Why I’m taking a second to do the proof is to emphasize that it works for ALL metrizable spaces, not just those for those having the usual assumptions of covering space theory. 

But…can we get rid of the “pseudo?” In a pseudometric space it’s possible to have distinct points that have zero-distance from each other. Such a space is not even T_0.  This can certainly happen for \pi_{1}^{wh}(X,x_0) and we saw an instance of it in an earlier post. But it’s also true that a pseudometrizable space is metrizable if and only if it’s Hausdorff and we characterized the Hausdorff property for \pi_{1}^{wh}(X,x_0) in Part 2. Moreover, we showed that \pi_{1}^{wh}(X,x_0) is zero-dimensional whenever it’s Hausdorff.

Let’s summarize. Taking several of our established results, we have the following.

Theorem: Let X be a path-connected, metrizable, homotopically Hausdorff space and x_0\in X. Then \pi_{1}^{wh}(X,x_0) is a homogeneous, zero-dimensional, metrizable space.

This applies in lots of cases, include all one-dimensional and planar spaces.

Separability

Now I’m wondering about separability. Here, I’m still assuming (X,d) is a homotopically Hausdorff metric space so that we’re in the situation of the previous theorem. Clearly \pi_{1}^{wh}(X,x_0) is separable if it is countable. That’s not so interesting. What about in other cases? Honestly, I’m not too sure how to characterize this in general. Here’s how I think we can understand it. Let U_n be the 1/n-ball about x_0 in X. If S=\{s_1,s_2,s_3,\dots\} was a dense countable subset of \pi_{1}^{wh}(X,x_0), then for every homotopy class g\in \pi_{1}^{wh}(X,x_0) and neighborhood n\in\mathbb{N}, we’d have s_k\in B(g,U_n) for some k=k(g,n). Another way to say this is that there is a sequence s_{k(g,1)},s_{k(g,2)},s_{k(g,3)},\dots in S where the products gs_{k(g,1)}^{-1},gs_{k(g,3)}^{-1},gs_{k(g,3)}^{-1},\dots have representatives that get arbitrarily small. So we can take any element of the fundamental group and make it small by multiplying on the right by inverses of the elements of S.

In the earring group, a Cantor diagonalization argument should be possible to see it’s not separable. I don’t want to get hung up on this here and get caught up in examples, but it’s a curious question and maybe someone else wants to go down this rabbit hole. This could make a nice student project.

Question: If X is a homotopically Hausdorff metric space, when is \pi_{1}^{wh}(X,x_0) separable? What are some nice examples illustrating when it is and isn’t separable?

To finish the post, I’m going to go back to why I actually spent 3 posts writing about a \pi_1  topology I’ve never really cared much about.

Why am I starting to care more about the whisker topology?

I’ll be honest… I used to think the whisker topology wasn’t too interesting or useful. Here, I’ll spend a little time trying to explain why I’ve changed my tune on this. I want to be clear that I’m not so much trying to publicize my own results as must as I want to publicize the fact that I don’t really understand a certain naturally occuring group as well as I’d like to. 

Consider the one-point union Y=\mathbb{E}_1\vee\mathbb{E}_n of the earring space and the n-dimensional earring as seen below for n=2.

The one-point union \mathbb{E}_1\vee\mathbb{E}_2.

The group \pi_n(Y) is fairly complicated because \pi_1(Y)=\pi_1(\mathbb{E}_1) is uncountable and infinitary. Unlike with S^1\vee S^2, the usual \pi_1-action is not enough to really understand \pi_2. Recently, I proved that \pi_n(Y) canonically embeds into the group \prod_{j\in\mathbb{N}}\oplus_{\pi_1(\mathbb{E}_1)}\mathbb{Z}. I’d like to understand the image in this product of \mathbb{Z}‘s better.

We can think of elements of G= \prod_{j\in\mathbb{N}}\oplus_{\pi_1(\mathbb{E}_1)}\mathbb{Z} as functions g:\mathbb{N}\times \pi_1(\mathbb{E}_1)\to\mathbb{Z}, which has finite support in the second variable: for each j\in\mathbb{N}, the set \{[\alpha]\in\pi_1(\mathbb{E}_1)\mid g(j,[\alpha])\neq 0\} is finite.

The canonical homomorphism \Psi:\pi_n(Y)\to G is not onto! Here’s the characterization of the image: an element g\in G is in the image of \Psi and therefore uniquely represents a homotopy class in \pi_n(Y) if and only if \{[\alpha]\in \pi_1(\mathbb{E}_1)\mid g(j,[\alpha])\neq 0\} is countable and has compact closure in \pi_{1}^{wh}(\mathbb{E}_1). So elements of \pi_n(Y) are classified by functions g:\mathbb{N}\times\pi_1(\mathbb{E}_1)\to\mathbb{Z} with countable support and such that that g is non-trivial on only a “bounded” set in the second variable (but only with the whisker topology!). Even though I have a geometric understanding of this group and feel this description is good enough for most any use I can imagine, there is topology embedded in the indexing sets. My instinct tells me that it’s built-in topological nature can’t be done away with but this idea is not really something I know how to formalize. Anyway, I never would have guessed the whisker topology would become relevant to wild higher homotopy theory.

So what are the compact subsets?

That stuff about homotopy groups is why I was interested in understanding compact subsets of \pi_{1}^{wh}(X,x_0). The theorem we worked out in these posts says that \pi_{1}^{wh}(X,x_0) is often zero-dimensional and metrizable. In this case, a compact subset K of \pi_{1}^{wh}(X,x_0) will be a zero-dimensional compact metric space.

This leaves open the possibility K could be homeomorphic to a Cantor set, which does seem reasonable. Here’s an example.

Example: X=\prod_{n=1}^{\infty}\mathbb{RP}_2 is the infinite direct product of copies of the projective plane, then \pi_1(X,x_0) is abelian and therefore \pi_{1}^{wh}(X,x_0)\cong \prod_{n=1}^{\infty}\mathbb{Z}/2\mathbb{Z} is a topological group by results in Post 1. But we can say more. We can identify \pi_{1}^{wh}(\mathbb{RP}^2,x) with the discrete group \mathbb{Z}/2\mathbb{Z} for any choice of basepoint. Now, a basic neighborhood of the basepoint in \prod_{n=1}^{\infty}\mathbb{RP}_2 can be taken to be of the form V= \prod_{i=1}^{k}U\times \prod_{i > k}\mathbb{RP}_2 where U is a contractible neighborhood of the basepoint in \mathbb{RP}_2 .Therefore, if \alpha=(\alpha_i) is a loop in X, then B([\alpha],V)=\prod_{i=1}^{k}\{[\alpha_i]\}\times \prod_{i > k}\mathbb{Z}/2\mathbb{Z}. This is the key idea needed to check that \pi_{1}^{wh}(X,x_0) is canonically isomorphic to the topological group \prod_{n=1}^{\infty}\mathbb{Z}/2\mathbb{Z} with the product topology. In particular, X is a metric space for which \pi_{1}^{wh}(X,x_0) is homeomorphic to a Cantor set. \square

So, yes, a compact subset K of \pi_{1}^{wh}(X,x_0) can be as complicated as a Cantor set. But descriptive set theory tells us it can’t be much worse. Every Polish space is the disjoint union of a countable scattered subspace and (possibly) a perfect set. In our situation, zero-dimensionality ensures that if there is a perfect subset, it must be a Cantor set.

Theorem: Let X be a path-connected, metrizable, homotopically Hausdorff space and x_0\in X. If K\subseteq \pi_{1}^{wh}(X,x_0) is a compact, then K is either homeomorphic to a countable compact ordinal or the union of a countable scattered space and a Cantor set.

I don’t know that this solves the entirety of my lack of understanding of my representation of \pi_n(Y) using the whisker topology but it does give me a way to handle compact sets in \pi_{1}^{wh}(X,x_0) if I need to.

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.

Posted in Covering Space Theory, Fundamental group, Higher Homotopy groups, homotopically Hausdorff, Infinite Group Theory, Infinite products, topological fundamental group, Topological groups | 1 Comment