Topologized Fundamental Groups: The Quotient Topology Part 4 (Subgroup Classifications)

Posted on October 7, 2023 by Jeremy Brazas

This is the last in a sequence of posts about the quotient topology. This one is about how the topological structure of \pi_{1}^{qtop}(X,x_0) can be used to classify certain generalizations of covering maps for locally complicated spaces. Sometimes it still amazes me how this works out. For a sneak peak, see the table at the end of the post.

Ordinary covering maps

One of the things we love about covering space theory is that covering spaces over a locally nice (path-connected, locally path connected, and semilocally simply connected) space X are completely classified by the subgroups of \pi_1(X,x_0). In particular, for every subgroup H\leq \pi_1(X,x_0), there is a based covering map p:(E,e_0)\to (X,x_0) such that p_{\#}(\pi_1(E,e_0))=H. Moreover, p is unique up to based homeomorphism for given H and the covering space can be built using a “standard construction.” In particular, if H=1 is the trivial subgroup, then E is simply connected and we call p a universal covering map.

Now, let’s say we drop the semilocally simply connectedness assumption. So X is still path-connected and locally path connected but it’s not semilocally simply connected so there is some point in X where arbitrarily small loops are not be null-homotopic. In this case, X won’t have a simply connected covering space. In extreme cases like the harmonic archipelago or Griffith twin cone, the only covering map over X will be the identity map. However, in general, X is likely to still have lots of non-trivial covering spaces. Standard arguments still show that if p_1:E_1\to X and p_2:E_2\to X are based covering maps that both correspond to the same subgroup H\leq \pi_1(X,x_0) under the usual correspondence. That is, if (p_i)_{\#}(\pi_1(E_i,e_i))=H for i=1,2, then there is a based homeomorphism h:E_1\to E_2 such that p_2\circ h=p_1. Yep, go back and check your algebraic topology textbook. The “semilocally simply connectedness” condition is only needed for the existence of covering spaces. It has nothing to do with uniqueness up to homeomorphism. Now…if you thow out the locally path-connected condition, then you could start getting into trouble. But let’s not go down that path.

Uniqueness still working out means that the covering spaces over X are still classified by the subgroups of \pi_1(X,x_0) it’s just that not ALL subgroups of \pi_1(X,x_0) correspond to covering spaces. So maybe there is something special about these particular subgroups of \pi_1(X,x_0) that do happen to admit a corresponding covering map. It sure would be cool if they are classified by a topological property related to \pi_1(X,x_0). But what could it be? Does a subgroup H\leq \pi_1(X,x_0) correspond to a covering map if and only if H is open in \pi_{1}^{qtop}(X,x_0)? Or closed? Or some other topological property a subgroup might have?

It turns out that there is such a classification. However, we need a definition.

Deifnition: The core of a subgroup H of G is the largest normal subgroup of G contained in H. You can define this explicitly as Core(H)=\bigcap_{g\in G}gHg^{-1}.

Apprently this construction is important in finite group theory.

Theorem: Suppose X is path connected and locally path connected. A subgroup H\leq \pi_1(X,x_0) admits a based covering map p:(E,e_0)\to (X,x_0) with p_{\#}(\pi_1(E,e_0))=H if and only if H has an open core in \pi_{1}^{qtop}(X,x_0). Moreover, there is a canonical Galois correspondence:

{Equiv. classes of based coverings over X} \leftrightarrow {subgroups of \pi_{1}^{qtop}(X,x_0) with open core}.

Semicovering maps

A while back, I wrote a couple of posts about semicovering maps. This is a weak version of covering map so every covering map is a semicovering map. The two notions agree on “locally nice spaces” but start to pick out intricacies for non semilocally simply connected spaces. It turns out that semicoverings are classified by the open subgroups of \pi_1(X,x_0). I realized this when I was working on my thesis after defining semicoverings and, honestly, I was totally shocked!

Theorem: Suppose X is path connected and locally path connected. A subgroup H\leq \pi_1(X,x_0) admits a based semicovering map p:(E,e_0)\to (X,x_0) with p_{\#}(\pi_1(E,e_0))=H if and only if H is open in \pi_{1}^{qtop}(X,x_0). Moreover, there is a canonical Galois correspondence:

{Equiv. classes of based semicoverings over X} \leftrightarrow {open subgroups of \pi_{1}^{qtop}(X,x_0)}.

Continuous path-lifting maps

After seeing the classification of covering maps and semicovering maps for locally path-connected spaces, you might say well… what about other properties of subgroups? Do they correspond to some special class of generalized covering maps. I’d say, that this does often have a positve answer but there are some limitations. For instance, every closed subgroup of \pi_{1}^{qtop}(X,x_0) does admit a Serre fibration with totally path-disconnected (t.p.d.) fibers that corresponds to it. However, not all such maps come from closed subgroups – the class of Serre fibrations with t.p.d. fibers is super useful but is too big for this classification and it’s not known exactly what is topologically special about the Serre fibrations that do correspond from closed subgroups. This is a good problem.

However, here’s another example. For a based space (X,x), let P(X,x) be the space of paths in X starting at x with the compact-open topology. A map f:E\to X has continuous path-lifting (we’ll call it a CPL-map) if for each e\in E, the induced map P(f):P(E,e)\to P(X,p(e)), P(f)(\alpha)=f\circ\alpha is a homeomorphism. That is, f uniquely lifts paths and if a sequence or net of paths (starting at the same point) converges downstairs, then the lifts upstairs also converge. These maps classify subgroups H\leq  \pi_1(X,x_0) with a totally path-disconnected coset space!

Theorem: Suppose X is path connected and locally path connected. A subgroup H\leq \pi_1(X,x_0) admits a based CPL-map p:(E,e_0)\to (X,x_0) with p_{\#}(\pi_1(E,e_0))=H if and only if the coset space \pi_{1}^{qtop}(X,x_0)/H is totally path-disconnected. Moreover, there is a canonical Galois correspondence:

{weak equiv. classes of based CPL-maps over X} \leftrightarrow {subgroups H with \pi_{1}^{qtop}(X,x_0)/H tot. path-disc.}.

Note that a certain notion of “weak” equivalence classes needs to be used here but this is pretty superficial. Instead of using homeomorphisms, you use bijective weak homotopy equivalences…so it’s still pretty rigid. This becomes equivalence up to homeomorphism if you restrict to a certain category. See https://arxiv.org/abs/2006.03667 for more details.

Here’s a table summarizing the classifications that are known and a couple that are relevant but not fully understood

In the table, classes of based maps over a given space (X,x_0) appear in the left column. Next to each one in the middle column is the topological property a subgroup may or may not have with respect to the quotient topology on the fundamental group. The subgroups having that property completely classify the type of maps in the left column.

Type of Map Property of classifying subgroup H in \pi_{1}^{qtop}(X,x_0) Classificaiton up to equivalence?
Covering map H has open core Yes
Semicovering map H is open Yes
Inverse limit of covering maps H is an intersection of open normal subgroups Yes
Inverse limit of semicovering maps H is an intersection of open subgroups Yes
CPL (continuous path-lifting) map \pi_{1}^{qtop}(X,x_0)/H is totally path disconnected Yes*, in a certain category
Hurewicz fibrations with unique path lifting ? ?
(mystery class ?? of Serre fibrations with unique path lifting) H is closed ?

What’s your favorite property a subgroup might have? Assuming it at least implies “closed,” we could ask what kind of generalized covering maps correspond to subgroups with that property.

Go back to Part I, Part II, or Part III.

Posted in Algebraic Topology, core of a subgroup, Fundamental group, Generalized covering space theory, Griffiths twin cone, harmonic archipelago, Quasitopological groups, quotient topology, semicovering, topological fundamental group, Uncategorized | 7 Comments

Topologized Fundamental Groups: The Quotient Topology Part 3 (Why isn’t it always a topological group?)

Posted on April 18, 2023 by Jeremy Brazas

In Part 1, I mentioned that one of the surprising things about the natural quotient topology on the fundamental group is that the resulting group with topology \pi_{1}^{qtop}(X,x_0) often fails to be a topological group. In fact, I’d say it’s usually not a topological group when \pi_{1}^{qtop}(X,x_0) is not discrete. This surprises a lot of folks because the concatenation function c:\Omega(X,x_0)\times \Omega(X,x_0)\to \Omega(X,x_0), (\alpha,\beta)\mapsto \alpha\cdot\beta IS continuous and it descends to the multiplication operation \mu:\pi_{1}^{qtop}(X,x_0)\times \pi_{1}^{qtop}(X,x_0)\to \pi_{1}^{qtop}(X,x_0), \mu([\alpha],[\beta])=[\alpha\cdot\beta]. In this post, I’ll detail out an important example showing why \mu isn’t always continuous. 

The reason why the “obvious” proof doesn’t work is actually not to hard to see: If q:\Omega(X,x_0)\to \pi_{1}^{qtop}(X,x_0) is the natural quotient map and the product q\times q is a quotient map, then because \mu \circ (q\times q)=q\circ c, the universal property of quotient maps ensures that \mu is continuous. However, in genreal topology the product of two quotient maps is NOT always a quotient map!

Ok, so the obvious proof doesn’t work….but this doesn’t really constitute a counterexample, does it? In fact, it’s not really known if, for given space X, q\times q being quotient is equivalent to \mu being continuous. So let’s talk through a counterexample. This one is a slightly modified version of the example given by Paul Fabel (MS State) in the 2011 paper [1].

We don’t have to go too far in our search for examples. Consider the usual infinite earring space \mathbb{E}=\bigcup_{n\in\mathbb{N}}C_n where C_n\subseteq \mathbb{R}^2 is the circle of radius \frac{1}{n} centered at (\frac{1}{n},0). We take the basepoint x_0 to be the origin. For brevity, let’s let L=\Omega(\mathbb{E}_n,x_0) denote the loop space and G=\pi_{1}^{qtop}(\mathbb{E},x_0) be the earring group with the natural quotient topology.

the infinite earring space

The infinite earring space

Let \ell_n:[0,1]\to C_n be the standard paramterization of C_n going once around counterclockwise and x_n=[\ell_n]\in G. The group G is locally free and although we can form infinite words in the letters x_n in G, we only need to use finite words to work through this example. If a and b are elements of L or G, we’ll use [a,b] to denote the commutator aba^{-1}b^{-1} and w^n will be the n-fold product www\cdots w of a word w with itself. We’ll write e for both the constant loop at x_0 and it’s homotopy class, which is the identity element of G.

We’re going to define two doubly indexed sequences and make some observations about them. Some of these observations are going to be something of a warm-up for the real argument later.

First doubly indexed sequence: Let w_{m,n}=[x_m,x_n]^{m} for m,n\in\mathbb{N}.

  1. Note that w_{n,n}=e and the word length otherwise is |w_{m,n}|=4m.
  2. Fix n and suppose n<m_1<m_2<m_3<\cdots. Then w_{m,n_k}=[x_{m_k},x_{n}]^{m_k} is a sequence of reduced words of unbounded length. In fact, the number of appearrances of the letter x_n is unbounded as k\to\infty. Hence, if we considering representing loops \alpha_k\in w_{m_k,n}, then, for each k, the loop \alpha_k may not be reduced (in the sense of not having any null-homotopic subloops) but because w_{m_k,n} is a reduced word, the loop \alpha_k must go around the circle C_n (in one direction or the other) at least 2m_k-many distinct times. Since the topology of L agrees with the topology of uniform convergence, there is no way for the sequence \{\alpha_k\} with to converge to any loop in L.
  3. Fix m. Then [\ell_m,\ell_n]^{m} has fixed “word length” as n\to \infty and converges to the null-homotopic loop [\ell_m,e]^m. Therefore, \{w_{m,n}\}_{n}\to e when m is fixed, showing that e is a limit point of the set W=\{w_{m,n}\mid m,n>1, m\neq n\}. In fact, we show something stronger next.
  4. Let U be an open neighborhood of e in G. Then q^{-1}(U) is an open neighborhood of the constant loop at x_0 in L. Since L has the compact-open topology, we can find an open neighborhood O of x_0 in \mathbb{E} such that all loops ([0,1],\{0,1\})\to (O,x_0) lie in q^{-1}(U). Find N such that C_N\subseteq O. Let m,n\geq N with m\neq n. Since \ell_m and \ell_{n} have image in O, so does [\ell_{m},\ell_{n}]^{m}. Thus [\ell_{m},\ell_{n}]^{m}\in q^{-1}(U) which gives w_{m,n}=q([\ell_{m},\ell_{n}]^{m})\in U for all m,n\geq N.

Second doubly indexed sequence: Let v_{m,n}=[x_1,x_m]^{n} for m,n\in\mathbb{N}.

  1. Note that v_{1,n}=e is trivial and the word length otherwise is |v_{m,n}|=4n.
  2. Fix m>1 and suppose n_1<n_2<n_3<\cdots. Then v_{m,n_k}=[x_1,x_{m}]^{n_k} is a sequence of reduced words of unbounded length such that the number of appearrances of x_1 is unbounded as k\to\infty. Hence, if we considering representing loops \alpha_k\in v_{m,n_k}, then \alpha_k must have at least n_k-many copies of the loop \ell_{1}^{\pm} as a subloop. Thus there is no way for the sequence \{\alpha_k\} to a loop in L.
  3. Fixing n, note that since [\ell_1,\ell_{m}]^{n} has fixed length, it converges uniformly to [\ell_1,e]^{n}, which is null-homotopic. The continuity of q, then ensures that \{v_{m,n}\}_{m>1}\to e in G.
  4. It follows from 3. that the identity element e \in G is a limit point of V=\{w_{m,n}\mid m>1, m\neq n\}.

Now, we use the sets W=\{w_{m,n}\mid m,n>1,m\neq n\} and V=\{w_{m,n}\mid m,n>1,m\neq n\} to define the set we really want to focus on. In particular, let

C=WV=\{w_{m,n}v_{m,n}\mid m,n>1,m\neq n\}

By definition C consists of the doubly indexed sequences:

w_{m,n}v_{m,n}=[x_m,x_n]^{m}[x_1,x_m]^{n}.

We add the restrictions m,n>1 and m\neq n just to ensure that these words are, in fact, reduced words. In particular, e\notin C. Let’s show that C is a desired counterexample, that is, it’s a closed subset C of G such that \mu^{-1}(G) is not closed in G\times G. First the easy direction.

Proposition 1: \mu^{-1}(C) is not closed in G\times G.

Proof. Since e\notin C and \mu(e,e)=e, the pair (e,e) cannot be an element of \mu^{-1}(C). However, by construction of C, we have W\times V\subseteq \mu^{-1}(C). The fourth observation for each doubly-indexed sequence above tells us that e is both a limit point of W and V in G. Thus (e,e) is a limit point of W\times V in G\times G. Since W\times V\subseteq \mu^{-1}(C), (e,e) is a limit point of \mu^{-1}(C). Since \mu^{-1}(C) is missing at least one limit point in G, it is not closed in G\times G. \square

For the other direction, we’ll need some observations. The first one is from genreal topology.

Definition 2: A subset C of a topological space X is sequentially closed if C is closed under limits of convergent sequences, that is, if whenever \{x_n\}\to x in X and x_n\in C for all n\in\mathbb{N}, we have x\in C.

Definition 3: A topological space X is a sequential space if a set C\subseteq X is closed if and only if C is sequentially closed.

All metrizable spaces are sequential spaces and all quotient spaces of sequential spaces are also sequential spaces. In particular, since the loop space L is metrizable and G is a quotient of L, both L and G are sequential spaces.

The world would be a different place if whenever we have a quotient map q:X\to Y and a convergent sequence \{y_n\}\to y in Y, we could lift it to a convergent sequence in X that maps to \{y_n\}\to y. “Biquotient maps” have this lifting property but ordinary quotient maps do not. However, even when lifting all convergent sequences is not at our disposal, we can often at least find a subsequence that lifts.

Lemma 4: If X is a Hausdorff sequential space and q:X\to Y is a quotient map where Y is Hausdorff, then for every convergent sequence \{y_n\}\to y in Y, there exists n_1<n_2<n_3<\cdots and a convergent sequence \{x_{n_k}\}\to x in X such that q(x_{n_k})=y_{n_k} and q(x)=y.

Proof. If there is a subsequence of y_n that is constant at y, then the conclusion is easy to acheive. Suppose y\notin C=\{y_n\mid n\in\mathbb{N}\}. Then C is not closed in Y. Since q is quotient, q^{-1}(C) is not closed in X. Since X is assumed to be sequential, q^{-1}(C) is not sequentially closed in X. Thus there exists a convergent sequence \{x_k\}\to x in X such that x_k\in q^{-1}(C) for all k\geq 1 and x\notin q^{-1}(C). We have q(x_k)=y_{n_k} for some n_k\in\mathbb{N}. If the sequence of integers \{n_k\} is bounded by M, then \{x_k\} has image in the finite set F=\{x_1,x_2,\dots,x_M\}. But since X is Hausdorff, F is closed and so we must have x\in F\subseteq q^{-1}(C); a contradiction. Thus \{n_k\} must be unbounded. In particular, we may replace \{n_k\} with an increasing subsequence. Doing so gives n_1<n_2<n_3<\cdots where q(x_{n_k})=y_{n_k}. The continuity of q ensures \{y_{n_k}\}\to q(x) and since Y is Hausdorff and \{y_n\}\to y, we must have q(x)=y.

We’re going to apply this lemma to the quotient map q:L\to G. Since L is metrizable, it is sequential and Hausdorff. The only hypothesis that may be a little unclear is that G is Hausdorff.

Theorem 5: G is Hausdorff.

Proof. Let X_n=\bigcup_{k\leq n}C_k be the finite wedge of circles so that \pi_1(X_n,x_0) is the free group F_n on the letters x_1,x_2,\dots,x_n. From Part 2, we know that F_n is discrete. Also, we have a canonical retraction r_n:\mathbb{E}\to X_n that collapse \bigcup_{k>n}C_k to x_0 and, which induce continuous homomorphisms R_n:G\to F_n. We also have bonding retractions r_{n+1,n}:X_{n+1}\to X_n, which collapse C_{n+1} to x_0. These induce maps R_{n+1,n}:F_{n+1}\to F_n on the discrete free groups. The inverse limit \varprojlim_{n}(F_n,R_{n+1,n}) is topologized as a subspace of the product \prod_{n}F_n. Since this product is Hausdorff, so is \varprojlim_{n}(F_n,R_{n+1,n}). Finally, note that since R_{n+1,n}\circ R_{n+1}=R_n, we have an induced continuous homomorphism \phi:G\to \varprojlim_{n}(F_n,R_{n+1,n}) given by \phi(g)=(R_1(g),R_2(g),R_3(g),\dots). We proved in a previous post that \phi is injective (warning! \phi is not a topological embedding). Therefore, since G continuously injects into a Hausdorff group, G must be Hausdorff. \square

Finally, we have the ingredients to finish our argument. I love this next part of the argument. It highlights why doubly indexed sequences are the thing to use.

Lemma 6: The set C=\{w_{m,n}v_{m,n}\mid m,n>1,m\neq n\} is closed in G.

Proof. Since G is the quotient of the sequential space L, G is sequential. Therefore, it suffices to prove that C is sequentially closed. Let g_{k}\to g be a convergent sequence in G such that g_k\in C for all k\geq 1. In particular, we have

g_k=w_{m_k,n_k}v_{m_k,n_k}=[x_{m_k},x_{n_k}]^{m_k}[x_1,x_{m_k}]^{n_k}

for integers m_k,n_k>1 with m_k\neq n_k. We must check that g\in C. Note that by Lemma 4, there exists integers k_1<k_2<k_3<\cdots and a convergent sequence of loops \{\alpha_j\}\to \alpha in L such that [\alpha_j]=g_{k_j} and [\alpha]=g.

Case I: Suppose both sequence of integers \{m_k\} and \{n_k\} are bounded. Then F=\{g_k\mid k\geq 1\} is a finite subset of C. Since G is Hausdorff by Theorem 5, F is closed and it follows that g\in F\subseteq C.

Case II: Suppose \{n_k\} is unbounded. Then \{n_{k_j}\}_{j\geq 1} is unbounded and the (possibly unreduced) loop \alpha_j must go around the circle C_1 in one-direction or the other at least 2n_{k_j}-many times (this is contributed by the subword v_{m_{k_j},n_{k_j}} of g_k). But this makes it impossible for the sequence \{\alpha_j\} to converge uniformly to any loop; a contradiction. Therefore, this case is impossible.

Case III: Suppose \{m_k\} is unbounded and \{n_k\} is bounded. Then \{m_{k_j}\}_{j\geq 1} is unbounded and \{n_{k_j}\}_{j\geq 1} is bounded. By the Pigeon-Hole Principle, there must exist some N such that n_{k_j}=N for infinitely many j. By replacing k_j with the subsequence of k_j for, which n_{k_j}=N, we may assume that n_{k_j}=N for all j\geq 1. Recall that (the possibly unreduced loop) \alpha_{k_j} represents [x_{m_{k_j}},x_{N}]^{m_{k_j}}[x_1,x_{m_{k_j}}]^{N} and so \alpha_j must go around the circle C_N in one-direction or the other at least (2m_{k_j})-many times (this is contributed by the subword w_{m_{k_j},N} of g_k). But this makes it impossible for the sequence \{\alpha_j\} to converge uniformly to any loop in L; a contradiction. Therefore, this case is impossible.

Thus only Case I is possible and in this case, we verified that g\in C. \square

Theorem [Fabel]: The infinite earring group G equipped with the natural quotient topology is not a topological group.

Proof. By Lemma 6, C=\{w_{m,n}v_{m,n}\mid m,n>1,m\neq n\} is closed in G. However, by Proposition 1, \mu^{-1}(C) is not closed in G\times G where \mu:G\times G\to G is the group operation. Thus \mu is not continuous and G is not a topological group.

Despite not being a topological group, we discussed in Part 1 that \pi_{1}^{qtop}(X,x_0) is always a quasitopological group. Thus G is a nice example of a quasitopological group that is “close” to being a topological group but doesn’t quite make the cut.

Deeper discussion: Even if you make it through the logic of the proof I presented here, you might feel that you still don’t really have a good feel for what’s going on. This is a common feeling about this continuity fiasco, so I’m going to offer another perspective that is a little informal but which I think could be made formal. The reason that G fails to be a topological group is because L and G fail to be locally compact in a certain way.

Our initial observations about the words w_{m,n} tells us that we can topologically identify W\cup \{e\} with the planar subset B=\{(0,0)\}\cup \{(\frac{1}{n},\frac{1}{mn})\mid m,n\geq 1\}. Here I’m fudging the fact that we needed to get rid of the diagonal and the first row and column in V but that’s superficial.

The planar set $latex B$ consisting of points clustering toward the origin.

The planar set B consisting of the origin and points (\frac{1}{n},\frac{1}{mn}).

This space B is important because every non-locally compact metric space contains a homeomorphic copy of B as a subspace. In the same manner, we’ve found a copy of B within G.

The elements of V=\{v_{n,m}\mid m,n>1, m\neq n\} work a little differently. If we lay out these points on the integer grid in the same way (so v_{m,n} is at (m,n)), then none of the vertical sequences converge. In fact, any sequence which meets infinitely many rows cannot converge. However, along the fixed row at height n, \{v_{m,n}\}_{m} converges to the identity e=[x_1,e]^{n}. This essentially means that V\cup\{e\} is a copy of the Frechet Uryshon Fan, which I’ll denote by A=\{a_0\}\cup\{a_{m,n}\mid m,n\in\mathbb{N}\}. The Frechet Urysohn Fan can be constructed by infinitely many disjoint convergent sequences S_m=\{a_0\}\cup \{a_{m,n}\mid n\geq 1\} with their limits points a_0 identified. We then give A the weak topology with respect to the subspaces S_m.

The Frechet-Uryshon Fan, a fan of converging sequences

The Frechet-Uryshon Fan

The set V\cup\{e\} is a copy of A inside of G. We’ve managed to play these embeddings of A and B off of each other to win the game. In particular, we’ve embedded these non-locally compact subspaces in G in a way that (1) doesn’t allow for them to algebraically intermingle and (2) lifts to similar behavior in L. In my view, that is what really is causing discontinuity. Admittedly, this is a little informal but all of the discontinuity proofs I’ve seen, even for higher homotopy groups [2] and the \pi_1-action [3] all kind of look like this one. So, I suspect there is a deeper thread here to unravel. 

Go to Part I, Part II, or Part IV.

References

[1] Paul Fabel, Multiplication is discontinuous in the Hawaiian earring group (with the Quotient Topology). Bulletin of the Polish Academy of Sciences 59 (2011), no. 1, 77-83.

[2] Paul Fabel, Compactly generated quasitopological homotopy groups with discontinuous multiplication, Topology Proc. 40 (2012) 303–309.

[3] J. Brazas, On the discontinuity of the pi_1-action, Topology Appl. 247 (2018) 29-40.

Posted in compact-open topology, earring group, Free groups, Fundamental group, quotient topology, topological fundamental group, Topological groups, Uncategorized | Tagged , , , , , , , , , , | 5 Comments

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.

Go to Part I, Part III, or Part IV.

References.

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

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