The “tau topology” on the fundamental group

In the multipart series on topologies on fundamental groups, we’ve discussed the fundamental group with the quotient topology: $\pi_{1}^{qtop}(X,x_0)$ . This is defined as having the quotient topology with respect to the map $q:\Omega(X,x_0)\to \pi_{1}(X,x_0)$ , $q(\alpha)=[\alpha]$ that identifies homotopic loops. Here, $\Omega(X,x_0)$ is the space of loops $S^1\to X$ based at $x_0$ with the compact-open topology. While $\pi_{1}^{qtop}(X,x_0)$ is a quasitopological group, it often fails to be a topological group.

Here, we’ll finally construct the “tau topology” which is the topological group “fix” to the quotient topology construction. This is, in a sense, dual to the “fix” in which one uses a coreflection functor to push the quotient topology into a Cartesian closed subcategory of spaces, e.g. group objects in k-spaces. Rather, we employ a reflection functor that the forces the quotient topology construction directly into category of topological groups. What is rather remarkable (to me) is that many of the main computational theorems about fundamental groups have analogues in the topological group category.

The universal property of quotient maps implies that $\pi_{1}^{qtop}(X,x_0)$ has the finest topology on the fundamental group such that the map $q:\Omega(X,x_0)\to \pi_{1}(X,x_0)$ is continuous. Why would we want $q$ to be continuous? Well the topology on the fundamental group should remember something about the geometry of representing loops. For instance, if $q$ is continuous, then whenever a sequence of loops $\{\alpha_n\}_{n\geq 1}$ converges uniformly to a loop $\alpha$ , then we might desire this convergence of maps implies the that the sequence of corresponding homotopy classes $\{[\alpha_n]\}_{n\geq 1}$ converges to $[\alpha]$ in the fundamental group. When we make a topology finer, we make it easier to distinguish points using open sets. That $\pi_{1}^{qtop}(X,x_0)$ has the finest topology on $\pi_1$ with $q$ quotient means that we have maximized our ability to distinguish fundamental group elements topologically while retaining the geometric information provided via $q$ .

In the past two posts (Part I and Part II), I’ve written about the topological group reflection functor $\tau$ that takes in a group with topology $G$ and outputs a topological group $\tau(G)$ with the same underlying group as $G$ by removing the fewest number of open sets from the topology of $G$ until a topological group is obtained. This construction has the following universal property: the identity homomorphism $G\to \tau(G)$ is continuous and if $H$ is a topological group and $f:G\to H$ is a continuous group homomorphism, then $f:\tau(G)\to H$ is continuous. The main idea is to apply this universal construction to the quasitopological group $\pi_{1}^{qtop}(X,x_0)$ .

Definition: Let $\pi_{1}^{\tau}(X,x_0)$ denote the topological group $\tau(\pi_{1}^{qtop}(X,x_0))$ . We refer to the topology of this group as the $\tau$ -topology (read “tau-topology”) on the fundamental group.

Since $\pi_{1}^{qtop}:\mathbf{Top_{\ast}}\to \mathbf{qTopGrp}$ and $\tau:\mathbf{qTopGrp}\to \mathbf{TopGrp}$ are both functors (here we restrict the domain of $\tau$ to the full subcategory of quasitopological groups), the composition $\pi_{1}^{\tau}=\tau\circ \pi_{1}^{qtop}:\mathbf{Top_{\ast}}\to \mathbf{TopGrp}$ is a functor. If I had to choose a construction to call the “topological fundamental group” of a space, it’d be this one.

Lemma 1: The function $q:\Omega(X,x_0)\to \pi_{1}^{\tau}(X,x_0)$ that identifies homotopy classes of loops is continuous.

Proof. Since $q:\Omega(X,x_0)\to \pi_{1}^{qtop}(X,x_0)$ is continuous and the identity function $\pi_{1}^{qtop}(X,x_0)\to \tau(\pi_{1}^{qtop}(X,x_0))=\pi_{1}^{\tau}(X,x_0)$ is continuous. The composition of these two maps is the map in the lemma. $\square$

Theorem 2: The $\tau$ -topology is the finest topology on $\pi_1(X,x_0)$ such that

$\pi_1(X,x_0)$ is becomes a topological group,
$q:\Omega(X,x_0)\to \pi_{1}(X,x_0)$ is continuous.

Proof. By construction, $\pi_{1}^{\tau}(X,x_0)$ is a topological group and 2. was verified in Lemma 1. Let $G$ denote $\pi_1(X,x_0)$ equipped with some topology such that $G$ is a topological group and $q:\Omega(X,x_0)\to G$ is continuous. Since $q:\Omega(X,x_0)\to \pi_{1}^{qtop}(X,x_0)$ is quotient, the universal property of quotient maps gives that the identity function $id:\pi_{1}^{qtop}(X,x_0)\to G$ is continuous. Since $G$ is a topological group, the universal property of the $\tau$ -construction gives that the reflection $id:\tau(\pi_{1}^{qtop}(X,x_0))\to G$ is continuous. Thus $\tau(\pi_{1}^{qtop}(X,x_0))=\pi_{1}^{\tau}(X,x_0)$ has a finer topology than $G$ . $\square$

Is the $\tau$ -topology really different that the quotient topology? Theorem 2 implies the following.

Corollary 3: $\pi_{1}^{qtop}(X,x_0)$ is a topological group if and only if $\pi_{1}^{qtop}(X,x_0)= \pi_{1}^{\tau}(X,x_0)$ .

Hence, whenever $\pi_{1}^{qtop}(X,x_0)$ fails to be a topological group (and this is often), then the two cannot be the same.

Example 4: In a previous post about the quotient topology, we showed that if $\mathbb{E}$ is the infinite earring space, then $\pi_{1}^{qtop}(\mathbb{E},b_0)$ fails to be a topological group. Hence, $\pi_{1}^{\tau}(\mathbb{E},b_0)$ has a strictly coarser topology than that of $\pi_{1}^{qtop}(\mathbb{E},b_0)$ . Exactly which sets are open in $\pi_{1}^{qtop}(\mathbb{E},b_0)$ but not in $\pi_{1}^{qtop}(\mathbb{E},b_0)$ remains very much a mystery. The difficulty here is because the construction $G\mapsto \tau(G)$ is carried out through transfinite recursion and it’s not clear at all how large of an ordinal is required until the recursion stabilizes.

From Example 4, we might be worried that there is no way to understand anything about the $\tau$ -topology. However, the fact that $\pi_{1}^{qtop}(X,x_0)$ is a quasitopological group (has continuous left and right translations and continuous inversion) and not an arbitrary group with topology, provides some help.

Lemma 5 [1, Cor. 3.9]: If $G$ is a quasitopological group and $H\leq G$ , then $H$ is open in $G$ if and only if $H$ is open in $\tau(G)$ .

Hence, when we delete open sets from the quotient topology of $\pi_{1}^{qtop}(X,x_0)$ in order to construct $\pi_{1}^{\tau}(X,x_0)$ , we never delete any subgroups. Also, from generalized covering space theory (see Part IV) the open (normal) subgroups of $\pi_{1}^{qtop}(X,x_0)$ classify the semicoverings (coverings) of $X$ for all locally path-connected spaces. We have the same Galois correspondences for the $\tau$ -topology $ as well:

$\{\text{open subgroups of }\pi_{1}^{\tau}(X,x_0) \}\leftrightarrow \{\text{semicovering maps over }X\}$

$\{\text{open normal subgroups of }\pi_{1}^{\tau}(X,x_0)\}\leftrightarrow \{\text{covering maps over }X\}$

So while we might lose some information when passing from the quotient topology to the $\tau$ -topology we won’t lose a ton of information. In particular, we’ll never lose these covering-map type classifications.

Those curious enough for more might take a look at my initial paper on the construction or the proof of a topological Nielsen-Schreier theorem [2], which says that every open subgroup of a free topological group is also a free topological group. This is a theorem that filled a long-standing gap in general topological group theory. The only known proof uses topological group fundamental groups, specifically the $\tau$ -topology$. I haven’t done much with the tau-topology in a while but there are several related open questions and unfinished ideas out there.

Topological Analogues of Classical Theorems

The proofs of the following are way too technical for a blog post. So I’ll just share some results that have always surprised me a bit. They serve as evidence that the $\tau$ -topology is probably the most natural extension of the fundamental group to the category of topological groups. All of these are proved in [1].

Classical Theorem A (fundamental group of a wedge of circles is a free group): Let $X$ be a discrete set and $X_+=X\cup\{\ast\}$ be the space with one additional isolated basepoint. If $\Sigma X_+$ is the reduced suspension (which is a wedge of circles indexed by $X$ ), then there is a canonical isomorphism $F(X)\to \pi_1(\Sigma X_+)$ from the free group on the set $X$ .

Topological Theorem A (topological fundamental group of a generalized wedge of circles is a free topological group): Let $X$ be any space and $X_+=X\cup\{\ast\}$ be the space with an additional isolated basepoint. If $\Sigma X_+$ is the reduced suspension (which is a generalized wedge of circles paramterized by $X$ ), then there is a canonical topological group isomorphism $F_{M}(\pi_0(X))\to \pi_{1}^{\tau}(\Sigma X_+)$ from the free Markov topological group on the quotient space $\pi_0(X)$ of path-components of $X$ .

Classical Theorem B: Every group is isomorphic to the fundamental group $\pi_1(X,x_0)$ of some space $X$ obtained by attaching 2-cells to a wedge of circles.

Topological Theorem B: Every topological group is isomorphic to the topological fundamental group $\pi_{1}^{\tau}(X,x_0)$ of some space $X$ obtained by attaching 2-cells to a generalized wedge of circles (see Theorem A).

van Kampen Theorem: If $X=U\cup V$ where $U,V,U\cap V$ are path connected, then there is a canonical isomorphism $\pi_1(X)\to \pi_1(U)\ast_{\pi_1(U\cap V)}\pi_1(V)$ to the pushout in the category of groups.

Topological van Kampen Theorem: If $X=U\cup V$ where $U,V,U\cap V$ are path connected and $U\cap V$ is locally path-connected (or a slightly weaker condition), then there is a canonical isomorphism $\pi_{1}^{\tau}(X)\to \pi_{1}^{\tau}(U)\ast_{\pi_{1}^{\tau}(U\cap V)}\pi_{1}^{\tau}(V)$ to the pushout in the category of topological groups.

References.

[1] J. Brazas, The fundamental group as a topological group, Topology Appl. 160 (2013) 170-188. Open Access.

[2] J. Brazas, Open subgroups of free topological groups, Fundamenta Mathematicae 226 (2014) 17-40.

The $\tau$ -topology was extended to the set of path-homotopy classes of paths starting at a point (the usual universal covering space construction) in the following paper.

[3] Z. Virk, A. Zastrow, A new topology on the universal path space. Topology Appl. 231 (2017) 186-196.