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

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 $U_{s_j}$ .

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)!