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}.

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.



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

This entry was posted in Fundamental group, Uncategorized and tagged , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s