In Part 1, I described the construction of , the fundamental group equipped with the quotient topology and some of the drama around failing to always be a topological group. In this second post, I plan to connect back to spaces with “nicer” local structure by discussing when is discrete. The theorem we’ll prove is the main result from .
If you’ve got a CW-complex, manifold, simplicial complex, or some other locally contractible space, you should very much hope that is a discrete group. If is NOT discrete, then it’s because the topological part of is detecting some non-trivial local structures in . In this post, we’ll explore what non-discreteness is really telling you.
Reminder: is the loop space with the compact-open topology and is the fundamental group with the quotient topology inherited from the map , 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 is not discrete. Recall that a space is semilocally simply connected at if there exists an open neighborhood of such that the homomorphism induced by the inclusion map is trivial, that is, if every loop in based at a contracts by a null-homotopy in . We say is semilocally simply connected if it has this property at all of its points.
Lemma 1: If is path connected and is discrete, then is semilocally simply connected.
Proof. Suppose is not semilocally simply connected at some point . Fix a path from to and let be the set of open sets in containing . Since is not semilocally simply connected at, for every , there exists a loop based at such that is not null-homotopic in , that is, in . Thus in . However, is a directed set (by subset inclusion) and so is a net in . Moreover, converges to in the compact-open topology where denotes the constant loop at . Since is continuous, the net of non-trivial homotopy classes converges to the identity element in . Since a net of non-identity elements converges to the identity element, there is no way the trivial subgroup can be open. Thus is not discrete.
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 is discrete.
Lemma 2: If is locally path connected and semilocally simply connected, then is a discrete group.
Proof. To show that is discrete, we must show that for any loop , the 1-point set is open. Since is a quotient map, is open in if and only if is open in . Remember that is the homotopy class of , that is, the set of all loops path-homotopic to . In order to do this we must show that all loops “nearby” are homotopic to where “nearby” really means “in some compact-open neighborhood.”
For each , let be an open neighborhood of such that every loop in is null-homotopic in (here, we are using the semilocally simply connected property). Using the Lebesgue Number Lemma, we may find an integer such that if (as in Part 1), then for each , we have for some . To simplify notation, we’ll write for .
Now we have . However, it may not be the case that the intersection is path connected. We can address this issue in the following way. Since is locally path connected, for each , find a path-connected neighborhood latex of such that .
Now we are ready to define the neighborhood
Recall that the notation denotes the set consisting of all loops that map the compact set into the open set . So we can think of above as the set of all loops that follow an ordered list of instructions. If , then must first proceed through and end somehwere in . It must then proceed through and end in , etc.
Our remaining job is to show that is open and this will be done if we can show that , that is every loop in is homotopic to .
Let . We’ll construct a homotopy . For , both and lies in and so we may find a path from to .
- is a loop in ,
- is a loop in when ,
- is a loop in .
By our choice of the sets , all of these loops are null-homotopic in . In particular, this means that
- when ,
Composing these homotopies “horizontally” gives that is homotopic to
cancelling the inverse pairs gives a homotopy with . Thus . We conclude that .
Lemmas 1 and 2 tell us that for locally path-connected spaces, being semilocally simply connected is equivalent to being a discrete group.
Theorem: Suppose is locally path connected. Then is discrete if and only if is semilocally simply connected.
This tells us that, for locally path-connected spaces, non-discreteness of 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 is semilocally simply connected but is not locally path-connected. This space includes the sequence of circles that all meet at a point and includes the limit circle too. If is the loop going once around the n-th circle (parameterized in a standard way) and goes once around the limit circle, then in the compact-open topology and so in even though none of these homotopy classes are the same. Since 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)!
 J. Calcut, J. McCarthy, Discreteness and homogeneity of the topological fundamental group, Topology Proc. 34 (2009) 339-349.