Local n-connectivity vs. the LC^n Property, Part 2

Posted on October 17, 2023 by Jeremy Brazas

In Part 1, I defined local n-connectivity, which says that a space has a basis of n-connected open sets, and the LC^n porperty, which says that for all 0\leq k\leq n, “small” maps from the k-sphere contract using “small” homotopies of “relatively the same size.” The difference is subtle and often motivates the use of LC^n over local n-connectivity in metric geometry, shape theory, and wild topology.

In Part 1, I went ahead and explained the difference for paths, i.e. in dimension n=0. In that case, the pointwise or based definitions (having the property at a given point) differ but the global properties (having the local property at all points) are equivalent: Locally 0-connected \Leftrightarrow LC^0. I also pointed out some examples that showed things can get a bit wild if you don’t assume first countability. In this post, we’ll explore the higher dimensional situation, i.e. where n\geq 1.

In dimension n=1, I learned about this difference from this MathOverflow question asked back in 2019 which is really asking for an example of a LC^1 space that is not locally 1-connected. Patrick Gillespie (now a PhD student at UTK and who has been a guest author on this blog) was in my topology course and asked about it. I told the class I’d offer extra credit to anyone who solved it. We had several discussions about it and Patrick came up with the space appearing in Figure 1, which I illustrated using Mathematica:

Figure 1: An example of a LC^1 space that is not locally 1-connected. Although it is not pictured, you want to imagine the large central circle filled in with a disk in this image. It becomes more difficult to see the other levels when it is included in the picture.

Construction: To build this space explicitly, the idea is to glue together a sequence of shrinking disks together in a certain way. Actually, it’s easier to visualize if you think of these disks as cones over circles. For m\in\mathbb{N}, let A_m be a circle with basepoint a_m and \alpha_m:[0,1]\to A_m be a loop based at a_m that parameterizes the circle. Let CA_m=[0,1]\times A_m/\{1\}\times A_m be the cone over A_m. We take the image of (0,a_m) to be the basepoint of CA_m and we identify A_m with the “base” of the cone, i.e. the image of A_m\times\{0\}. Let \ell_m:[0,1]\to CA_m be the path that runs up the “spine” of the cone, that is where \ell_m(t) is the image of (a_m,t) in CA_m.

Now, consider the wedge sum/based one-point union X=\bigvee_{n\geq 1}CA_m where the points a_m are identified to a single wedge-point x_0. We give X the topology of a shrinking wedge determined by the following: U\subseteq X is open if and only if U\cap CA_m is open in CA_m for all m and if whenever x_0\in U, we have CA_m\subseteq U for all but finitely many m (see Figure 2).

Figure 2: A shrinking one-point union of cones over the the circle. The base circles form an infinite earring and the spines of each cone runs up to the vertex.

Remark 1: X is contractible by a null-homotopy that fixes the basepoint x_0. To construct a contraction of X one only need glue together null-homotopies of each cone CA_m that leaves the basepoint of CA_m (on the base circle) fixed.

Now we do some gluing. Recall that \alpha_m:[0,1]\to A_m runs around the m-th base circle and \ell_m:[0,1]\to CA_m runs up the m-th spine (these are both bolded in Figure 2). Let \sim be the smallest relation on X generated by \ell_m(t)\sim\alpha_{m+1}(t) for all m\geq 1 and t\in[0,1]. The resulting quotient space Y=X/\mathord{\sim} the 2-dimensional Peano continuum Y from Figure 1 but where the large central circle is filled in by a disk or formally the cone CA_1. Let q:X\to Y denote the quotient map. It does seem to me like this space can be embedded in \mathbb{R}^3 although the pattern used to create Figure 1 might need to be modified a little.

Let’s make some observations about this space. Let y_0 be the image of x_0 in Y. This is obviously the point of interest since Y is locally contractible at all points in Y\backslash\{y_0\}. Moreover, this space is a Peano continuum and is therefore both locally 0-connected at all points. Therefore, we can focus on what’s going on with loops.

Let’s once again identify A_m with it’s image in Y. So \bigcup_{m\geq 1}A_m still forms an infinite earring space in Y (that is bolded in Figure 1). Let’s also write D_m for the image of the cone CA_m in Y. Note that D_m is not a cone but is a cone where it’s vertex has been attached to one of the points on the base circle. Most importantly, we have A_m=D_m\cap D_{m+1} for all m\geq 1.

Proposition 2: Y is simply connected.

Proof. By modifying standard cellular approximation methods, any given loop in Y based at y_0 is path-homotopic to a loop that has image in the earring space \bigcup_{m\geq 1}A_m. The idea here is to start with a loop and, for all m\geq 1, “push it off” a given point in the interior of D_m. This makes it easy to deformation retract the original loop to a loop L with image in \bigcup_{m\geq 1}A_m. But every loop L:[0,1]\to \bigcup_{m\geq 1}A_m lifts to a loop L':[0,1]\to X such that q\circ L'=L. Since X is contracible, L' is null-homotopic. It follows that L is null-homotopic. \square

Why isn’t Y locally 1-connected? Take any open neighborhood U of the basepoint y_0 which does not contain the largest cone-image D_1. Find the largest m such that D_m\nsubseteq U. Since A_m\subseteq D_{m+1}\subseteq U, the loop \alpha_m:[0,1]\to Y has image in U but it does not contract in U because the image of a contraction of \alpha_m would necessarily contain D_m. Thus U is not simply connected.

Why is Y an LC^1 space? Take any open neighborhood U of the basepoint y_0. Find the smallest m such that D_k\subseteq U for all k\geq m. Let V be a path-connected open neighborhood of y_0 such that V\subseteq U and such that A_k\nsubseteq V for all 1\leq k\leq m. Any loop in V based at y_0 can be deformed onto \bigcup_{k\geq m+1}A_k. But the argument used to prove Proposition 2 shows any such loop is null-homotopic in the subspace \bigcup_{k\geq m+1}D_k, which is a subset of U. Therefore, any loop in V contracts in U (even though it does not contract in V).

Theorem 3: There exists a 2-dimensional Peano continuum, which is LC^1 and locally 0-connected but not locally 1-connected.

So the equivalence of the global properties no longer holds when n\geq 1. I don’t really know if there is someone I should credit this Theorem to. No doubt it was known a long time ago so I’ll call it “folklore” until I learn otherwise.

Here are some brain teasers.

Question 4: Is Y contractible?

Question 5: Is there a LC^1 space that is not locally 1-connected at any point?

The answer to both of these questions should be “yes” but I haven’t thought too much about the details.

What about even higher dimensions? If you’re looking for an example of an LC^n space that is not locally n-connected you can mimic the above example by defining A_m to be an n-sphere instead of a circle. But the reason why the above example worked so well is because the paths \alpha_m are surjective – in order to contract \alpha_m in Y, you need to use all of D_m but doing so requires that you use all of the image of \alpha_{m-1}, which is A_{m-1}. Then your homotopy would have to use A_{m-2},A_{m-3},...,A_1, which is clearly a problem. To make this same thing happen in higher dimensions (using cones), we must realize that we don’t have a canoncial way to map the spine of a cone (which is an arc) onto a higher-dimensional sphere. How do we make \alpha_m:[0,1]\to A_m surjective like we did when n=1? If we choose \alpha_m to be a simple close curve, we won’t get you the kind of example we want. This is where we are thankful for space-filling curves! We can take \alpha_m:[0,1]\to A_m to be any continuous surjection, i.e. space-filling curve (such maps exist due to results in Continuum Theory). Once this choice is made, pretty much all of the arguments are the same for the n=1 case work the same. This leads us to the following extension of Theorem 3.

Theorem 6: For each n\geq 1, there exists a n-dimensional Peano continuum, which is LC^n and locally (n-1)-connected but not locally n-connected.

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