Inverse limits of planar Peano continua are not always planar

Recently, I ran into a situation where I had an inverse sequence of *planar Peano continua. I wondered naively if the limit of such a sequence always had to be planar. It turns out the answer is no, even in the one-dimensional case, but I don’t this fact is particularly obvious. After a literature search I found a really nice example that I can’t help but share.

*A Peano continuum is a connected, locally connected, metrizable space (equivalently, a continuous image of $[0,1]$ ) and a space $X$ is planar if there exists a topological embedding $X\to \mathbb{R}^2$ . For instance, the Sierpinski Carpet is a planar Peano continuum.

Kuratowski’s Theorem [3] says that a connected graph $G$ is planar if and only if it contains a topological copy of $K_{3,3}$ or $K_5$ (see below).

Let’s use an infinite null-sequence of copies of $K_{3,3}$ but without the large overarching arc. We’re going to connect these and make them limit to a point. We’ll also extend the connecting line a little past the limit point. We end up with the following space $L_1$ . Kuratowski actually mentioned this space in [5] and suggested it’s potential importance.

Let’s see why $L_1$ is not planar. Let’s call the long diagonal edges in the each partial copy of $K_{3,3}$ the “crossing edges” since these are involved in a crossing-over. The sequence of crossing edges is null in the sense that the diameter of these edges approaches $0$ . We could take finitely many crossing edges and push them over the left end. But we can’t move all of them over the left end or we wind up with a space that is not homeomorphic to $L_1$ . Similarly, because we have extended the connecting line a little past the limit point on the right, we can only push finitely many of the connecting edges over the right side of the space. Hence, any embedding into $\mathbb{R}^2$ would have infinitely many crossings somewhere and we’d have a contradiction.

However, if we delete all but finitely many of the partial copies of $L_1$ , we would end up with a finite planar graph:

A finite approximation to $L_1$ with only finitely many partial copies of $K_{3,3}$ .

A finite approximation to $L_1$ with only finitely many partial copies of $K_{3,3}$ .

This finite approximation graph is planar since there are only finitely many crossing edges and we can move all of them over the left side. Here’s a planar embedding of this finite approximation.

These finite approximations (where we keep only finitely many partial copies of $K_{3,3}$ can be put into an inverse system whose limit is homeomorphic to $L_1$ . To do this correctly, you need to define the bonding maps carefully. Let $X_n\subseteq L_1$ be the finite graph consisting of the entire horizontal arc and $n$ of the partial copies of $K_{3,3}$ , let’s call them $G_1,G_2,G_3,\dots$ . To ensure that the subspaces $G_n$ shrinking toward the limit point on the arc as $n\to \infty$ , we need to define the bonding map $f_{n+1,n}:X_{n+1}\to X_n$ to map $G_{n+1}$ and the arc connecting $G_{n+1}$ to the limit point to the limit point. We map $G_k$ to itself by the identity for all $1\leq k\leq n$ , including the arcs connecting them. The arc connecting $G_{n+1}$ and $G_{n}$ we stretch out and map onto the arc connecting $G_n$ and the limit point in $X_n$ . I’ve illustrated the maps $f_{5,4}$ and $f_{4,3}$ below.

It’s not too hard to see that $L_1\cong \varprojlim_{n}(X_n,f_{n+1,n})$ . Hence, we have the following answer to my original naive question: The inverse limit of one-dimensional planar Peano continua need not be planar.

You can also perform a similar construction with $K_5$ . In particular, the following space, which we’ll call $L_2$ is built out of partial copies of $K_5$ in an analogous way.

I learned about these spaces in the paper [1]. It also taught me about the following remarkable statement proved by S. Claytor in 1937.

Claytor’s Theorem [2]: A Peano continuum embeds in $\mathbb{R}^2$ if and only it does not contain a homeomorphic copy of $K_{3,3}, K_5, L_1$ , or $L_2$ .

This completely characterizes the planarity of Peano continua in a way that directly extends Kuratowski’s Theorem! Amazing!

References

[1] R. Ayala; M. Chávez; A. Quintero, On the planarity of Peano generalized continua: An extension of a theorem of S. Claytor, Colloquium Mathematicae 75 (1998) no. 2, 175-181.

[2] S. Claytor, Peanian continua not imbeddable in a spherical surface, Ann. of Math. 38 (1937), 631–646.

[3] K. Kuratowski, Sur le probl`eme des courbes gauches en topologie, Fund. Math. 15 (1930), 271–283.