CW structure on 2-sphere with $K_5$ and $K_{3,3}$ as 1-skeletons with face edges self-identified

Question

The following is an exercise from Hatcher's Algebraic Topology:

Suppose we build $\mathbb S^2$ from a finite collection of polygons by identifying edges in pairs. Show that in the resulting CW structure on $\mathbb S^2$ the $1$-skeleton cannot be either of the two graphs $K_5$ (the complete graph with $5$ vertices) or $K_{3,3}$ (the complete bipartite graph between sets of $3$ vertices).

This question is answered here and here, but both solutions make a crucial assumption: we do not identify edges from the same polygon. So for example, the classic CW structure on the torus with one face (a square), two edges (opposite sides of the square), and one vertex, resulting in the $1$-skeleton $\mathbb S^1 \vee \mathbb S^1$, would not be permitted.

My question: Why can't we construct a CW structure on $\mathbb S^2$ with edges of the same face identified, and for which the resulting $1$- skeleton is $K_5$ or $K_{3,3}$?

If $K_5$ is the $1$-skeleton of a weird CW complex on $\mathbb S^2$, then it would have to have $7$ faces by considering the Euler characteristic. There are certain kinds of polygons with self-identified edges that wouldn't be allowed; a triangle with an edge identified, for example, would include self-loops or vertices connected with two edges, which don't exist in $K_5$. But the casework needed to look at each type of self-identification is incredibly tedious, and I'm not sure how I'd do that systematically anyways (especially if we identify vertices and not edges, creating "parachutes" as sub-CW-complexes).

Any thoughts?

Answer 1

The missing link --- which requires the Jordan-Schonflies Theorem --- is that for every embedded graph $G \subset S^2$ that is homeomorphic to either $S_5$ or $S_{3,3}$, the graph $G$ is the $1$-skeleton of a CW structure on $S^2$ having the following property:

Embedded 2-cells: The characteristic map $D^2 \mapsto S^2$ of each $2$-cell is an embedding.

It follows that the attaching map $S^1 \mapsto S^2$ of each 2-cell is an embedding, and therefore we clearly do not "identify edges from the same polygon".

I'll prove that the Embedded 2-cells property holds for all nonempty, finite, connected graphs $G$ embedded in $S^2$ that satisfy the following hypothesis, which is clearly true for the cases $G=S_5$ and $G=S_{3,3}$:

Lots of circles: $G$ is a finite connected graph For every edge $e \subset G$ there exists an embedded circle in $G$ that contains $e$.

Assuming this hypothesis, to describe the desired CW structure on $S^2$, for any component $K$ of $S^2-G$ it suffices to produce an embedding $D^2 \mapsto S^2$ that takes $S^1 = \partial D^2$ homeomorphically to a circle in $G$ and that takes $\text{interior}(D^2)=D^2-S^1$ homeomorphically onto $K$.

For the proof, we first consider any embedded circle $C \subset G \subset S^2$. By applying the Jordan-Schönflies Theorem we conclude the following:

• $C$ separates $S^2$ into two components,
• the closure of each such component $D$ may be written as $\overline D = D \sqcup C$,
• there exists a homeomorphism $\overline D \approx D^2$ with respect to which $C \approx S^1$.

Let's refer to this conclusion by saying that $\overline D$ is a disc with boundary $C$ in $G$.

The set of discs with boundary in $G$ is partially ordered by inclusion.

I'll prove that the 2-cells of the desired CW structure are precisely those discs with boundary in $G$ that are minimal elements of the partial ordering. To prove this, consider any component $K$ of $S^2-G$. For every embedded circle $C \subset G$, clearly $K$ is contained in one of the two discs with boundary $C \subset G$, and so $K$ is contained in some innermost disc with boundary in $G$ that we denote $\overline D_K$. The interior of $\overline D_K$ is disjoint from $G$, for otherwise we could find a smaller disc with boundary in $G$ that contains $K$. It follows that the interior of $\overline D_K$ is equal to $K$, and that the homeomorphism $\overline D_K \approx D^2$ coming from the Jordan-Schönflies theorem satisfies the requirements of a characteristic map for $K$ as an open 2-cell. Doing this for every such $K$, we obtain the collection of characteristic maps needed to define the CW structure.