The n-Category Café

Jesse wrote:

Just to clarify your question for people knowing better than me: are you asking primarily to follow what people mean when they don’t spell out their sense of polygonal decomposition, or so you know what to call them for other folk to read?

I want to know 1) what to call them in a paper I’m writing, and 2) a reference to point to, that works out the theory so I don’t have to.

There seems to be a well-established literature on ‘triangulated surfaces’, so that you can say this phrase and expect people to understand. But the literature on ‘decompositions of surfaces into polygons’ seems diffuse and murky: I feel I haven’t found the locus classicus.

I really enjoyed Hog-Angeloni, Metzler, and Sieradski’s Two-Dimensional Homotopy and Combinatorial Group Theory back when I was studying ‘spin foams’. They had a lot of nice terminology for quite general spaces made of polygons, not necessarily manifolds. That’s what I needed then.

But forcing my readers through this stuff now, when I just want to talk about a compact surface chopped into polygons, seems a bit sadistic.

Hmm, this looks promising:

• Seifert and Threlfall, A Textbook on Topology, Chapter 6: Surface Topology.

This sounds almost like what I want. However, I don’t want to have to prove all the supposedly obvious consequences of this definition: someone should have done it, and I should just refer to them.

I have a couple issues with the concept here. First, what does ‘properly’ embedded mean?

Second, I can take a sphere and embed a graph in it with two vertices $x,y$, one edge $f: x \to y$ and one edge $g: x \to x$. Both components of the complement are homeomorphic to open disks. To walk around one of these components I follow the loop $g$, but to walk around the other I follow the loop $g f f^{-1}$, which is a bit irritating.

In other words, if I treat the sphere as made of vertices, edges and faces, one ‘face’ is a unigon while the other is a triangle with two edges glued together.

I don’t need unigons for what I’m doing, but I can live with them. For what I’m doing, faces with edges glued together will not work: I’ll need to rule them out.

Okay, thanks! I looked up ‘proper embedding, and guessed something like this. I’m not doing surfaces with boundary, so I can avoid this nuance. Right now my paper says this:

Suppose we have a graph $X$ embedded in a compact surface $S$. The complement $S - X$ is the disjoint union of finitely many open sets whose closures we call faces. We demand that the faces are closed disks embedded in $S$.

But I’ll probably add a bit of clarification. Everything in the paper is topological, not smooth, and I’ve said how a graph is a topological space, but it may allay some worries about ‘wildness’ to assure people that the edges are mapped in smoothly.

I don’t think this actually matters for the combinatorial structure: that is, I think that whether we interpret my words above smoothly or topologically, we get the same class of combinatorial maps. But I want to make it clear that nothing nasty is allowed.

Thanks!