Find all surfaces that can be obtained from an octagon by identifying edges in pairs.

Question

Find all surfaces that can be obtained from an octagon by identifying edges in pairs.

I think there are many many surfaces. Can anyone give some hints for the question?Thanks.

Answer 1

You're right that there are a very large number of edge symbols that you can get by pairwise identifying the edges of an octagon. However, many of them yield homeomorphic surfaces. Here are a few facts that might help you in your thinking:

• The classification theorem for surfaces states that any compact 2-manifold is homeomorphic to either a sphere, a connected sum of $n$ tori, or a connected sum of $n$ projective planes.
• A torus is obtained from an edge symbol with four letters. A projective plane is obtained from an edge symbol with two letters.
• The connected sum of two surfaces with given edge symbols can be obtained from a polygon whose edge symbol is just the concatenation of the original two edge symbols.

Based on these three facts, you should be able to make a very short list of the possible surfaces that can be obtained. I'm purposefully omitting the exact number of surfaces that I think there are, so that I don't give the whole game away. I will say that if I'm right, it's less than 10.

(Background: I'm following the discussion of edge symbols and polygon identification in Massey's A Basic Course in Algebraic Topology. It's also possible to understand this all in terms of quotients of free groups on a finite number of generators, but I'm a little rustier on how that's done, so I'll leave that discussion to someone else.)