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