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## Main Question or Discussion Point

Problem: Let P be a polygon with an even number of sides. Suppose that the sides are identified in pairs in accordance with any symbol whatsoever. Prove that the quotient space is a compact surface.

Proof:

Ok, here are some of my thoughts about the proof.

I believe that one would need to use the following theorem(while it is possible that it can be done in other ways as well).

Thm.

The part that throws me off is "Suppose that the sides are identified in pairs in accordance with

Proof: If it is a 2-gon, then there are really only two ways one could identify its sides in pairs. If identified in the same direction, then the quotient space of this 2-gon would be a sphere, thus a compact surface. If they are identified in the opposite direction, then it would be a projective plane, hence a compact surface.

Now, suppose that we are talking about a 2n-gon. Now, i know that the quotient space of a 2n-gon with sides identified as follows:

[tex]a_1a_1a_2a_2...a_na_n[/tex] is homeomorphic to a connected sum of n projective planes, thus by the above theorem it is compact.

I believe, the proof would be concluded if we could show that any other identification could somehow be transformed into either the connected sum of n projective planes or n tori.

Any suggestions would be greatly appreciated!

Thnx in advance.

Proof:

Ok, here are some of my thoughts about the proof.

I believe that one would need to use the following theorem(while it is possible that it can be done in other ways as well).

Thm.

*Any compact surface is either homeomorphic to a sphere, or to a connected sum of tori, or to a connected sum of projective planes.*The part that throws me off is "Suppose that the sides are identified in pairs in accordance with

**any symbol whatsoever**". Do they mean that any two sides of the n-gon can be identified with one another or?Proof: If it is a 2-gon, then there are really only two ways one could identify its sides in pairs. If identified in the same direction, then the quotient space of this 2-gon would be a sphere, thus a compact surface. If they are identified in the opposite direction, then it would be a projective plane, hence a compact surface.

Now, suppose that we are talking about a 2n-gon. Now, i know that the quotient space of a 2n-gon with sides identified as follows:

[tex]a_1a_1a_2a_2...a_na_n[/tex] is homeomorphic to a connected sum of n projective planes, thus by the above theorem it is compact.

I believe, the proof would be concluded if we could show that any other identification could somehow be transformed into either the connected sum of n projective planes or n tori.

Any suggestions would be greatly appreciated!

Thnx in advance.