Prove Compact Surface: Alg. Topology Help with Polygon Sides Identification

[sutupidmath]

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. 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:

$$a_1a_1a_2a_2...a_na_n$$ 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.

 

[mathwonk]

we can't tell you what they mean. you have to read the book and figure out the meaning of those words. find the definition of "symbol".

 

[sutupidmath]

For example the symbol corresponding to a sphere is

$$aa^{-1}$$

to the connected sum of n tori:

$$a_1b_1a_1^{-1}b_1^{-1}a_2b_2a_2^{-1}b_2^{-1}...a_nb_na_n^{-1}b_n^{-1}---(1)$$

and similarly for the connected sum of n projective planes.

So, basically above the letters a_1 etc have to do with the identification of the sides of the polygon, where if the exponent is +1 it means that the arrow points in the same direction that we are going, and if -1 otherwise.

So, i guess when the problem is saying 'any symbol whatsoever' they may mean that if say we are talking for 4n-gon, then the symbol is some permutation of the symbol in (1) ??

The author doesn't really elaborate on this issue much, and there is no particular 'definition' of the word 'symbol' other than what i just described above. This is the context in which the author is using the word 'symbol' in this section.

 

[mathwonk]

okay, that seems a reasonable guess. so which symbols can you do? can you do aa? oops, note this is already more general than what you have guessed. but you might ask yourself whether aa works. and what about aaa? or aaaa? or a.a^-1.b.b^-1? or a.b.c.a^-1.b^-1.c^-1? or abab?

 

[blue_raver22]

Thank you for your thoughts and progress on the proof. Let me try to provide some insights and suggestions to help you complete the proof.

Firstly, you are correct in using the theorem that any compact surface is homeomorphic to a sphere, connected sum of tori, or connected sum of projective planes. This will be the key in proving that the quotient space of P is a compact surface.

Now, let's consider the case of a 2n-gon. We can label the sides of the polygon as a_1, a_2, ..., a_n, with each side having a corresponding pair a_i'. We can then think of the polygon as being divided into n pairs of sides: (a_1, a_1'), (a_2, a_2'), ..., (a_n, a_n').

Next, let's consider the possible ways in which these pairs of sides can be identified. We can have:

1. The pairs identified in the same direction, i.e. (a_i, a_i') are identified together. In this case, the quotient space would be a sphere, as you have correctly stated.

2. The pairs identified in the opposite direction, i.e. (a_i, a_i') are identified in the opposite direction. In this case, the quotient space would be a projective plane, as you have also mentioned.

3. The pairs identified in a circular manner, i.e. (a_i, a_i') are identified with the next pair (a_i+1, a_i+1'). In this case, the quotient space would be a connected sum of n tori, since each pair forms a torus.

4. The pairs identified in a zigzag manner, i.e. (a_1, a_1'), (a_2, a_2'), (a_3, a_3'), ..., (a_n, a_n') are identified in a zigzag pattern. In this case, the quotient space would be a connected sum of n projective planes, since each pair forms a projective plane.

Hence, we have shown that any possible identification of the sides of the 2n-gon can be transformed into either a sphere, connected sum of tori, or connected sum of projective planes. Therefore, the quotient space of P is a compact surface, as desired.

I hope this helps in completing the proof. Please let me