Alg. Topology! Help?

  • #1
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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. 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.
 

Answers and Replies

  • #2
mathwonk
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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".
 
  • #3
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For example the symbol corresponding to a sphere is

[tex] aa^{-1}[/tex]

to the connected sum of n tori:

[tex]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)[/tex]

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.
 
  • #4
mathwonk
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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?
 
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