Applying the Borsuk-Ulam theorem to the torus: A helpful hint

  • Thread starter qinglong.1397
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In summary, the Borsuk-Ulam theorem does not hold for the torus, as shown by Hatcher's proof for the map f: S^2\rightarrow \mathbb{R}^2. The method used for S^2 may not be applicable for this new problem, but it is still worth exploring. A helpful hint is to consider whether the theorem holds for S^1, which can provide some insight into the solution for the torus.
  • #1
Please give me some hint!

Homework Statement



Does the Borsuk-Ulam theorem hold for the torus? In other words, for every map [tex]f: S^1\times S^1\rightarrow \mathbb{R}^2[/tex] must there exist [tex](x, y) \in S^1\times S^1[/tex] such that [tex] f(x,y)=f(-x, -y)[/tex]?

Homework Equations





The Attempt at a Solution


In Hatcher's book, he gave the proof for the map [tex]f: S^2\rightarrow \mathbb{R}^2[/tex].

For this new problem, I really do not know what to do. It seems that Hatcher's method is useful, but I just do not know how to use it. So, please, tell me some hint. Thank you very much!
 
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  • #2


Here's a hint. It doesn't hold for S^1, does it?
 
  • #3


Dick said:
Here's a hint. It doesn't hold for S^1, does it?

Thanks for your hint! I know the answer now!:biggrin:
 

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