Please give me some hint

  • #1
qinglong.1397
108
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!
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,263
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Here's a hint. It doesn't hold for S^1, does it?
 
  • #3
qinglong.1397
108
1


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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