- #1
qinglong.1397
- 108
- 1
Please give me some hint!
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]?
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!
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!