Does Borsuk-Ulam theorem hold for the torus?

[qinglong.1397]

Please give me some hint!

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

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!

 

[Dick]

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

 

[qinglong.1397]

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

Thanks for your hint! I know the answer now!