- #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!