Is the Borsuk-Ulam theorem valid for a torus? In other words, for any map $f: S^1 \times S^1 \rightarrow \mathbb{R^2}$ there is a point $(x,y) \in S^1 \times S^1$ which $f(x,y)=f(-x,-y)$
I'm very stuck on this task. Can someone give a hint? Or there can be a detailed solution, if suddenly this task is easy enough.
Thank you in advance for help!
No. With the usual torus embedded in $\mathbb{R}^3$, lying on the $OXY$ plane, one has a natural projection onto that plane, $p:S^1×S^1\to \mathbb{R}^2$, which is continuous.
Two points on the torus have the same image if they are one above the other, in the same vertical line. In particular, they are in the same meridian of the torus, i.e. they have the same first coordinate. So, if $p(a,b) = p(c,d)$, $a = c$. This implies that the Borsuk-Ulam theorem fails on the torus because if $x=-x$, and then $x=0\notin S^1$.