Does the Borsuk–Ulam theorem hold for the torus? In other words, for every map \(f: S^1 \times S^1 \longrightarrow \mathbb{R}^2\) must there exist \((x, y) \in S^1 \times S^1\) such that \(f(x, y) = f(-x, -y)\)?
Solution
No.
1. Regard \(S^1\) as the unit circle in \(\mathbb R^2\) and define\[ f:S^1\times S^1\to\mathbb R^2, \qquad f(x,y)=x. \]
This is the composition of the projection \[ S^1\times S^1\to S^1, \qquad (x,y)\mapsto x, \] with the inclusion \(S^1\hookrightarrow\mathbb R^2\), so it is continuous.
2. For every \((x,y)\in S^1\times S^1\),\[ f(-x,-y)=-x\ne x=f(x,y). \]
If \(x=-x\) as a vector in \(\mathbb R^2\), then \(2x=0\), hence \(x=0\). But \(x\in S^1\) has norm \(1\), so this is impossible.
3. Therefore the stated Borsuk–Ulam analogue fails for the torus.
The continuous map in 1 has no point satisfying the required equality by 2.