Exercise 1.1.8

Question

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

Amit Awasthi · Accepted Answer

\begin{proof}[Solution]
  We consider the torus $S^1 	imes S^1$ embedded into $\mathbb{R}^3$ via an
  immersion $\iota\colon S^1 	imes S^1 \hookrightarrow \mathbb{R}^3$, such that
  the torus is symmetric about the $z$-axis and across the $xy$-plane. Then, let
  $f(x,y,z) = (x,y)$ be the projection map from torus to the $xy$-plane. It is
  continuous since the projection $\mathbb{R}^3 	o \mathbb{R}^2$ is continuous,
  and since the topology on the torus is then the same as the subspace topology
  inherited from $\mathbb{R}^3$. Since $(f \circ \iota)(-x,-y) 
e
  (f \circ \iota)(x,y)$, we see that the Borsuk-Ulam theorem does not hold for this
  map.
\end{proof}

Exercise 1.1.8

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Exercise 1.1.8

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