TheoremProved

Borsuk–Ulam theorem

Statement

For every continuous map $f: S^n \to \mathbb{R}^n$ there is a point $x \in S^n$ with $f(x) = f(-x)$ . Equivalently, there is no continuous map $S^n \to S^{n-1}$ satisfying $f(-x) = -f(x)$ for every $x$ .

Why is it true?

At every moment, there exist two antipodal points on Earth's surface (opposite ends of a diameter through the centre) with exactly the same temperature and exactly the same barometric pressure simultaneously, because temperature and pressure vary continuously and $n=2$ here.

Proof sketch

The theorem can be derived from a covering lemma of Lusternik and Schnirelmann: if $S^n$ is covered by $n+1$ closed sets, at least one of them must contain a pair of antipodal points. Applying this to sets built from $f$ (partitioning $S^n$ according to which coordinate of $f(x)$ is largest in absolute value) forces $f(x)=f(-x)$ for some $x$ ; the theorem can also be proved directly via a degree/index argument on the map $g(x)=f(x)-f(-x)$ .

Topics that use this theorem

Homology and cohomology

Related theorems

Brouwer fixed-point theorem Jordan curve theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Karol Borsuk (1933). Drei Sätze über die n-dimensionale euklidische Sphäre
Jiří Matoušek (2003). Using the Borsuk–Ulam Theorem