Brouwer fixed-point theorem

Remarks

The idea of the proof: suppose $f(x) \neq x$ for all $x$. Send $x$ to the point where the ray from $f(x)$ through $x$ leaves the ball; this is a continuous retraction of $D^n$ onto the sphere $S^{n-1}$ that fixes $S^{n-1}$. No such retraction exists by the no-retraction theorem, because $H_{n-1}(S^{n-1}) \neq 0$ for $n \ge 2$ while $D^n$ is contractible; for $n = 1$ the intermediate value theorem applied to $f(x) - x$ suffices. The theorem is also the case of a closed ball in the Lefschetz fixed-point theorem, since $L(f) = 1$ for every self-map of a contractible compact polyhedron. In the language of the fixed-point property, it says that $D^n$ has this property. So does every space homeomorphic to $D^n$, such as every compact convex set of $\R^n$ with nonempty interior. Each hypothesis matters: the translation $x \mapsto x + 1$ of the non-compact space $\R$ has no fixed point. The rotation $z \mapsto -z$ of the circle $S^1$, which is compact but has a hole, has none either.

Going deeper

For a student

The theorem is the basic existence tool behind many other fixed-point results. Schauder's theorem extends it to compact convex sets in normed spaces, and Kakutani's theorem to set-valued maps, which is how Nash proved the existence of equilibria in games. Its proofs connect several subjects: degree theory, homology, the no-retraction theorem, and combinatorial arguments through Sperner's lemma.