ProvedTopology·1911

Brouwer Fixed Point Theorem

Every continuous map from a closed ball to itself has at least one fixed point.

Stir your coffee however you like — at least one molecule returns to where it started. Brouwer's theorem says every continuous self-map of Dⁿ has a fixed point. Equivalently, there is no continuous retraction from the ball onto its boundary. The Lefschetz number of any such map is 1 ≠ 0, forcing a fixed point purely by topology. Kakutani's 1941 extension to set-valued maps gave Nash the tool to prove that every finite game has an equilibrium.

Formula

Theorem

f: D^n \to D^n \text{ continuous} \quad\Longrightarrow\quad \exists\, x_0 \in D^n,\; f(x_0) = x_0

Every continuous self-map of the closed unit ball has at least one fixed point.

No-retraction (equivalent)

\nexists\; r: D^n \to S^{n-1} \text{ continuous with } r|_{S^{n-1}} = \mathrm{id}

If a fixed-point-free map existed, the ray from f(x) through x would define a retraction — but no such retraction can exist.

Lefschetz number

\Lambda_f = \sum_{k \ge 0} (-1)^k \operatorname{tr}(f_{*k}) = 1 \neq 0 \quad\Longrightarrow\quad \text{fixed point}

Since Dⁿ is contractible, Λ_f = 1 for every self-map. The Lefschetz theorem then guarantees a fixed point.

Summary

Sources

Über Abbildung von Mannigfaltigkeiten — Brouwer's original proof via mapping degree

1911

Analytic proofs of the 'hairy ball theorem' and the Brouwer fixed point theorem

1978

A generalization of Brouwer's fixed point theorem — Kakutani's set-valued extension

1941

Videos

Proving Brouwer's Fixed Point Theorem | Infinite Series

PBS Infinite Series

Brouwer's fixed point theorem

Mathemaniac