Brouwer & Borsuk–Ulam

Stir the coffee: a group with nothing in it forbids the escape, so a fixed point has to be there.

The idea

The Brouwer fixed point theorem and the Borsuk–Ulam theorem are two statements about continuous maps, and homology proves both.

Theorem (Brouwer's fixed point theorem).

Every continuous map $f \colon D^{n} \to D^{n}$ of the closed disc $D^{n} = \{x \in \mathbb{R}^{n} : |x| \le 1\}$ to itself has a fixed point: a point $x$ with $f(x) = x$.

Theorem (Borsuk–Ulam theorem).

Every continuous map $f \colon S^{n} \to \mathbb{R}^{n}$ agrees at some pair of antipodal points: $f(x) = f(-x)$ for some $x \in S^{n}$.

To illustrate the second, send each point of the Earth's surface to its temperature and barometric pressure; both vary continuously, so at this moment two opposite points of the planet agree in both.

Both are pure existence statements: each guarantees that a point with the stated property is there and says nothing about where it is. Neither statement mentions a group, and neither has an elementary proof.

Ways to work on it

• Walkthrough. The two theorems, the obstruction behind them, and their classical consequences.
• Proof. Brouwer from the no-retraction lemma: a fixed-point-free map builds a retraction, and homology forbids it.
• Practice. Quick applications of both theorems and their classical consequences.
• Hardest. Prove the Borsuk–Ulam theorem.

Not sure where to start? Take the ten-question placement test.