Brouwer fixed-point theorem

This article describes a theorem about spheres

Statement

In the language of spheres

Any continuous map from a disc to itself must have a fixed point. In other words, if $D^n$ denotes the spherical disc in ${\mathbb{R}}^n$, any continuous map $f:D^n\to D^n$ must have a point $x$ such that $f(x)=x$.

In the language of simplices

Any continuous map from the standard $n$-simplex, to itself, has a fixed point.

Facts used

1. No-retraction theorem

Proof

The Brouwer fixed-point theorem follows easily from the no-retraction theorem. Suppose $f:D^n\to D^n$ is a continuous map with no fixed points. Define a map $g:D^n\to S^{n-1}$, that sends $x\in D^n$ to the unique point on $S^{n-1}$ that is colllinear with $x$ and $f(x)$ in such a way that $x$ lies between that point and $f(x)$. We can see that:

• Since $f(x)$ is never equal to $x$, and $x$ is inside the unit disc, $g$ is well-defined throughout $D^n$
• $g$ is continuous
• $g$ is a retraction because it fixes every point on $S^{n-1}$