There is an clever and interesting combinatorial (homology-free) approach to the proof of the well-known Brouwer fixed point theorem. Recall that this theorem states that:

Theorem 1 Any continuous ${f: B^n \rightarrow B^n}$ (for ${B^n}$ the unit ball in euclidean ${n}$-space ${\mathbb{R}^n}$) has a fixed point.

The first idea that suggests that a combinatorial approach might tackle the Brouwer theorem is that the set

$\displaystyle \left\{ f: B^n \rightarrow B^n \ \mathrm{with} \ \mathrm{no} \ \mathrm{fixed \ pt } \right\}$

is open in the set of continuous maps ${B^n \rightarrow B^n}$ (with the uniform topology). So if we can show that any continuous map ${f: B^n \rightarrow B^n}$ can be uniformly approximated by maps that do have fixed points to an arbitrary degree, then it will follow that ${f}$ itself has a fixed point.

Now one way you could take this is to assume that ${f}$ is differentiable. And indeed, there are differential-topological proofs of Brouwer’s theorem. This is not the purpose of the present post, though. We will replace the continuous ball ${B^n}$ with a simplicial complex. (more…)