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…)