I really like the following result, which allows one to drop the usual compactness assumption.
Okhezin's theorem: For a polyhedron $K$ and a continouous map $f\colon K\to K$ at least one of the following conditions is true:
- $f$ has a fixed point;
- $f$ is not nullhomotopic;
- $K$ contains a closed subset homeomorphic to $[0,\infty)$ (a closed ray).
Since $[0,\infty)$ is an absolute retract without fixed point property, no polyhedron containing it as a closed subset has the fixed point property. This gives the following corollary.
Corollary (Okhezin): A contractible polyhedron has the fixed point property if and only if it is rayless, i.e. contains no closed subset homeomorphic to $[0,\infty)$.
Okhezin also proved some fixed point theorems that apply to other classes of rayless spaces, including some Lefschetz-type results. However, it is not known whether a rayless, acyclic polyhedron has the fixed point property.