I really like the following result, which allows one to drop the usual compactness assumption.

**[Okhezin's theorem][1]**:
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 $[0,\infty)$.

This was not noticed by Okhezin, but the following stronger result is implied.

**Corollary:** An acyclic polyhedron has the fixed point property if and only if it is rayless.

*Proof:* As noted above, the "only if" part is obvious. For the "if" part, let $f\colon K\to K$ be a self-map of an acyclic, rayless polyhedron. The suspension $SK$ is a [contractible](https://mathoverflow.net/a/73702/2578), rayless polyhedron. Thus, by the results of Okhezin, the map $\tilde{f}\colon SK\to SK$ that extends $f$ and swaps the two added cones has a fixed point, which must also be a fixed point of $f$.

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Okhezin also proved some fixed point theorems that apply to other classes of rayless spaces, including some Lefschetz-type results.

  [1]: http://www.tmna.ncu.pl/files/v05n1-05.pdf