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

**Okhezin's theorem<sup>1</sup>**:
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.

<sup>1</sup><cite authors="Okhezin, Vladimir P.">_Okhezin, Vladimir P._, **On the fixed-point theory for noncompact maps and spaces. I**, Topol. Methods Nonlinear Anal. 5, No. 1, 83-100 (1995). DOI: [10.12775/TMNA.1995.005](https://doi.org/10.12775/TMNA.1995.005), [projecteuclid](https://projecteuclid.org/journals/topological-methods-in-nonlinear-analysis/volume-5/issue-1/On-the-fixed-point-theory-for-non-compact-maps-and/tmna/1479287021.full); [ZBL0917.54046](https://zbmath.org/?q=an:0917.54046), [MR1350346](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1350346).</cite>

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