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 $[0,\infty)$.

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, 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 $f$.

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

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

Okhezin's theorem¹: 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, 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 $f$.

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

¹Okhezin, 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, projecteuclid; ZBL0917.54046, MR1350346.

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.

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 $[0,\infty)$.

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, 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 $f$.

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