Rationally acyclic compact polyhedron has fixed-point property

Statement

Suppose $X$ is a topological space that is a rationally acyclic compact polyhedron (where polyhedron means it arises as the geometric realization of a simplicial complex, which in this case would have a finite set of vertices because of compactness). Then, $X$ has the Fixed-point property (?): any continuous map from $X$ to itself has a fixed point.

Related facts

• Lefschetz fixed-point theorem
• Brouwer fixed-point theorem applies this to the special case of disks.
• Euler characteristic of compact connected nontrivial Lie group is zero

Facts used

1. Lefschetz fixed-point theorem