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 fact about::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) uses::Lefschetz fixed-point theorem