# On the existence of a domination map of a finite polyhedron

A continuous map $$d:X\to A$$ is called domination if there exists a map $$u:A\to X$$ so that $$d\circ u\simeq 1_A$$.

Is there a domination map $$d:P\to P$$ of a finite polyhedron $$P$$ so that $$d$$ is not a homotopy equivalence?

For instance, if $$P$$ is a manifold [Bernstein-Ganea, 1959] or, more generally, a Poincaré complex [Kwasik, 1984], then every domination of $$P$$ is a homotopy equivalence. The same is true for polyhedra $$P$$ with polycyclic-by-finite fundamental group [Kołodziejczyk, 2005].