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?



On the existence of a domination map of a finite polyhedron
This question apparently goes back to Karol Borsuk, at least in spirit. An interesting discussion together with a history of the problem can be found in a paper of Danuta Kołodziejczyk, *Polyhedra for which every homotopy domination over itself is a homotopy equivalence*, [arxiv:1411.1032](https://arxiv.org/abs/1411.1032). 


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].


