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?


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.

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].

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