The fixed point property in every weak homotopy type - Barmak, Jonathan Ariel (2016)
We prove that for any connected compact CW-complex K there exists aspace X weak homotopy equivalent to K which has the fixed point property, that is,every continuous map X -> X has a fixed point. The result is known to be false if werequire X to be a polyhedron. The space X we construct is a non-Hausdorff space withfinitely many points.