Let M m ( m ⩾ 3 ) {M^m}\;(m \geqslant 3) be a compact, connected PL manifold and let X ⊆ M X \subseteq M be a proper, closed subset of the interior of M such that for each open, connected subset U ⊆ M U \subseteq M either U − ( X ∩ U ) U - (X \cap U) is connected or X ∩ bd ( U ) ≠ ∅ X \cap {\text {bd}}(U) \ne \emptyset . Let P be a connected and simply connected polyhedron with dim ⁡ P ⩾ 3 \dim P \geqslant 3 . There exists a monotone mapping f from M onto P with each component of X being a point-inverse of f. In the case with M oriented and P the m-sphere, there exists such a monotone mapping of each degree.