Extending monotone decompositions of manifolds

Abstract:

Let ${M^m}\;(m \geqslant 3)$ be a compact, connected PL manifold and let $X \subseteq M$ be a proper, closed subset of the interior of M such that for each open, connected subset $U \subseteq M$ either $U - (X \cap U)$ is connected or $X \cap {\text {bd}}(U) \ne \emptyset$. Let P be a connected and simply connected polyhedron with $\dim P \geqslant 3$. 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.