Monotone and open mappings on manifolds. I

Sufficient conditions are given for the existence of open mappings from a p. 1. manifold ${M^m},m \geqslant 3$, onto a polyhedron $Q$. In addition, it is shown that a mapping $f$ from ${M^m},m \geqslant 3$, to $Q$ is homotopic to a monotone mapping of $M$ onto $Q$ iff ${f_ \ast }:{\pi _1}(M) \to {\pi _1}(Q)$ is onto. Finally, it is shown that a monotone mapping of ${M^m},m \geqslant 3$, onto $Q$ can be approximated by a monotone open mapping of $M$ onto $Q$.