Monotone and open mappings on manifolds. I

Abstract

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