Isotoping mappings to open mappings

Abstract:

Let f be a quasi-monotone mapping from a compact, connected manifold ${M^m} (m \geqslant 3)$ onto a space Y; then there is an open mapping g from M onto Y such that, for each $y \in Y, {g^{ - 1}}(y)$ is not a point and ${g^{ - 1}}(y)$ and ${f^{ - 1}}(y)$ are equivalently embedded in M (in particular, ${g^{ - 1}}(y)$ and ${f^{ - 1}}(y)$ have the same shape). Applying the result with f equal to the identity mapping on M yields a continuous decomposition of M into cellular sets each of which is not a point.