Let f be a quasi-monotone mapping from a compact, connected manifold M m ( m ⩾ 3 ) {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 ∈ Y , g − 1 ( y ) y\, \in \,Y,\,{g^{ - 1}}(y) is not a point and g − 1 ( y ) {g^{ - 1}}(y) and f − 1 ( y ) {f^{ - 1}}(y) are equivalently embedded in M (in particular, g − 1 ( y ) {g^{ - 1}}(y) and f − 1 ( y ) {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.