  We study the topological invariant $\phi$ of Kwieci\'nski and Tworzewski,
particularly beyond the case of mappings with smooth targets. We derive a lower
bound for $\phi$ of a general mapping, which is similarly effective as the
upper bound given by Kwieci\'nski and Tworzewski. Some classes of mappings are
identified for which the exact value of $\phi$ can be computed. Also, we prove
that the variation of $\phi$ on the source space of a mapping with a smooth
target is semicontinuous in Zariski topology.
