On topological classification of complex mappings

TL;DR

This paper investigates the topological invariant  of complex mappings, providing bounds, exact computations for specific classes, and establishing semicontinuity properties in the Zariski topology.

Contribution

It extends the understanding of the invariant  beyond smooth targets, offering new bounds, exact values for certain mappings, and topological continuity results.

Findings

Derived a lower bound for  of general mappings

Identified classes where  can be exactly computed

Proved semicontinuity of  variation in Zariski topology

Abstract

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.