Cross-caps, triple points and a linking invariant for finitely determined germs

TL;DR

This paper provides a new direct proof that certain counts of singularities in finitely determined germs are topological invariants, linking them to a known linking invariant of an associated immersion.

Contribution

It offers a new, direct proof of the topological invariance of the combination of Whitney umbrella points and triple values, and clarifies the relation between different definitions of the linking invariant.

Findings

The combination C(Φ)-3T(Φ) is a topological invariant.

A new, direct proof of the invariance is provided.

Clarification of the relation between various definitions of the linking invariant.

Abstract

It was recently proved that for finitely determined germs $ \Phi: ( \mathbb{C}^2, 0) \to ( \mathbb{C}^3, 0) $ the number $C(\Phi)$ of Whitney umbrella points and the number $T(\Phi)$ of triple values of a stable deformation are topological invariants. The proof uses the fact that the combination $C(\Phi)-3T(\Phi)$ is topological since it equals the linking invariant of the associated immersion $S^3 \looparrowright S^5$ introduced by Ekholm and Sz\H{u}cs. We provide a new, direct proof for this equality. We also clarify the relation between various definitions of the latter invariant.