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

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 Szűcs. We provide a new, direct proof for this equality. We also clarify the relation between various definitions of the latter invariant.

arXiv comment13 page, 2 figures