Topological invariants and Milnor fibre for $\mathcal{A}$-finite germs $\mathbb{C}^2\to\mathbb{C}^3$

TL;DR

This paper shows that key analytic invariants of finitely determined map germs from a2a2 to a2a3 depend only on the topological type of their image, establishing topological invariance of certain Smale invariants for these germs.

Contribution

It demonstrates the topological invariance of several analytic invariants and the sign-refined Smale invariant for finitely determined map germs from a2a2 to a2a3, correcting previous computational errors.

Findings

Invariants depend only on the embedded topological type of the image.

Topological invariance of the sign-refined Smale invariant is established.

Corrected computation of a key topological quantity, leading to revised proofs.

Abstract

This note is the observation that a simple combination of known results shows that the usual analytic invariants of a finitely determined multi-germ $f\colon (\mathbb{C}^2,S)\to(\mathbb{C}^3,0)$ ---namely the image Milnor number $\mu_I$, the number of crosscaps and triple points, $C$ and $T$, and the Milnor number $\mu(\Sigma)$ of the curve of double points in the target--- depend only on the embedded topological type of the image of $f$. As a consequence one obtains the topological invariance of the sign-refined Smale invariant for immersions $j\colon S^3\looparrowright S^5$ associated to finitely determined map germs $(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$. This note is a corrected version of a previous homonymous work containing an error. A previous wrong computation of $b_1(\mathbb{F})$, spotted by Siersma, has been replaced by the correct statement, due to Van Straten. This has forced the proofs for the mentioned topological invariances to differ significantly from the previous version.