TOPOLOGICAL INVARIANTS AND MILNOR FIBRE FOR \(\mathcal{A}\)-FINITE GERMS \(C^2\) to \(C^3\)

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 \(f : (C^2 , S) → (C^3 , 0) \)—namely, the image Milnor number , the number of cross-caps and triple points, \(C\) and \(T\), and the Milnor number \(μ(Σ)\) 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 \(j : S^3 \looparrowright S^5\) associated to finitely determined map germs \((C^2 , 0) → (C^3 , 0)\).

Keywords

• milnor fiber
• topological invariance.