TOPOLOGICAL INVARIANTS AND MILNOR FIBER FOR A-FINITE GERMS C 2 → 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 # S 5 associated to finitely determined map germs (C 2 , 0) → (C 3 , 0).