A Lê-Greuel type formula for the image Milnor number

Abstract

Let $f:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p:({\mathbb{C}}^{n+1},0)\to (\mathbb{C},0)$ we denote by $g:({\mathbb{C}}^{n-1},0)\to ({\mathbb{C}}^n,0)$ the transverse slice of $f$ with respect to $p$. We prove that the sum of the image Milnor numbers ${\mu }_I(f)+{\mu }_I(g)$ is equal to the number of critical points of the stratified Morse function $p{|}_{X_s}:X_s\to \mathbb{C}$, where $X_s$ is the disentanglement of $f$ (i.e., the image of a stabilisation $f_s$ of $f$).

Accepted in Hokkaido Mathematical Journal

References in corpus (1)

• Multiple point spaces of finite holomorphic maps