A L\^e-Greuel type formula for the image Milnor number

TL;DR

This paper establishes a formula relating the image Milnor number of a corank 1 finitely determined map germ to the sum of the Milnor number of its transverse slice and the critical points of a stratified Morse function, extending classical results.

Contribution

It introduces a Lê-Greuel type formula connecting the image Milnor number with stratified Morse theory for corank 1 map germs.

Findings

Proves the sum of image Milnor numbers equals the number of critical points of a stratified Morse function.

Extends classical Lê-Greuel formulas to the setting of image Milnor numbers.

Provides a new tool for computing invariants of map germs via stratified Morse theory.

Abstract

Let $f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C}^{n+1},0)$ 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 $g:(\mathbb{C}^{n-1},0)\rightarrow (\mathbb{C}^{n},0)$ 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 $f$).