Critical Points of Functions on Singular Spaces

TL;DR

This paper explores different definitions of the critical locus for complex analytic functions on singular spaces, focusing on a topological variant and its implications for conditions like Thom's a_f.

Contribution

It introduces a topological notion of the critical locus and generalizes key results linking Milnor number constancy to Thom's a_f condition.

Findings

The topological critical locus is justified as a primary notion.

Constant Milnor number implies Thom's a_f condition under the new framework.

The work bridges classical invariants with topological perspectives on singularities.

Abstract

We compare and contrast various notions of the "critical locus" of a complex analytic function on a singular space. After choosing a topological variant as our primary notion of the critical locus, we justify our choice by generalizing L\^e and Saito's result that constant Milnor number implies that Thom's a_f condition is satisfied.