On the Mather stability theorem for smooth maps

TL;DR

The paper analyzes the Mather stability framework for smooth maps $f: M\to N$, establishing the equivalence of stability, infinitesimal stability, and the Mather normal crossing condition for proper maps. It presents and leverages a short, algebraic argument inspired by Golubitsky–Guillemin to derive the Mather stability theorem from existing results, emphasizing the role of mutually transversal subspaces and Mather's Lemma MaIII. The key contribution is a concise, coordinate-based reduction showing that the Mather normal crossing condition implies infinitesimal stability at every finite subset of inverse images, thereby proving stability. This provides a streamlined route to the stability criterion for Morin maps and related singularities, with implications for understanding the global topology of singularities in low and higher dimensions.

Abstract

In [MaII] Mather proved that a smooth proper infinitesimally stable map is stable. This result is the key component of the Mather stability theorem [MaV], which can be reformulated as follows: a smooth proper map $f: M\to N$ is stable if and only if it is infinitesimally stable if and only if it satisfies the Mather normal crossing condition. The latter condition, roughly speaking, means that all map germs of $f$ are stable and $f$ maps the singular strata of $f$ to $N$ in a mutually transversal manner. In this note we adapt a short argument from the book by Golubitsky and Guillemin to derive the Mather stability theorem presented in [MaV] from the theorem in [MaII].

Theorems & Definitions (11)

• Theorem 1.1: Mather
• Theorem 1.2: Mather stability theorem
• Corollary 1.3
• Lemma 2.1
• proof
• Theorem 3.1: Mather
• proof
• Proposition 3.2
• proof : Sketch of the proof.
• Theorem 3.3