Non-density of stable mappings on non-compact manifolds

TL;DR

This paper extends Mather's stability theory by proving that stable mappings are never dense on non-compact manifolds, and provides a characterization of density conditions for all manifolds.

Contribution

It demonstrates the non-density of stable mappings on non-compact manifolds and completes the characterization of stability density for all manifold types.

Findings

Stable mappings are never dense on non-compact manifolds.

Provides a complete characterization of stability density for all manifolds.

Extends Mather's results to non-compact cases.

Abstract

Around 1970, Mather established a significant theory on the stability of $C^\infty$ mappings and gave a characterization of the density of proper stable mappings in the set of all proper mappings. The result yields a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is compact. The aim of this paper is to complement Mather's result. Namely, we show that the set of stable mappings in the set of all mappings is never dense if the source manifold is non-compact. Moreover, as a corollary of Mather's result and the main theorem of this paper, we give a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is not necessarily compact.