Non-density of stable mappings on non-compact manifolds

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 noncompact. 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.

Keywords

stable mapping, Whitney $C^\infty$ topology, strong conjecture

2010 Mathematics Subject Classification

58K25, 58K30

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This work was supported by JSPS KAKENHI Grant Number JP21K13786.

Received 5 November 2021

Received revised 4 July 2022

Accepted 22 July 2022

Published 7 April 2023