nLab Mather's stability theorem

Ingredients

Definition

A smooth map $f \,\colon\, X \to Y$ between smooth manifolds is called stable if every nearby smooth function, hence every function in some open neighbourhood of $f$ inside the mapping space $C^\infty(X,Y) \subset C^0(X,Y)$, is equal to $f$ up to conjugation with a pair of diffeomorphisms.

Similarly, $f$ is infinitesimally stable if this statement holds to first order in derivatives, in a suitable sense.

(due to Mather 68, see Ruas 22, Def. 2.2, Def. 3.3).

Statement

Proposition

A proper smooth map between smooth manifolds is “stable” (Def. ) if and only if it is infinitesimally stable.

(Mather 70, Thm. 4.1, Ruas 22, Thm. 3.11)

Related statements

• inverse function theorem

References

The original articles:

• John N. Mather, Stability of $C^\infty$ mappings: I. The division theorem, Annals of Mathematics 87 1 (1968) 89 $[$doi:10.2307/1970595$]$

• John N. Mather, Stability of $C^\infty$ Mappings: II. Infinitesimal Stability Implies Stability, Annals of Mathematics 89 2 (1969) 254 $[$doi:10.2307/1970668$]$

• John N. Mather, Stability of $C^\infty$ mappings: V. Transversality, Advances in Mathematics 4 3 (1970) 301-336 (doi:10.1016/0001-8708(70)90028-9)

Review:

• Maria Aparecida Soares Ruas, Old and new results on density of stable mappings $[$arXiv:2201.03888$]$

A proof in synthetic differential topology is provided in section 7.3 of

• Marta Bunge, Felipe Gago, Ana Maria San Luis Fernández, Synthetic Differential Topology, 2018, (CUP) (excerpt)

following

• Ana Maria San Luis Fernández, Estabilidad transversal de gérmenes representables infinitesimalmente, Ph.D. Thesis, Universidad de Santiago de Compostela (1999) (abstract page, on GoogleBooks)