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nLab > Latest Changes: Mather's stability theorem

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 17th 2022

    Created:

    Statement

    A proper smooth map between smooth manifolds is stable if and only if it is infinitesimally stable.

    References

    • John N. Mather, Stability of C ∞C^\infty mappings: I. The division theorem, Annals of Mathematics 87:1 (1968), 89. doi.

    • John N. Mather, Stability of C ∞ Mappings: II. Infinitesimal Stability Implies Stability, Annals of Mathematics 89:2 (1969), 254. doi.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2022

    I have added some formatting, and have added the definition of “stable”, following and pointing to:

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

    In an attempt to cross-link this with anything, to make it findable (eg. by search engines), I have added cross-link with differential geometry and inverse function theorem. Best to add more

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2022

    added added pointer to

    It’s theorem 4.1 in here that Ruas 22, Thm. 3.11 refers to for the stability theorem.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 17th 2022

    Added references

    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)

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2022

    I can see inside Fernández’ thesis on books.google.ae here. (tried changing to books.google.us and it failed to work.)

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 17th 2022

    I can’t see inside using that link. I tried the .au version, and I only get snippet view of that, too.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2022

    But did you try to enter good keywords? You need to “search” for good generic words, then you get taken inside.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 17th 2022

    I do not get “taken inside”, no. I get tiny snippet views in response to searches, on both domains.

1 to 8 of 8

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