On the Mather stability theorem for smooth maps
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909837814661120 |
|---|---|
| author | Sadykov, Rustam |
| author_facet | Sadykov, Rustam |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10305 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Mather stability theorem for smooth maps Sadykov, Rustam Geometric Topology Dynamical Systems 58K30, 58K65, 57R45 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]. |
| title | On the Mather stability theorem for smooth maps |
| topic | Geometric Topology Dynamical Systems 58K30, 58K65, 57R45 |
| url | https://arxiv.org/abs/2510.10305 |