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date: 2021-04-28 17:23:07
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Tags:
#projects/notes/reading #geomtop/symplectic-topology 
Refs:
[Unsorted/Stein](Unsorted/Stein.md)

# Stein Manifolds

Goal: set up Stein and Weinstein cobordisms, get a Weinstein analog of the [h-cobordism](h-cobordism.md) theorem.

Supporting concepts

- Normal forms of functions near critical points
- Vector fields near zeros (?)
- [Lyapunov pair](Lyapunov%20pair) : a function and a gradient-like vector field
- Smale cobordisms and homotopies
- Elementary cobordisms
- Profiles and [holonomy](holonomy.md)

- Review Classical h-cobordism theorem
- [Morse Lemma](Morse%20Lemma.md) : Nondegenerate index $k$ for $\phi: M^n \to \RR$ implies $\phi(u) = \phi(p) - \vector{u}_k^t A \vector{u}_{n-k}$
- Definitions: embryonic, birth-death types, Cerf diagram
- Theorem: Critical points in 1-parameter families of diffeos are nondegenerate and only birth-death type
- Definition: Morse index
- Definition: Stable and unstable manifolds
- Definition: Lyapnuov functions