---
date: 2022-04-05 23:42
modification date: Wednesday 6th April 2022 17:08:28
title: "Weinstein"
aliases: [Weinstein]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #geomtop/symplectic-topology 
- Refs:
	- #todo/add-references
- Links:
	- [Stein](Unsorted/Stein.md)
	- [surgery](Unsorted/surgery.md)

---

# Weinstein

# Definition


**Definition (Weinstein Surgery):**
Let $(W, \lambda)$ be a [Liouville domain](Liouville%20domain) (although we won't need compactness).

> Recall: $(W, \lambda)$ is a $2n\dash$dimensional exact [symplectic manifold](Unsorted/symplectic.md) with contact-type boundary $\bd W$ such that the Liouville vector field $X$ points outwards along $\bd W$.

**Weinstein surgery** takes 

- $(W, \lambda)$ a $2n\dash$dimensional manifold
- That is exact and symplectic
- With contact-type boundary $\bd W$
- Where the Liouville vector field points outward along $\bd W$

and produces a distinct manifold $(W(\Lambda), \lambda')$ with the above properties which is obtained by [surgery](surgery.md) along $\Lambda$ an isotropic embedded sphere.
Thus $W(\Lambda)$ is obtained from attaching a $k\dash$handle to $W$ along $\Lambda$. 

# Why Care About Weinstein Surgery

#why-care 
**Theorem:** 
Every compact [3-manifold](Unsorted/Three-manifolds%20MOC.md) arises  as  a  combination  of  (2  different  variants  of) Weinstein surgeries on $S^3$.

Compare to theorem: 
Every compact 3-manifold arises as [surgery](surgery.md) on a [link](link).

Theorem (Gromov, Eliashberg)
Any Stein manifold of dimension $n$ embeds holomorphically into $\CC^{\floor{3n \over 2} + 1}$, and this is optimal.

**Theorem:** 
[overtwisted contact structure](overtwisted%20contact%20structure).

**Theorem:** 
[contact manifold](Subjects/Contact%20Topology%20(Subject%20MOC))


# Notes

## Weinstein

Aside:



Moral: flexible, symplectic object.


**Definition**
A Weinstein manifold is the data of

- $M^{2n}$ a smooth manifold,
- $\omega$ a symplectic form,
- $\phi: M\to \RR$ an exhausting generalized Morse function
- $\xi$ a complete Liouville vector field which is gradient-like for $\phi$.

Subdefinitions:

- Exhausting: proper and bounded from below
- Generalized Morse function: non-degenerate critical points of only birth-death type
- Liouville: $\mathcal L_X \omega = \omega$, i.e. the Lie derivative preserves the symplectic form.
  - Recall
  $$
  \mathcal L: \Gamma(TM)\cross \Gamma(TM^{\tensor k}) \to \Gamma(TM^{\tensor k}) \quad \mathcal (\xi, E) \mapsto \mathcal L_\xi(E)
  $$ acts on vector fields and arbitrary tensor fields, in particular alternating tensor fields, i.e. $n\dash$forms.
  - Measures change of a tensor field wrt a vector field, giving a new tensor field. Reduces to lie bracket when $k=1$.
- Complete: flow curves of $\xi$ exist for all time.
  - Recall that the gradient operator takes scalar fields (functions!) to vector fields.
- Gradient-like:
  - $\nabla \phi(q) \xi(q) > 0 \in \RR$ for $q\in M\setminus \crit(\phi)$ (so $\xi$ "points in the same direction" as $\nabla \phi$)
  - Near $p\in \crit(\phi)$, we have $\phi(\vector x) =\vector x^t A \vector x$ where $A = \diag(-1, -1, \cdots, -1_k, 1, \cdots, 1_{n})$.

[Flow Curves](figures/2020-03-31-18:35.png)\