Weinstein

Definition

Definition (Weinstein Surgery): Let \((W, \lambda)\) be a Liouville domain (although we won’t need compactness).

Recall: \((W, \lambda)\) is a \(2n{\hbox{-}}\)dimensional exact symplectic manifold with contact-type boundary \({{\partial}}W\) such that the Liouville vector field \(X\) points outwards along \({{\partial}}W\).

Weinstein surgery takes

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

and produces a distinct manifold \((W(\Lambda), \lambda')\) with the above properties which is obtained by surgery along \(\Lambda\) an isotropic embedded sphere. Thus \(W(\Lambda)\) is obtained from attaching a \(k{\hbox{-}}\)handle to \(W\) along \(\Lambda\).

Why Care About Weinstein Surgery

#why-care Theorem: Every compact 3-manifold arises as a combination of (2 different variants of) Weinstein surgeries on \(S^3\).

Compare to theorem: Every compact 3-manifold arises as surgery on a link.

Theorem (Gromov, Eliashberg) Any Stein manifold of dimension \(n\) embeds holomorphically into \({\mathbf{C}}^{{\left\lfloor 3n \over 2 \right\rfloor} + 1}\), and this is optimal.

Theorem: overtwisted contact structure.

Theorem: contact manifold

Notes

Weinstein

Aside:

Moral: flexible, symplectic object.

Definition A Weinstein manifold is the data of

\(M^{2n}\) a smooth manifold,
\(\omega\) a symplectic form,
\(\phi: M\to {\mathbf{R}}\) 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 \begin{align*} \mathcal L: \Gamma(TM)\times\Gamma(TM^{\otimes k}) \to \Gamma(TM^{\otimes k}) \quad \mathcal (\xi, E) \mapsto \mathcal L_\xi(E) \end{align*} acts on vector fields and arbitrary tensor fields, in particular alternating tensor fields, i.e. \(n{\hbox{-}}\)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:
- \(\nabla \phi(q) \xi(q) > 0 \in {\mathbf{R}}\) for \(q\in M\setminus \operatorname{crit}(\phi)\) (so \(\xi\) “points in the same direction” as \(\nabla \phi\))
- Near \(p\in \operatorname{crit}(\phi)\), we have \(\phi(\mathbf{x}) =\mathbf{x}^t A \mathbf{x}\) where \(A = \operatorname{diag}(-1, -1, \cdots, -1_k, 1, \cdots, 1_{n})\).

Flow Curves\