# Tuesday, February 08

:::{.remark}
Topics for talks:

- Thom-Pontryagin 
- Brieskorn spheres
- Milnor fibrations
- Lens spaces

:::

:::{.theorem title="?"}
Every closed oriented 3-manifold $Y$ admits a (positive) contact form.
:::

:::{.remark}
Three proofs:

- Lickorish-Wallace, using that $Y$ is Dehn surgery on a link in $S^3$,
- Birman-Hildon, using that $Y$ is a branched cover of $S^3$,
- Alexander, using that $Y$ admits an open book decomposition.

:::

:::{.remark}
Dehn surgery for slope $p/q$: for $K \injects S^3$, cut out $\nu(K) \cong S^1\times \DD$ and re-glue by a map $\bd(S^1\times \DD) \to \bd \nu(K)$ such that $[\ts{0} \times \bd \DD] = p[m] + q[\ell] \in H^1(\bd \nu (K) )$.
Use that $\nu(K) \cong S^1\times \DD$ and $\bd \nu(K) \cong S^1\times S^1 = T^2$.
Idea: wrapped $p$ times longitudinally, $q$ times around the meridian.
:::

:::{.remark}
Recall:

- Every knot $K$ can be $C^0$ approximated by a transverse knot
- Every link $L$ can be $C^0$ approximated by a transverse link
- Neighborhood theorem: for every transverse knot $K$, there is a $w(K)$ and a contactomorphism to a standard model: $S^1\times \DD$ in coordinates $(\phi, r, \theta)$ with $0\leq r\leq \delta$ and $\alpha = d\phi + r^2\dtheta$.
  Re-gluing corresponds to the map $\tv{0, \delta, \theta}\mapsto \tv{q\theta, \delta, p\theta}$.

\[
\tv{0, \delta, \bar\theta} &\mapsto \tv{q\bar\theta, \delta, p\bar \theta} \\
\tv{\bar\pi, \bar r, \bar\theta} &\mapsto \tv{\phi,r,\theta}
.\]
If $p, q$ are coprime there exist $m,n$ with $pm-qn = 1$.
So define
\[
\psi: \tv{\bar\pi, \bar r, \bar\theta} &\mapsto \tv{\phi,r,\theta} 
,\]
so
\[
\psi^*(\alpha) = d(\alpha\bar\theta + m\bar\phi) + r^2d(p\bar\theta + n\bar\phi) = (q+r^2p)d\bar\theta + (m+r^2n)d\bar\phi
.\]
We want $\alpha = h_1(r) d\bar\theta + h_2(r) d\bar\theta$ to be contact and satisfy $(h_1, h_2) = (r^2, 1)$ near $r=0$ and $(q+r^2 p, m+r^2 n)$ near $r=\delta$.
This requires
\[
d\alpha = h_1' \dr \wedge d\bar\phi + h_2' \dr \wedge d\bar\theta = (h_2 h_1' - h_1 h_2') dr \wedge d\bar \theta \wedge d\bar\phi
,\]
which happens iff
\[
\det 
\begin{bmatrix}
h_2 & h_2' 
\\
h_1 & h_1'
\end{bmatrix}
> 0
.\]

Think of $\tv{h_2, h_1}$ as a path with tangent vector $\tv{h_2', h_1'}$.
This requires moving counterclockwise.

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![](figures/2022-02-08_11-57-58.png)

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:::{.definition title="?"}
An **open book decomposition** of $Y$ is a pair $(B, \pi)$ where

- $B$ is a link in $Y$, called the **binding**
- $\pi: Y\sm B\to S^1$ is a locally trivial fibration of relatively compact fibers **pages**

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![](figures/2022-02-08_12-09-13.png)

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:::{.remark}
An open book decomposition is determined by its monodromy map $\phi: \Sigma_0\to \Sigma_0$, which determines a class $[\phi] \in \MCG(\Sigma_0)$.
Form 
\[
Y\sm \nu(B) \cong {\Sigma \times I \over \phi(x) \cross \ts{0} \sim x\cross \ts{1}}
,\]
which is a glued cylinder:

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![](figures/2022-02-08_12-14-04.png)

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:::{.definition title="Open book decompositions supporting a contact structure"}
An open book decomposition **supports** a contact structure $\xi$ iff there exists a contact form $\alpha$ such that $d\alpha$ is an area form on each page and $B$ is a transverse link in $(B, \xi)$.
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:::{.theorem title="Thurston-Winkelnkemper"}
Every open book decomposition admits a contact structure.
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:::{.theorem title="Giroux"}
Every $(Y^3, \xi)$ with $Y$ closed has a supporting open book decomposition.
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:::{.proposition title="?"}
If an open book decomposition supports $\xi_1$ and $\xi_2$, then $\xi_1$ is isotopic to $\xi_2$.
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:::{.proof title="?"}
Two steps:

- Form a mapping cylinder of the monodromy map $\phi$,
- Extend over the binding, using the same idea as in Dehn surgery.

Choose an area form $\omega$ on $\Sigma$ and a primitive $\beta$ with $d\beta = \omega$.
Let $\beta_1 \da \phi^*\beta$ and $\beta_0 = \beta$, then set
\[
\beta_t = t\beta_1 + (1-t)\beta_0
.\]
This yields a 1-form on $\Sigma\times I$ that extends to the mapping cylinder.
Moreover $d\beta_t = td\beta_1 + (1-t)d\beta_0$ is an area form on $\sigma\times \ts{t}$ and $\alpha = \dt + \eps \beta_t$ is a contact form for small $\eps > 0$.
Then $d\alpha = \eps d\beta_t + \eps \dt \wedge \dot{\beta}_t$ and $\alpha \wedge d\alpha = \eps dt \wedge d\beta_t + \bigo(\eps^2)$.
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