# Thursday, April 21

> Note: student talks in previous weeks!

:::{.remark}
Possible topics for the remainder of the class:

- Open book decompositions
- Every $(Y^3, \xi)$ is homotopic to a contact structure.
- Seifert fibered spaces

:::

## Seifert Fibered Spaces

:::{.remark}
Brieskorn spheres $\Sigma(p,r,q) \da \ts{x^p + y^q + z^r = 0} \intersect S_\eps^5 \subseteq \CC^3$ are 3-manifolds *foliated* by $S^1$.
Note that $S_1\to X \to S^2$ for $X = S^3$ or $L(p, q)$ are actual fibrations.
Idea: a foliation by $F$'s is a decomposition $X = \disjoint F \to B$ which is a fibration with ramification in some fibers.
:::

:::{.definition title="Seifert fibered spaces"}
A **Seifert fibered space** associated to $(\Sigma, (p_1/q_1, \cdots, p_n/q_n))$ with $p_i/q_i\in \QQ$ and $\Sigma$ an orbifold surface is a 3-manifold $Y$ and knots $L_1,\cdots, L_n$ with neighborhoods $\nu L_i$ such that 

- $Y\sm\union_i\nu(L_i) = (\Sigma\sm\ts{\pt_1,\cdots, \pt_n}) \times S^1$
- $\nu L_i = S^1\times \DD^2$ is glued in by $p_i/q_i$ Dehn surgery.

:::

:::{.example title="?"}
$L(p, q)$ is $-p/q$ surgery on $S^1$, or by a slam-dunk move:

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![](figures/2022-04-21_11-46-26.png)


$\Sigma(p,q,r)$:

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Spring/ContactTopology/sections/figures/2022-04-21_11-46.xoj -->


![](figures/2022-04-21_11-48-50.png)


:::

:::{.exercise title="?"}
Show that for $\Sigma(p,q,r)$, removing the axes in $\CC^3$ yields a trivial fibration by copies of $S^1$ over $S^2\sm{\pt_1,\pt_2,\pt_3}$ and check the surgery slopes.
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:::{.exercise title="?"}
Prove that $\Sigma(p,q,r)$ comes from the plumbing diagram for the Milnor fibration using Kirby calculus.
:::