# Tuesday, March 15

> See <https://arxiv.org/pdf/math/9910127.pdf>

:::{.remark}
Last time: classifying tight contact structures on $T^3$.
Some contact structure:
\[
\xi_n = \ker( \cos(2\pi n z) \dx - \sin(2\pi n z)\dy )
.\]
Realize $T^3$ as a cube with faces glued, then moving in the $z$ direction twists $n$ times as you traverse the cube.
We can reduce this to $\xi_1$ using $\tv{x,y,z}\mapsto \tv{x,y,nz}$.
:::

:::{.remark}
Goal: classify tight contact structures on lens spaces $L_{p, q} = T^2\times I/\sim$.
We can discretize the contact structure on $\Sigma\times I$ into a finite number of *bypass moves* on the dividing sets.
The basic move:

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![](figures/2022-03-15_11-30-20.png)
:::

:::{.definition title="Basic slice"}
A **basic slice** is a contact structure on $T^2$ such that 

- $T^2\times\ts{0}$ is convex with 2 dividing curves of slope 0
- $T^2\times\ts{1}$ is convex with 2 dividing curves of slope -1
- $\xi$ is tight
- $\xi$ is minimally twisting, so if $T^2 \subseteq T^2\times I$ is convex then $\slope(r) \in [-1, 0]$.

:::

:::{.proposition title="?"}
There are exactly 2 basic slices.
Both embed in $(T^3, \xi_1) = \ker(\cos(2\pi z)\dx - \sin(2\pi z)\dy) = T^2\times I/\sim$, and are given by

- $(T_2\times [0, 1/8], \xi_1)$
- $(T_2\times [1/2, 5/8], \xi_1)$

:::


:::{.proof title="?"}
Step 1:
There are at most 2 basic slices.
Reduce to $S^1\times D^2$ by removing a convex annulus.
Note that $T^2\times I\sm (S^1\times I) \cong S^1\times I^2 \cong S^1\times D^2$.

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![](figures/2022-03-15_11-40-54.png)

Since the boundary is convex, we can make the foliations on both of the ruling curves of slope $\infty$.


> ?

Take an annulus $A$ with some condition on $\bd A$, perturb to be convex?
Something contradicts the "minimally twisting" assumption, involving these pics:

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![](figures/2022-03-15_11-51-21.png)

Smooth corners?

> ?

:::



:::{.definition title="Relative Euler class"}
Let $(M,\xi)$ be a contact 3-manifold with $\ro{\xi}{\bd M}$ trivial.
Let $s$ be a nonvanishing section of $\ro{\xi}{\bd M}$, then the **relative Euler class** $e(\xi, s) \in H^2(M, \bd M;\ZZ) \cong H_1(M)$ (by Lefschetz duality) is the dual of the vanishing set of an extension of $s$ to a section of $\xi$ on $M$.
:::


:::{.remark}
In this case $\dim s\inv = \dim M - \dim \xi$.
:::


:::{.lemma title="?"}
If $\Sigma \embeds (M,\xi)$ is a properly embedded convex surface and $s$ is a section of $\ro{\xi}{\bd M}$ that is tangent to $\bd \Sigma$ with the correct orientation, then
\[
\inp{e(\xi, s)}{\Sigma} = \chi(\Sigma_+) - \chi(\Sigma_-)
.\]
where $\inp\wait\wait: H^2(M, \bd M;\ZZ) \times H_2(M, \bd M; \ZZ) \to \ZZ$.
:::


:::{.remark}
Note $H_2(T^2\times I, \bd; \ZZ) = \gens{[\alpha\times I], [\beta\times I]}$ where $H_2(T^2) = \gens{\alpha, \beta}$.
:::