Giroux correspondence

Statement

Let $M$ be a closed oriented $3$-manifold. An open book decomposition of $M$ is a pair $(B, \pi)$ of the following kind. The binding $B \subset M$ is an oriented link, and $\pi \colon M \setminus B \to S^1$ is a fibration. Each fiber $\pi^{-1}(\theta)$ is the interior of a compact surface $\Sigma_\theta$, a page, with oriented boundary $\partial \Sigma_\theta = B$. A cooriented positive contact structure $\xi$ on $M$ is supported by $(B, \pi)$ if $\xi = \ker \alpha$ for some contact form $\alpha$ with two properties. The form $\alpha$ is positive on the oriented binding $B$, and $d\alpha$ is a positive area form on each page. Then:

1. Every open book decomposition of $M$ supports a contact structure, and any two contact structures supported by the same open book are isotopic.
2. Every positive contact structure on $M$ is supported by some open book decomposition.
3. Two open book decompositions support isotopic contact structures if and only if they become isotopic after finitely many positive stabilizations of each.

So supporting gives a bijection between two sets. The first is the set of positive contact structures on $M$ up to isotopy. The second is the set of open books of $M$ up to isotopy and positive stabilization.