Stabilizations via Lefschetz Fibrations and Exact Open Books

We show that if a contact open book $(Σ,h)$ on a $(2n+1)$-manifold $M$ ($n\geq1$) is induced by a Lefschetz fibration $π:W \to D^2$, then there is a one-to-one correspondence between positive stabilizations of $(Σ,h)$ and \emph{positive stabilizations} of $π$. More precisely, any positive stabilization of $(Σ,h)$ is induced by the corresponding positive stabilization of $π$, and conversely any positive stabilization of $π$ induces the corresponding positive stabilization of $(Σ,h)$. We define \emph{exact open books} as boundary open books of compatible exact Lefschetz fibrations, and show that any exact open book carries a contact structure. Moreover, we prove that there is a one-to-one correspondence (similar to the one above) between \emph{convex stabilizations} of an exact open book and \emph{convex stabilizations} of the corresponding compatible exact Lefschetz fibration. We also show that convex stabilization of compatible exact Lefschetz fibrations produces symplectomorphic completions.

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