Stabilizations via Lefschetz Fibrations and Exact Open Books

We show that if a contact open book $(\Sigma,h)$ on a $(2n+1)$-manifold $M$ ($n\geq1$) is induced by a Lefschetz fibration $\pi:W \to D^2$, then there is a one-to-one correspondence between positive stabilizations of $(\Sigma,h)$ and \emph{positive stabilizations} of $\pi$. More precisely, any positive stabilization of $(\Sigma,h)$ is induced by the corresponding positive stabilization of $\pi$, and conversely any positive stabilization of $\pi$ induces the corresponding positive stabilization of $(\Sigma,h)$. 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.