Relative trisection maps on 4-manifolds

We show that for an arbitrary compact orientable smooth $4$--dimensional manifold M with a given open book structure on the boundary ∂ M ≠ ∅, which may be disconnected, there exists a generic smooth map of M onto the $2$--disk which gives a relative trisection on M consistent with the given open book structure on ∂ M. This gives a new proof to the existence of a relative trisection on M. In fact, it turns out that every relative trisection can be obtained in this way. As a byproduct, we also show that M always admits a decomposition into a PALF (positive allowable Lefschetz fibration) over the $2$--disk D² and a fiber bundle over the circle with a sutured compression body fiber, consistent with the given open book structure on ∂ M and with the surface bundle structure of the PALF over ∂ D².