  We present explicit algorithms for simplifying the topology of indefinite
fibrations on 4-manifolds, which include broken Lefschetz fibrations and
indefinite Morse 2-functions. The algorithms consist of sequences of moves,
which modify indefinite fibrations in smooth 1-parameter families. In
particular, given an arbitrary broken Lefschetz fibration, we show how to turn
it to one with directed and embedded round (indefinite fold) image, and to one
with all the fibers and the round locus connected. We also show how to realize
any given null-homologous 1-dimensional submanifold with prescribed local
models for its components as the round locus of such a broken Lefschetz
fibration. These algorithms allow us to give purely topological and
constructive proofs of the existence of simplified broken Lefschetz fibrations
and Morse 2-functions on general 4-manifolds, and a theorem of
Auroux-Donaldson-Katzarkov on the existence of broken Lefschetz pencils with
directed embedded round image on near-symplectic 4-manifolds. We moreover
establish a correspondence between broken Lefschetz fibrations and Gay-Kirby
trisections of 4-manifolds, and show the existence of simplified trisections on
all 4-manifolds. Building on this correspondence, we provide several new
constructions of trisections, including infinite families of genus-3
trisections with homotopy inequivalent total spaces, and exotic same genera
trisections of 4-manifolds in the homeomorphism classes of complex rational
surfaces.
