On nonorientable $4$--manifolds

We present several structural results on closed, nonorientable, smooth $4$--manifolds, extending analogous results and machinery for the orientable case. We prove the existence of simplified broken Lefschetz fibrations and simplified trisections on nonorientable $4$--manifolds, yielding descriptions of them via factorizations in mapping class groups of nonorientable surfaces. With these tools in hand, we classify low genera simplified broken Lefschetz fibrations on nonorientable $4$--manifolds. We also establish that every closed, smooth $4$--manifold is obtained by surgery along a link of tori in a connected sum of copies of CP², S¹ × S³ and S¹× S³. Our proofs make use of topological modifications of singularities, handlebody decompositions, and mapping classes of surfaces.