Lefschetz fibrations on nonorientable 4-manifolds

Let $W$ be a nonorientable $4$-dimensional handlebody without $3$- and $4$-handles. We show that admits Lefschetz fibration over the $2$-disk, whose regular fiber is surface with nonempty boundary. This an analogue of result Harer obtained in orientable case. As corollary, we obtain proof fact every closed $3$-manifold open book decomposition, which was first proved by Berstein Edmonds using branched coverings. Moreover, monodromy for given belongs to twist subgroup mapping class group page. In particular, construct explicit minimal connected sum arbitrarily many copies product circle real projective plane. also relative trisection diagram $W$, based on construct, similar case studied Castro. get diagrams some $4$-manifolds, e.g. $2$-sphere plane, doubling $W$. if $X$ $4$-manifold $2$-sphere, equipped section square $\pm 1$, then $X$, determined vanishing cycles fibration. Finally, include simple observations about low-genus fibrations $4$-manifolds.