Realizing crosscap transpositions as monodromies of singular fibrations

We introduce a new type of singularity for smooth maps from $4$-manifolds to surfaces, called an $M$-singularity, whose critical locus is a circle contained in a single fiber. We show that the monodromy around an $M$-singularity is a crosscap transposition in the mapping class group of a non-orientable surface. We also introduce $M$-fibrations, namely smooth maps whose singularities consist only of $M$-singularities, and prove that relations among crosscap transpositions give rise to such fibrations on non-orientable $4$-manifolds. We then study handle decompositions associated with $M$-fibrations and their orientation double coverings. In particular, we describe the attaching circles and framings of the two $2$-handles arising from the orientation double cover of an $M$-singularity. Using this description, we construct a closed non-orientable $4$-manifold which admits an $M$-fibration but admits no Lefschetz fibration. We further discuss singularity-theoretic properties of the local model of an $M$-singularity, namely its infinite $\mathcal{A}_e$-codimension and an explicit stable perturbation.