Pin ^± -structures on non-oriented 4-manifolds via Lefschetz fibrations

Abstract We study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a ${\text{Pin}}^{\pm}$ -structure, extending the work of A. Stipsicz in the orientable setting. As a corollary, we get existence results of ${\text{Pin}}^{+}$ and ${\text{Pin}}^-$ -structures on closed non-orientable 4-manifolds and on Lefschetz fibrations over the 2-sphere. In particular, we show via three explicit examples how to read-off ${\text{Pin}}^{\pm}$ -structures from the Kirby diagram of a 4-manifold. We also provide a proof of the well-known fact that any closed 3-manifold M admits a ${\text{Pin}}^-$ -structure and we find a criterion to check whether or not it admits a ${\text{Pin}}^+$ -structure in terms of a handlebody decomposition. We conclude the paper with a characterization of ${\text{Pin}}^+$ -structures on vector bundles.