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

We study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a Pin±-structure, extending the work of A. Stipsicz in the orientable setting. As a corollary, we get existence results of Pin+ and 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 Pin±-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 Pin−-structure and we find a criterion to check whether or not it admits a Pin+-structure in terms of a handlebody decomposition. We conclude the paper with a characterization of Pin+-structures on vector bundles.