$\text{Pin}^{\pm}$-structures on non-oriented 4-manifolds via Lefschetz fibrations

TL;DR

This paper characterizes when non-orientable 4-manifolds and Lefschetz fibrations admit $ ext{Pin}^{ ext{±}}$-structures, providing explicit examples, criteria, and extending previous orientable results to the non-orientable setting.

Contribution

It extends the theory of $ ext{Pin}^{ ext{±}}$-structures to non-orientable 4-manifolds and Lefschetz fibrations, including explicit examples and criteria for existence.

Findings

Criteria for $ ext{Pin}^{ ext{±}}$-structures on Lefschetz fibrations

Explicit methods to read $ ext{Pin}^{ ext{±}}$-structures from Kirby diagrams

Proof that all closed 3-manifolds admit $ ext{Pin}^-$-structures

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.