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Pin±-structures on non-oriented 4-manifolds via Lefschetz fibrations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 11 November 2025, pp. 1-20
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- Article
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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.
