On nonorientable $4$--manifolds

TL;DR

This paper extends structural results for nonorientable 4-manifolds, introducing fibrations and classifications, and showing all such manifolds can be constructed via surgeries on specific link configurations.

Contribution

It develops new tools for nonorientable 4-manifolds, including fibrations and classifications, and demonstrates their construction through surgeries on links in connected sums.

Findings

Existence of simplified broken Lefschetz fibrations on nonorientable 4-manifolds

Classification of low genus fibrations in this setting

Every closed nonorientable 4-manifold can be obtained by surgery on a link of tori

Abstract

We present several structural results on closed, nonorientable, smooth $4$--manifolds, extending analogous results and machinery for the orientable case. We prove the existence of simplified broken Lefschetz fibrations and simplified trisections on nonorientable $4$--manifolds, yielding descriptions of them via factorizations in mapping class groups of nonorientable surfaces. With these tools in hand, we classify low genera simplified broken Lefschetz fibrations on nonorientable $4$--manifolds. We also establish that every closed, smooth $4$--manifold is obtained by surgery along a link of tori in a connected sum of copies of $\mathbb{CP}^2$, $S^1 \times S^3$ and $S^1\widetilde{\times} S^3$. Our proofs make use of topological modifications of singularities, handlebody decompositions, and mapping classes of surfaces.