On nonorientable $4$--manifolds

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.

💡 Research Summary

The paper extends several foundational tools from the theory of smooth, closed, orientable 4‑manifolds to the non‑orientable setting. Its main contributions are four theorems that together give a comprehensive picture of how non‑orientable 4‑manifolds can be described, classified, and constructed.

Theorem 1 (Existence of simplified broken Lefschetz fibrations).
For any closed, connected, non‑orientable 4‑manifold (X) there exists a map (f\colon X\to S^{2}) that is a simplified broken Lefschetz fibration (SBLF). The authors adapt the Baikur‑Saeki algorithm, originally designed for orientable manifolds, to allow achiral Lefschetz critical points (the local model ( (z_{1},z_{2})\mapsto z_{1}z_{2}) without orientation constraints) together with indefinite fold singularities. The SBLF is required to satisfy four extra conditions: injectivity on the set of Lefschetz points and fold arcs, connectivity of the fold set, connectivity of every fiber, and placement of Lefschetz images on the “higher‑genus side” of each fold. This yields a concrete factorization of the monodromy in the mapping‑class group of a non‑orientable surface (N_{k}).

Theorem 2 (Existence of simplified trisections).
Every closed, non‑orientable 4‑manifold admits a simplified trisection, i.e. a ((g,k)) trisection whose associated trisection map has an embedded singular image. The construction again follows the adapted algorithm: one first produces an SBLF, then modifies it to obtain a trisection map whose critical set consists only of indefinite folds and achiral Lefschetz points. The resulting decomposition (X=X_{1}\cup X_{2}\cup X_{3}) has each sector diffeomorphic to a 4‑dimensional 1‑handlebody, pairwise intersections to 3‑dimensional 1‑handlebodies, and triple intersection to a non‑orientable surface (N_{g}). Stable equivalence of such trisections holds exactly as in the orientable case.

Theorem 3 (Classification of low‑genus SBLFs).
The authors give a complete classification of relatively minimal SBLFs of non‑orientable genus two. Let (C_{f}) denote the set of Lefschetz points and (Z_{f}) the set of indefinite folds. Depending on whether these sets are empty, the total space (X) falls into one of four families:

• (C_{f}=Z_{f}=\varnothing): (X) is a Klein‑bottle bundle (K_{n}) over (S^{2}).
• (C_{f}=\varnothing,;Z_{f}\neq\varnothing): (X) is either (N_{n}) or (N’_{n}), two explicit non‑orientable bundles described by Kirby diagrams.
• (C_{f}\neq\varnothing,;Z_{f}=\varnothing): (X) is a manifold (M_{m,n}) obtained by attaching (m) Lefschetz singularities to a genus‑two fibration without folds.
• Both sets non‑empty: this case cannot be relatively minimal; after blowing down the appropriate spheres or real projective planes one reduces to one of the previous three families.

The classification is expressed in terms of Kirby diagrams and factorizations in (\operatorname{Mod}(N_{2})), using Dehn twists and the non‑orientable analogue of Y‑homeomorphisms. Concrete examples are provided, and the authors show how the classification yields minimal‑genus trisections for many of the listed manifolds.

Theorem 4 (Surgery description).
Every closed, non‑orientable 4‑manifold can be obtained from a connected sum
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