Small genus-4 Lefschetz fibrations on simply-connected 4-manifolds. (English) Zbl 07582156
Summary: We consider simply connected 4-manifolds admitting Lefschetz fibrations over the 2-sphere. We explicitly construct nonhyperelliptic and hyperelliptic Lefschetz fibrations of genus 4 on simply-connected \(4\)-manifolds which are exotic symplectic 4-manifolds in the homeomorphism classes of \(\mathbb{C}P^2\#8\overline{\mathbb{C}P^2}\) and \(\mathbb{C}P^2\#9\overline{\mathbb{C}P^2}\), respectively. From these, we provide upper bounds for the minimal number of singular fibers of such fibrations. In addition, we prove that this number is equal to 18 for \(g = 3\) when such fibrations are hyperelliptic. Moreover, we discuss these numbers for higher genera.
MSC:
| 57K40 | General topology of 4-manifolds |
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 20F38 | Other groups related to topology or analysis |
| 53Dxx | Symplectic geometry, contact geometry |
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