Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory

2025/11/01 by Konno, Hokuto, Lin, Jianfeng, Mukherjee, Anubhav +1
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2511.00601

Abstract

We establish constraints on the topology of smooth Lefschetz fibrations with 4-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on (-2)-spheres in closed 4-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic 4-manifolds admitting Torelli symplectomorphisms which are smoothly non-trivial. In particular, their symplectic Torelli mapping class group is not generated by squared Dehn-Seidel twists on Lagrangian spheres -- providing a negative answer to a question of Donaldson. (2) We provide the first examples of irreducible closed 4-manifolds (both symplectic and non-symplectic) that admit exotic diffeomorphisms given by Seifert-fibered Dehn twist.

Citations

• Exotic diffeomorphisms on a contractible 4-manifold surviving two stabilizations
• Irreducible 4-manifolds can admit exotic diffeomorphisms
• The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory
• Boundary Dehn twists on Milnor fibers and Family Bauer--Furuta invariants
• Exotic Dehn twists and homotopy coherent group actions
• On four-dimensional Dehn twists and Milnor fibrations
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• Exotic Dehn twists on 4-manifolds
• Symplectic Torelli groups of rational surfaces
• Symplectic mapping class groups of blowups of tori
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• Six Lectures on Four 4-Manifolds
• A gluing theorem for the relative Bauer-Furuta invariants
• Lectures on four-dimensional Dehn twists
• A stable cohomotopy refinement of Seiberg-Witten invariants: II
• A stable cohomotopy refinement of Seiberg-Witten invariants: I
• Knots, Links, and 4-Manifolds
• Rational Blowdowns of Smooth 4-Manifolds

Related

• Broken Lefschetz fibrations and mapping class groups
• Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds
• On four-dimensional Dehn twists and Milnor fibrations
• Families of Simply Connected 4-Manifolds with the Same Seiberg-Witten Invariants
• Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds