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

TL;DR

This work develops a framework to constrain smooth Lefschetz fibrations and diffeomorphisms on 4-manifolds by connecting Seiberg–Witten theory, framed bordism of family moduli spaces, and Lefschetz-fibration data. Central to the approach is the spin-number invariant $\triangle(S_1,\ldots,S_n)\in \pi_1SO(b^+(X))$, defined from framings of the $H^+$-bundle and shown to control whether a homologically-trivial product of Dehn twists is smoothly trivial; $ riangle$ is preserved under mutations and relates to Lefschetz-fibration characteristic classes. The paper introduces and analyzes the family Bauer–Furuta invariant $\text{FBF}$, proves key vanishing and framing-change results (notably for Dehn twists), and uses excision to prove a main parity constraint: under suitable SW-conditions, Ind$(D^+(E,\frak{s}_E))$ modulo 2 equals $w_2(H^+(f))\cdot [\Sigma]$, which together with $\triangle$ yields obstructions to smooth isotopy of monodromies. Applied to explicit constructions (e.g., Milnor fibers from exceptional unimodal singularities and elliptic surfaces), these obstructions produce the first known examples of simply-connected symplectic 4-manifolds with Torelli elements not generated by squared Dehn–Seidel twists, as well as irreducible 4-manifolds admitting exotic Seifert-fibered Dehn twists; and they extend to non-symplectic irreducible manifolds. Overall, the results deepen understanding of the gap between smooth and topological/diffeomorphic structures in dimension four and expand the toolkit for detecting exotic diffeomorphisms via 6-manifold Lefschetz-fibration topology. $

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.

Figures (5)

• Figure 1: Schematic depiction of the spin number $\Delta (S_1 , \ldots , S_n )$ as a loop in $\mathcal{E}$ based at $e_0$.
• Figure 2: Gabrielov's Dynkin diagram for the exceptional unimodal singularities.
• Figure 3: $\Delta = [\eta ]$ for the exceptional unimodal singularities $E_{12}$ (e.g. $x^2 + y^3 + y^7 = 0$) and $E_{14}$ (e.g. $x^2 + y^3 + z^8 = 0$).
• Figure :
• Figure :

Theorems & Definitions (85)

• Theorem A
• Remark 1.1
• Definition 1.2
• Theorem B
• Corollary 1.3
• Example 1.4
• Theorem C
• Corollary 1.5
• Example 1.6
• Definition 2.1