On four-dimensional Dehn twists and Milnor fibrations

2024/09/18 by Hokuto Konno, Jianfeng Lin, Konno, Hokuto +5 · 3 citations
Mathematics · #57K41 #57R58 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2409.11961

openalex publication_date 2024/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More precisely, we establish infinite order non-triviality results for boundary Dehn twists on indefinite symplectic fillings of links of minimally elliptic surface singularities. Using this, we exhibit a wide variety of new phenomena in dimension four: (1) smoothings of isolated complex surface singularities whose Milnor fibration has monodromy with infinite order as a diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3) compactly supported exotic diffeomorphisms of exotic ℝ⁴'s and contractible manifolds.

Cited by

• Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory
• On absolutely exotic diffeomorphisms of 4-manifolds
• Exotic ℝ⁴'s with Compactly Supported Diffeomorphisms

Related

• The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory
• Boundary Dehn twists on Milnor fibers and Family Bauer--Furuta invariants
• Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory
• Exotic Dehn twists on 4-manifolds
• Floer Homology and Fractional Dehn Twists