The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory

TL;DR

This paper proves that the monodromy of certain complex hypersurface singularities has infinite order in the smooth mapping class group, using Seiberg--Witten Floer homology techniques, with implications for symplectic and contact topology.

Contribution

It establishes the infinite order of monodromy diffeomorphisms for weighted-homogeneous singularities, extending understanding of their topological and symplectic properties.

Findings

Monodromy diffeomorphism has infinite order for non-rational double point singularities.

Boundary Dehn twist of indefinite symplectic fillings has infinite order in the mapping class group.

Uses $Z/p$-equivariant Seiberg--Witten--Floer homology techniques.

Abstract

We prove that the monodromy diffeomorphism of a complex 2-dimensional isolated hypersurface singularity of weighted-homogeneous type has infinite order in the smooth mapping class group of the Milnor fiber, provided the singularity is not a rational double point. This is a consequence of our main result: the boundary Dehn twist diffeomorphism of an indefinite symplectic filling of the canonical contact structure on a negatively-oriented Seifert-fibered rational homology 3-sphere has infinite order in the smooth mapping class group. Our techniques make essential use of analogues of the contact invariant in the setting of $\mathbb{Z}/p$-equivariant Seiberg--Witten--Floer homology of 3-manifolds.