Seiberg-Witten invariants and surface singularities III. Splicings and cyclic covers

TL;DR

This paper proves a conjecture relating Seiberg-Witten invariants to the signature of Milnor fibers for certain surface singularities, using splicing formulas for topological invariants and computing invariants from Newton pairs.

Contribution

It introduces general splicing formulas for Casson-Walker invariant and Reidemeister-Turaev torsion, enabling computation of Seiberg-Witten invariants for surface singularities.

Findings

Verified the conjecture for suspension singularities of type g(x,y,z)=f(x,y)+z^n.

Derived splicing formulas for key topological invariants.

Computed invariants explicitly from Newton pairs and integer n.

Abstract

We verify the conjecture formulated in math.AG/0111298 for suspension singularities of type $g(x,y,z)= f(x,y)+z^n$, where $f$ is an irreducible plane curve singularity. More precisely, we prove that the modified Seiberg-Witten invariant of the link $M$ of $g$, associated with the canonical $spin^c$ structure, equals $-\sigma(F)/8$, where $\sigma(F)$ is the signature of the Milnor fiber of $g$. In order to do this, we prove general splicing formulae for the Casson-Walker invariant and for the sign refined Reidemeister-Turaev torsion (in particular, for the modified Seiberg-Witten invariant too). These provide results for some cyclic covers as well. As a by-product, we compute all the relevant invariants of $M$ in terms of the Newton pairs of $f$ and the integer $n$.