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

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

2002-07-02

arxiv.org/abs/math/0207018v1

Mathematics

Algebraic Geometry

Scientific paper

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 $n$.

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