Seiberg–Witten invariants and surface singularities: splicings and cyclic covers

Abstract

We verify the conjecture formulated in [36] 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 −σ(F)/8, where σ(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. 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.