Seiberg-Witten invariant of the universal abelian cover of $S^3_{-p/q}(K)$

TL;DR

This paper establishes an additivity property of Seiberg-Witten invariants under universal abelian covers for certain 3-manifolds derived from negative rational surgeries on algebraic knots, linking topology and complex singularity theory.

Contribution

It proves a new topological invariance property of Seiberg-Witten invariants related to universal abelian covers, using complex singularity theory techniques.

Findings

Additivity property holds for specific 3-manifolds from algebraic knots

Counterexamples show the property does not hold universally

Links between topological invariants and complex singularity theory

Abstract

We prove an additivity property for the normalized Seiberg-Witten invariants with respect to the universal abelian cover of those 3-manifolds, which are obtained via negative rational Dehn surgeries along connected sum of algebraic knots. Although the statement is purely topological, we use the theory of complex singularities in several steps of the proof. This topological covering additivity property can be compared with certain analytic properties of normal surface singularities, especially with functorial behaviour of the (equivariant) geometric genus of singularities. We present several examples in order to find the validity limits of the proved property, one of them shows that the covering additivity property is not true for negative definite plumbed 3-manifolds in general.