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Seiberg–Witten Invariant of the Universal Abelian Cover of \({S_{-p/q}^{3}(K)}\)

  • József Bodnár
  • András NémethiEmail author
Chapter

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.

Keywords

3-Manifolds Abelian coverings Geometric genus Lattice cohomology Links of singularities Normal surface singularities Plumbed 3-manifolds \(\mathbb{Q}\)-Homology spheres Seiberg–Witten invariants Superisolated singularities Surgery 3-manifolds 

2010 Mathematics Subject Classification:

Primary. 32S05 32S25 32S50 57M27 Secondary. 14Bxx 32Sxx 57R57 55N35 

Notes

Acknowledgements

The first author is supported by the ‘Lendület’ and ERC program ‘LTDBud’ at MTA Alfréd Rényi Institute of Mathematics. The second author is partially supported by OTKA Grants 100796 and K112735.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.A. Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

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