Geometry & Topology, Vol. 6 (2002) Paper no. 9, pages 269--328.

Seiberg--Witten invariants and surface singularities

Andras Nemethi Liviu I Nicolaescu

Abstract. We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally elliptic (including the cyclic quotient and `polygonal') singularities, and Brieskorn-Hamm complete intersections. Some of the verifications are based on a result which describes (in terms of the plumbing graph) the Reidemeister-Turaev sign refined torsion (or, equivalently, the Seiberg-Witten invariant) of a rational homology 3-manifold M, provided that M is given by a negative definite plumbing.
These results extend previous work of Artin, Laufer and S S-T Yau, respectively of Fintushel-Stern and Neumann-Wahl.

Keywords. (Links of) surface singularities, (Q)-Gorenstein singularities, rational singularities, Brieskorn-Hamm complete intersections, geometric genus, Seiberg-Witten invariants of Q-homology spheres, Reidemeister-Turaev torsion, Casson-Walker invariant

AMS subject classification. Primary: 14B05, 14J17, 32S25, 57R57. Secondary: 57M27, 14E15, 32S55, 57M25.

DOI: 10.2140/gt.2002.6.269

E-print: arXiv:math.AG/0111298

Submitted to GT on 11 January 2002. (Revised 25 April 2002.) Paper accepted 17 May 2002. Paper published 20 May 2002.

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Andras Nemethi Liviu I Nicolaescu
Department of Mathematics, Ohio State University
Columbus, OH 43210, USA
and
Department of Mathematics, University of Notre Dame
Notre Dame, IN 46556, USA

Email: nemethi@math.ohio-state.edu, nicolaescu.1@nd.edu
URL: http://www.math.ohio-state.edu/~nemethi/, http://www.nd.edu/~nicolae/
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