#### 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.

Notes on file formats
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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