Seiberg–Witten invariants and surface singularities

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, ($\mathbb{Q}$–)Gorenstein singularities, rational singularities, Brieskorn–Hamm complete intersections, geometric genus, Seiberg–Witten invariants of $\mathbb{Q}$–homology spheres, Reidemeister–Turaev torsion, Casson–Walker invariant

Mathematical Subject Classification 2000

Primary: 14B05, 14J17, 32S25, 57R57

Secondary: 57M27, 14E15, 32S55, 57M25

References

Forward citations

Publication

Received: 11 January 2002

Revised: 25 April 2002

Accepted: 17 May 2002

Published: 20 May 2002

Proposed: Walter Neumann

Seconded: Robion Kirby, Yasha Eliashberg

Authors

András Némethi
Department of Mathematics Ohio State University Columbus Ohio 43210 USA
http://www.ohio-state.edu/~nemethi/
Liviu I Nicolaescu
Department of Mathematics University of Notre Dame Notre Dame Indiana 46556 USA
http://www.nd.edu/~nicolae/

Volume 6, issue 1 (2002)

András Némethi and Liviu I Nicolaescu