Seiberg-Witten invariants of lens spaces
Seiberg-Witten theory on 3-manifolds that are rational homology spheres does not lead, right away, to topological invariants, due to the metric dependence of the algebraic counting of points in the moduli space of solutions of the monopole equations. However, after introducing a correction term given by a certain combination of eta invariants, the modified counting yields a topological invariant, which interpolates between Reidemeister torsion and the Casson-Walker invariant. In fact, Seiberg-Witten invariants provide a natural unifying framework for these classes of invariants: on integral homology 3-spheres one recovers Casson's invariant [\textit{Y. Lim}, Seiberg-Witten invariants for 3-manifolds in the case \(b_1=0\) or 1, Pac. J. Math. 195, No. 1, 179-204 (2000; Zbl 1015.57022)], on 3-manifolds with non-trivial rational homology one recovers Milnor torsion [\textit{G. Meng} and \textit{C. H. Taubes}, Math. Res. Lett. 3, No. 5, 661-674 (1996; Zbl 0870.57018)], hence, in particular, from an averaged version of the Seiberg-Witten invariants one obtains Casson-Walker-Lescop's invariant. On rational homology spheres an averaged version of the modified Seiberg-Witten invariants gives the Casson-Walker invariant [\textit{M. Marcolli} and \textit{B.-L. Wang}, Geom. Dedicata 91, 45-58 (2002; Zbl 0994.57027)], while the ``complement'' of this average recovers the torsion invariant [Nicolaescu, math.GT/0103020]. NEWLINENEWLINENEWLINEThis paper fits into a series of results for special classes of 3-manifolds, aimed at establishing the general relation described above between the modified Seiberg-Witten and the other classical invariants. Here the author considers the case of lens spaces, for which he describes a simple algorithm computing the modified Seiberg-Witten invariants. Since lens spaces have positive scalar curvature, the original monopole invariants vanish for that choice of the metric, and the problem can be reduced to computing the corresponding correction term. The author achieves this by considering geometric Seifert structures and Sasakian structures on lens spaces and deriving the computation of the eta invariants in terms of Dedekind-Rademacher sums. NEWLINENEWLINENEWLINEThe main result of this paper on lens spaces provides the first step of the inductive proof of the general result on the relation between modified Seiberg-Witten invariants and Casson-Walker invariants of rational homology 3-spheres in Marcolli and Wang [loc. cit.], which in turn provides the initial step for the inductive proof in [Nicolaescu, math.GT/0103020] of the general relation between modified Seiberg-Witten invariants, Casson-type invariants and torsion invariants.
scientific article; zbMATH DE number 1688935