Seiberg-Witten invariants of lens spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1001.57051
DOI: 10.4153/CJM-2001-032-7
author: Liviu I. Nicolaescu
published in: Canadian Journal of Mathematics
publication date: 2002-12-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9901071
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 58D27
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 57R15
zbMATH DE Number: 1688935
zbMATH Keywords: eta invariants
MaRDI profile type: Publication
arXiv ID: math/9901071
title: Seiberg-Witten invariants of lens spaces
review text: 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…
reviewed by: Matilde Marcolli
description: scientific article; zbMATH DE number 1688935