Seiberg-Witten and Casson-Walker invariants for rational homology 3-spheres
This paper considers a modified version of the Seiberg-Witten invariants for rational homology 3-spheres, obtained by adding to the original invariants a correction term which is a combination of \(\eta\)-invariants. Then it is shown that these modified invariants are topological invariants. This is achieved by showing how they behave under surgery on a knot in a rational homology 3-sphere, by using the results in [\textit{A. Carey, M. Marcolli} and \textit{B. L. Wang}, The geometric triangle for 3-dimensional Seiberg-Witten monopoles, Preprint math.DG/9907065]. The sum of these modified Seiberg-Witten invariants over all \(\text{spin}^c\) structures of the rational homology 3-sphere \(Y\) is \(|H_{1}(Y;Z)|/2\) times the Casson-Walker invariant of \(Y\).
scientific article; zbMATH DE number 1786837