Seiberg Witten Invariants of Rational Homology 3-spheres

In 1996 Meng and Taubes [16] have established a relationship between the Seiberg– Witten invariants of a (closed) 3-manifold with b1 > 0 and the Milnor torsion. A bit later Turaev, [29, 30], enhanced Meng–Taubes’ result, essentially identifying the Seiberg–Witten invariant with the refined torsion he introduced earlier in [27]. In [29] Turaev raised the question of establishing a connection between these two invariants in the remaining case, that of rational homology spheres. Around the same time, Lim [12] succeeded in providing a combinatorial description of the Seiberg–Witten invariants of integral homology spheres. Namely, in this case they coincide with the Casson invariant. In [19] we investigated a special class of rational homology spheres, the lens spaces, and we proved that the Seiberg–Witten invariants of such spaces are determined by the Casson–Walker invariant and the Reidemeister–Turaev torsion in a very explicit fashion. In that paper we also raised the question whether this is the case in general. Recently Marcolli and Wang [15] (see also the related work of Ozsváth–Szabó [23]) have shown that the Seiberg–Witten invariants of a QHS determine the Casson–Walker invariant. Additionally, they have proved a very general surgery formula involving the Seiberg–Witten invariants. The main result of the present paper is Theorem 2.4 where we prove that for rational homology spheres we have