The equivalence of Seiberg-Witten and Casson invariants for homology \(3\)-spheres
It was conjectured by P. Kronheimer that Casson's invariant \(\lambda\) for oriented integral homology \(3\)-spheres and the \(\tau\) invariant derived from the Seiberg-Witten equations are equal. This was verified for Brieskorn homology spheres [\textit{T. Mrowka, P. Ozsváth} and \textit{B. Yu}, Commun. Anal. Geom. 5, No. 4, 685-791 (1997; Zbl 0933.57030)] and proven modulo \(2\) by \textit{W. Chen} [Turk. J. Math. 21, No. 1, 61-81 (1997; Zbl 0891.57021)]. In this paper, the author proves the complete conjecture. Using considerable information from the Seiberg-Witten theory, he shows that \(\tau(Y_{\pm 1})=\tau(Y)\pm {1\over 2}\Delta_K''(1)\), where \(Y_{\pm 1}\) is the result of \(\pm 1\) surgery on the knot \(K\) in the homology \(3\)-sphere \(Y\), and \(\Delta_K(t)\) is the symmetrized Alexander polynomial of \(K\), normalized so that \(\Delta_K(1)=1\). Since this property (together with the single value \(\lambda(S^3)=\tau(S^3)=0\)) characterizes the Casson invariant, the equality of \(\lambda\) and \(\tau\) is established.
scientific article; zbMATH DE number 1489525