Casson's invariant of Seifert homology 3-spheres
Let \(\alpha_ 1,\alpha_ 2,...,\alpha_ n\) be pairwise coprime positive integers. \(\Sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) be the Seifert homology 3-sphere associated to \((\alpha_ 1,\alpha_ 2,...,\alpha_ n)\), and \(V(\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) be the 2-dimensional smooth affine variety in \({\mathbb{C}}^ n\) which is the Brieskorn complete intersection of weight \((\alpha_ 1,\alpha_ 2,...,\alpha_ n)\). It is known that \(\Sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) is diffeomorphic to \(V(\alpha_ 1,\alpha_ 2,...,\alpha_ n)\cap S_ r^{2n-1}\), where \(S_ r^{2n-1}\) is the (2n-1)-sphere centered at \(0\in {\mathbb{C}}^ n\) with radius r sufficiently large [\textit{H. Hamm}, Math. Ann. 197, 44-56 (1972; Zbl 0239.14003)]. Let \(\sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) denote the signature of \(V(\alpha_ 1,\alpha_ 2,...,\alpha_ n)\). Concerning the \(\lambda\)-invariant \(\lambda (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) of \(\Sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\), we prove the following theorem which extends Theorem 2.10 of \textit{Fintushel} and \textit{Stern} [Instanton homology of Seifert fibered homology three spheres (preprint)]: Theorem 1. \(\lambda (\alpha_ 1,\alpha_ 2,...,\alpha_ n)=(1/8)\sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\). More explicitly: \(\lambda (\alpha_ 1,\alpha_ 2,...,\alpha_ n)=(- 1/8)+(1/24\alpha)(1+\sum^{n}_{j=1}(\alpha /\alpha_ j)^ 2-(n- 2)\alpha^ 2)-(1/2)\sum^{n}_{j=1}s(\alpha /\alpha_ j,\alpha_ j),\) where \(\alpha =\prod^{n}_{j=1}\alpha_ j\) and \(s(\quad,\quad)\) denotes the Dedekind sum. Related results are obtained, too.
scientific article; zbMATH DE number 4103995
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