A geometric proof of the Fintushel-Stern formula
The purpose of this paper is to give a geometric explanation of an observation of Fintushel-Stern that the following formula holds for any pairwise relatively prime integers \(p,q,r\): \[ \lambda (\Sigma (p,g,r))=\textstyle{1\over 8}\sigma (M(p.q.r)) \] where the left side is the Casson invariant of the Brieskorn homology sphere defined as the link of the singularity at \(0\) of the equation \(x^p +y^q +z^r =0\) in \(\mathbb C^3\) and the right side is the signature of the Milnor fiber of that singularity. The explanation proceeds by showing the following: 1. \(\Sigma (p,q,r)\) is the \(r\)-fold branched cyclic cover over the \((p,q)\)-torus knot \(K_{p,q}\), and so has the structure of an orbifold, 2. A theorem of Viro shows that \(\sigma (M(p,q,r))\) is a sum of twisted signatures of \(K_{p,q}\), 3. The irreducible \(SU(2)\)-representations of \(\pi =\pi_1 (\Sigma (p,q,r))\) coincides, under restriction, with the irreducible representations of \(\tilde\pi =\) the orbifold fundamental group of \(\Sigma (p,q,r)\), which implies that the Floer homology of \(\Sigma (p,q,r)\) is isomorphic to the sum, over all eigenvalues, of the orbifold Floer homologies based upon \(SU(2)\)-representations of \(\tilde\pi\) which have that eigenvalue on the meridian element, 4. A theorem of Collin-Steer asserts that the orbifold Floer homology of the cyclic branched cover over a knot \(K\) at a particular eigenvalue has Euler characteristic equal to one-half of the twisted signature of \(K\) evaluated at that eigenvalue. Since only no. 3 uses the particular context, there is some discussion of a generalization of this result to branched covers over more general knots.
scientific article; zbMATH DE number 1417610
scientific article