The authors consider 3-dimensional homology spheres \(\Sigma\) which occur as the link of a complete intersection surface singularity and its Milnor fibre F, a 4-dimensional manifold with boundary \(\Sigma\). The Casson invariant \(\lambda\) (\(\Sigma\)) is an integer defined for any homology 3- sphere, and the paper is mainly concerned with the conjecture that for links of singularities we have \(\lambda (\Sigma)=(1/8)sign(F).\)
The paper extends the analysis of Fintushel and Stern, who proved the conjecture in the case of the singularity \(x^ p+y^ q+z^ r=0\) in \({\mathbb{C}}^ 3\), to deal with the cases of \(f(x,y)+{\mathbb{Z}}^ n=0\) in \({\mathbb{C}}^ 3\) as well as complete intersection singularities in \({\mathbb{C}}^ 4\). They also prove the result for weighted homogeneous surface singularities, and give a calculation of \(\lambda\) (\(\Sigma\)) in the further case of an arbitrary graph manifold \(\Sigma\). The techniques involve a study of the behaviour of \(\lambda\) under the ‘splicing’ operation of Eisenbud and Neumann on homology spheres.
Homology spheres which are links of surface singularities have been classified, and in simple cases the Casson invariant is readily found, but it is not known in general which are complete intersection singularities, nor is it easy to find the signature of the Milnor fibre in those cases, or the general case of the conjecture has not been resolved. The paper includes some speculation about relations with Taubes’ definition of the Casson invariant, and also extends the conjecture to the case of rational homology spheres, where it has recently proved possible to define a Casson invariant.