Topology of the links of cDV singularities of types $cA_n$ for $n>0$ and $cD_n$ for $n>4$

Abstract

We show that the second integral homology group of the link of an isolated compound Du Val (cDV, for short) singularity of type $cA_n$ is either trivial or a torsion-free abelian group. Consequently, by a result of Smale, it follows that the link is either $S^5$ or a connected sum of finitely many copies of $S^2\times S^3$. We also determine the rank of the second integral homology group of the link of a singularity of type $cD_n$ with $n>4$ under the assumption that the singularity is Newton non-degenerate. Furthermore, we focus on the weighted homogeneous case and determine the homology group of the link, including its torsion subgroup, under the assumption that the singularity is a Thom-Sebastiani sum of singularities of Brieskorn-Pham, cyclic, or chain type.