Cylindrical contact homology of the links of the simple singularities

We compute the cylindrical contact homology of the links of the simple singularities. These manifolds are contactomorphic to $S^3/G$ for finite subgroups $G\subset\text{SU}(2)$. We perturb the degenerate contact form on $S^3/G$ with a Morse function, invariant under the corresponding $H\subset\text{SO}(3)$ action on $S^2$, to achieve nondegeneracy up to an action threshold. The cylindrical contact homology is recovered by taking a direct limit of the action filtered homology groups. The rank of this homology is $|\text{Conj}(G)|$, demonstrating a form of the McKay correspondence