Cylindrical contact homology of links of simple singularities (V2)

TL;DR

This paper computes the cylindrical contact homology of links of simple singularities, revealing a connection to the McKay correspondence by relating homology ranks to conjugacy classes of finite subgroups of SU(2).

Contribution

It introduces a method to compute contact homology of these links by perturbing the contact form and taking a direct limit, linking topology with group theory.

Findings

Homology ranks relate to conjugacy classes of G.

Method involves perturbing the contact form with a Morse function.

Results demonstrate a form of the McKay correspondence.

Abstract

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 ranks of this homology are given in terms of $|\text{Conj}(G)|$, demonstrating a form of the McKay correspondence.