Cylindrical contact homology of links of simple singularities (V2)

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.

Double spaced, 143 pages, 15 figures. This is a doctoral thesis from Rice University and an expansion of a previous paper arXiv:2107.07102. This updated version includes new exposition, see sections 1.1, 1.2, 2, 3.3, and 4.5, and appendix A.2. This updated version now includes many new figures, including illustrations for the octahedral and icosahedral invariant Morse functions on $S^2$