No abstract

We obtain new sharp obstructions to symplectic embeddings of four-dimensional polydisks P (a, 1) into four-dimensional ellisoids E(bc, c) when 1 ≤ a < 2 and b is a halfinteger. We demonstrate that P (a, 1) symplectically embeds into E(bc, c) if and only if a + b ≤ bc, showing that inclusion is optimal and extending the result by Hutchings [Hu16] when b is an integer. Our proof is based on a combinatorial criterion developed by Hutchings [Hu16] to obstruct symplectic embeddings. We additionally show that the range of a and b cannot be extended further using the Hutchings criterion.

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 ⊂ SU(2). We perturb the degenerate contact form on S 3 /G with a Morse function, invariant under the corresponding H ⊂ 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 |Conj(G)|, demonstrating a form of the McKay correspondence. Contents