Exact orbifold fillings of contact manifolds

Abstract

We study exact orbifold fillings of contact manifolds using Floer theories. Motivated by Chen-Ruan's orbifold Gromov-Witten invariants, we define symplectic cohomology of an exact orbifold filling as a group using classical techniques, i.e. choosing generic almost complex structures. By studying moduli spaces of pseudo-holomorphic/Floer curves in orbifolds, we obtain various non-existence, restrictions and uniqueness results for orbifold singularities of exact orbifold fillings of many contact manifolds. For example, we show that exact orbifold fillings of $(\mathbb{RP}^{2n-1},\xi_{\mathrm{std}})$ always have exactly one singularity modeled on $\mathbb{C}^n/(\mathbb{Z}/2\mathbb{Z})$ if $n\ne 2^k$. Lastly, we show that in dimension at least $3$ there are pairs of contact manifolds without exact cobordisms in either direction, and that the same holds for exact orbifold cobordisms in dimension at least $5$.