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 $5$.

arXiv comment64 pp. Added some additional explanations and corrected some typos, following referee reports. Accepted for publication by the Journal of Topology

References in corpus (11)

• A New Cohomology Theory for Orbifold
• An exact sequence for contact- and symplectic homology
• Contact homology and virtual fundamental cycles
• Floer theory for negative line bundles via Gromov-Witten invariants
• Symplectic cohomology and q-intersection numbers
• $S^1$-equivariant symplectic homology and linearized contact homology
• Planar open book decompositions and contact structures
• A Boothby-Wang theorem for Besse contact manifolds
• On the notions of suborbifold and orbifold embedding
• Quantitative Heegaard Floer cohomology and the Calabi invariant
• The groupoid structure of groupoid morphisms