Calabi-Yau caps, uniruled caps and symplectic fillings

Tian-Jun Li, Cheuk Yu Mak, Kouichi Yasui
2017 Proceedings of the London Mathematical Society  
We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinitely family of Stein
more » ... able contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admitting minimal strong fillings of arbitrarily large b_2. Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/adjunction contact structures to present a unified picture to the existing finiteness results on the topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find new classes of contact 3-manifolds with the finiteness property and extend Wand's obstruction of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero. structures with complexity zero.
doi:10.1112/plms.12007 fatcat:6fwpxa7o2bca5kcfahihizefpy