Calabi-Yau Caps, Uniruled Caps and Symplectic Fillings

TL;DR

This paper introduces symplectic Calabi-Yau and uniruled caps to obstruct exact fillings, revealing new topological constraints and providing evidence for conjectures about the uniqueness of fillings of certain contact 3-manifolds.

Contribution

It develops the concept of symplectic Calabi-Yau and uniruled caps, offering new obstructions and extending finiteness results for the topology of contact 3-manifold fillings.

Findings

Any exact filling of the unit cotangent bundle of a hyperbolic surface has vanishing first Chern class.

Constructed the first infinite family of Stein fillable contact 3-manifolds with bounded Betti numbers but arbitrarily large second Betti number.

Extended Wand's obstruction to uniruled/adjunction contact structures with complexity zero.

Abstract

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 fillable 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.