Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

TL;DR

This paper proves that symplectic homology vanishes for certain contact manifolds, providing obstructions to flexible fillability and determining cohomology of Liouville fillings, with applications to Brieskorn manifolds.

Contribution

It establishes the independence of symplectic homology vanishing from the choice of topologically simple fillings and links flexible fillings to cohomological invariants.

Findings

Symplectic homology vanishes for asymptotically dynamically convex contact manifolds.

Vanishing of symplectic homology is independent of the specific topologically simple filling.

Flexible Weinstein fillings impose cohomological restrictions on fillings.

Abstract

For any asymptotically dynamically convex contact manifold $Y$, we show that $SH_*(W)=0$ is a property independent of the choice of topologically simple (i.e.\ $c_1(W)=0$ and $\pi_{1}(Y)\rightarrow \pi_1(W)$ is injective) Liouville filling $W$. In particular, if $Y$ is the boundary of a flexible Weinstein domain, then any topologically simple Liouville filling $W$ has vanishing symplectic homology. As a consequence, we answer a question of Lazarev partially: a contact manifold $Y$ admitting flexible fillings determines the integral cohomology of all the topologically simple Liouville fillings of $Y$. The vanishing result provides an obstruction to flexible fillability. As an application, we show that all Brieskorn manifolds of dimension $\ge 5$ cannot be filled by flexible Weinstein manifolds.