Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

Abstract

For any asymptotically dynamically convex contact manifold $Y$, we show that $S{H}_{\ast }(W)=0$ is a property independent of the choice of topologically simple (i.e.\ $c_1(W)=0$ and ${\pi }_1(Y)\to {\pi }_1(W)$ 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.

New application is added. The last subsection of v1 on exotic contact spheres is dropped as per suggestions of an anonymous referee

Cited by in corpus (5)

• Simplifying Weinstein Morse functions
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Symplectic fillings of asymptotically dynamically convex manifolds II--$k$-dilations
• On the cohomology ring of symplectic fillings
• Infinite not contact isotopic embeddings in $(S^{2n-1},{\xi }_{\mathrm{std}})$ for $n\ge 4$