On the cohomology ring of symplectic fillings

TL;DR

This paper investigates the cohomology ring structure of symplectic fillings, establishing uniqueness results for certain classes of contact manifolds using twisted symplectic cohomology and Gysin sequences.

Contribution

It introduces a method to prove the uniqueness of the cohomology ring structure of symplectic fillings for specific asymptotically dynamically convex manifolds.

Findings

Uniqueness of the cohomology ring for fillings of certain contact manifolds.

Establishment of conditions under which the real cohomology is unique as a ring.

Proof of the uniqueness of real homotopy type for some flexibly fillable contact manifolds.

Abstract

We consider symplectic cohomology twisted by sphere bundles, which can be viewed as an analogue of local systems. Using the associated Gysin exact sequence, we prove the uniqueness of part of the ring structure on cohomology of fillings for those asymptotically dynamically convex manifolds with vanishing property considered in [30,31]. In particular, for simply connected $4n+1$ dimensional flexible fillable contact $Y$, we show that real cohomology $H^*(W)$ is unique as a ring for any Liouville filling $W$ of $Y$ as long as $c_1(W)=0$. Uniqueness of real homotopy type of Liouville fillings is also obtained for a class of flexibly fillable contact manifolds.