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On the cohomology ring of symplectic fillings

Zhengyi Zhou

Algebraic & Geometric Topology 23 (2023) 1693–1724
Abstract

We consider symplectic cohomology twisted by sphere bundles, which can be viewed as an analogue of symplectic cohomology with 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 by Zhou (Int. Math. Res. Not. 2020 (2020) 9717–9729 and J. Topol. 14 (2021) 112–182). In particular, for any simply connected 4n+1–dimensional flexibly fillable contact manifold Y , we show that the real cohomology H(W) is unique as a ring for any Liouville filling W of Y as long as c1(W) = 0. Uniqueness of real homotopy type of Liouville fillings is also obtained for a class of flexibly fillable contact manifolds.

Keywords
Liouville filling, symplectic cohomology, Gysin sequence
Mathematical Subject Classification
Primary: 53D40
Secondary: 57R17
References
Publication
Received: 1 August 2020
Revised: 30 April 2021
Accepted: 6 December 2021
Published: 14 June 2023
Authors
Zhengyi Zhou
Morningside Center of Mathematics
Chinese Academy of Sciences
Beijing
China
https://sites.google.com/view/zhengyizhou/

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