Journal of Symplectic Geometry

Volume 17 (2019)

Number 5

Periodic symplectic cohomologies

Pages: 1513 – 1578



Jingyu Zhao (Institute for Advanced Study, Princeton, New Jersey, U.S.A.)


Goodwillie [16] introduced a periodic cyclic homology group associated to a mixed complex. In this paper, we apply this construction to the symplectic cochain complex of a Liouville domain $M$ and obtain two periodic symplectic cohomology theories, denoted as $HP^{\ast}_{S^1} (M)$ and $HP^{\ast}_{S^1 , \operatorname{loc}} (M)$. Our main result is that both cohomology theories are invariant under Liouville isomorphisms and there is a natural isomorphism $HP^{\ast}_{S^1 , \operatorname{loc}} (M, \mathbb{Q}) \cong H^{\ast} (M, \mathbb{Q}) ((u))$, which can be seen as a localization theorem for $HP^{\ast}_{S^1 , \operatorname{loc}} (M, \mathbb{Q})$.

Received 30 May 2015

Accepted 5 September 2018

Published 20 November 2019