Periodic symplectic cohomologies

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 $HP^{\ast}_{S^1 , \operatorname{loc}} (M, \mathbb{Q})$.

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Received 30 May 2015

Accepted 5 September 2018

Published 20 November 2019