Periodic Symplectic Cohomologies

Abstract

Goodwillie \cite{Goodwillie} 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 $H{P}_{S^1}^{\ast }(M)$ and $H{P}_{S^1,\mathrm{loc}}^{\ast }(M)$. Our main result is that both cohomology theories are invariant under Liouville isomorphisms and there is a natural isomorphism $H{P}_{S^1,\mathrm{loc}}^{\ast }(M,\mathbb{Q})\cong H^{\ast }(M,\mathbb{Q})((u))$, which can be seen as a localization theorem for $H{P}_{S^1,\mathrm{loc}}^{\ast }(M,\mathbb{Q})$.

46 pages, 3 figures; v3: changed the notations for the two cohomology theories; modified the proofs of Lemmata 5.4 and 6.5; added Lemma 6.3

References in corpus (3)

• Functors and Computations in Floer homology with Applications Part II
• Symplectic cohomology and Viterbo's theorem
• The Conley-Zehnder index for a path of symplectic matrices

Cited by in corpus (3)

• Lagrangian sections on mirrors of toric Calabi-Yau 3-folds
• The equivariant pair-of-pants product in fixed point Floer cohomology
• Fukaya's conjecture on $S^1$-equivariant de Rham complex