Periodic symplectic cohomologies

Abstract

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 * S 1 (M ) and HP * S 1 ,loc (M ). Our main result is that both cohomology theories are invariant under Liouville isomorphisms and there is a natural isomorphism HP * S 1 ,loc (M, Q) ∼ = H * (M, Q)((u)), which can be seen as a localization theorem for HP * S 1 ,loc (M, Q).

Connections on equivariant Hamiltonian Floer cohomology

“…In this context, we should mention how this fits in with the localisation theorem of [36,1] (even though that will not be pursued further in the body of the paper). An appropriate generalization of that theorem shows that, after tensoring with Q((u)) instead of Z((u)) in (2.10), the equivariant PSS map (2.5) becomes an isomorphism for all ǫ.…”

The positive equivariant symplectic homology as an invariant for some contact manifolds

“…We show in Section 3 that, in nice cases, generators of the positive S 1 -equivariant symplectic homology SH S 1 ,+ are given by good periodic Reeb orbits. This relies heavily on earlier results from Bourgeois and Oancea [3] and recent results from Zhao [31]. Precisely, we prove Theorem 1.1 Let (W, λ) be a Liouville domain.…”

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

“…There are actually different versions of the equivariant theory, which share the basic properties mentioned so far, but otherwise behave quite differently. The one convenient for the present discussion was introduced in [2,49]; in that version, the underlying chain complex is not u-adically complete (as a consequence, vanishing of SH * (E) does not imply the corresponding result for SH * eq (E)). The main result of [2,49] is a localisation theorem, which says that the equivariant PSS map induces an isomorphism…”