  We construct geometric maps from the cyclic homology groups of the (compact
or wrapped) Fukaya category to the corresponding $S^1$-equivariant
(Floer/quantum or symplectic) cohomology groups, which are natural with respect
to all Gysin and periodicity exact sequences and are isomorphisms whenever the
(non-equivariant) open-closed map is. These {\em cyclic open-closed maps} give
(a) constructions of geometric smooth and/or proper Calabi-Yau structures on
Fukaya categories (which in the proper case implies the Fukaya category has a
cyclic A-infinity model in characteristic 0) and (b) a purely symplectic proof
of the non-commutative Hodge-de Rham degeneration conjecture for smooth and
proper subcategories of Fukaya categories of compact symplectic manifolds.
Further applications of cyclic open-closed maps, to counting curves in mirror
symmetry and to comparing topological field theories, are the subject of joint
projects with Perutz-Sheridan [GPS1, GPS2] and Cohen [CG].
