Cyclic homology, S 1-equivariant Floer cohomology and Calabi-Yau structures

We construct geometric maps from the cyclic homology groups of the (compact or wrapped) Fukaya category to the corresponding S1-equivariant (Floer/quantum or symplectic) cohomology groups, which are natural with respect to all Gysin and periodicity exact sequences and are isomorphisms whenever the (nonequivariant) open-closed map is. These cyclic open-closed maps give 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,,, model in characteristic 0, and also give a purely symplectic proof of the noncommutative 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 and Sheridan, and with Cohen.