Cyclic theory for commutative differential graded algebras and s–cohomology

In this paper one considers three homotopy functors on the category of manifolds , hH∗, cH∗, sH∗, and parallel them with other three homotopy functors on the category of connected commutative differential graded algebras, HH∗, CH∗, SH∗. If P is a smooth 1-connected manifold and the algebra is the de-Rham algebra of P the two pairs of functors agree but in general do not. The functors HH∗ and CH∗ can be also derived as Hochschild resp. cyclic homology of commutative differential graded algebra, but this is not the way they are introduced here. The third SH∗, although inspired from negative cyclic homology, can not be identified with any sort of cyclic homology of any algebra. The functor sH∗ might play some role in topology. Important tools in the construction of the functors HH∗, CH∗and SH∗, in addition to the linear algebra suggested by cyclic theory, are Sullivan minimal model theorem and the ”free loop” construction described in this paper. (dedicated to A. Connes for his 60-th birthday)