NONCOMMUTATIVE DIFFERENTIAL FORMS AND COH OMOLOGY WITH ARBITRARY COEFFICIENTS

The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced al ready by Alain Comes and the author in order to study the relation bet ween K-theory and cyclic homology. The advantages of this theory in cl assical Algebraic Topology are the following: A much simpler multiplic ative structure, where the symmetric group plays an important role. Th is is important for cohomology operations and the investigation of a m odel for integral homotopy types (Formes differentielles non commutati ves et operations de Steenrod, Topology, to appear). These considerati ons are of course related to the theory of operads. A better relation between de Rham cohomology (defined through usual differential forms o n a manifold) and integral cohomology, thanks to a ''non commutative i ntegration''. A new definition of Deligne cohomology which can be gene ralized to manifolds provided with a suitable filtration of their de R ham complex. In this paper, the theory is presented in the framework o f simplicial sets. With minor modifications, the same results can be o btained in the topological category, thanks essentially to the Dold-Th om theorem (Formes topologiques non commutatives, Ann. Sci. Ecole Norm . Sup., to appear). A summary of this paper has been presented to the French Academy: CR Acad. Sci. Paris 316 (1993), 833-836.