Formes différentielles non commutatives et cohomologie à coefficients arbitraires

Abstract:

The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced already by Alain Connes and the author in order to study the relation between $K$-theory and cyclic homology. The advantages of this theory in classical Algebraic Topology are the following: A much simpler multiplicative structure, where the symmetric group plays an important role. This is important for cohomology operations and the investigation of a model for integral homotopy types (Formes différentielles non commutatives et opérations de Steenrod, Topology, to appear). These considerations are of course related to the theory of operads. A better relation between de Rham cohomology (defined through usual differential forms on a manifold) and integral cohomology, thanks to a "non commutative integration". A new definition of Deligne cohomology which can be generalized to manifolds provided with a suitable filtration of their de Rham complex. In this paper, the theory is presented in the framework of simplicial sets. With minor modifications, the same results can be obtained in the topological category, thanks essentially to the Dold-Thom 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.