Acyclic models and de Rham’s theorem

Abstract

The concept of acyclic models, developed by Eilenberg and MacLane [2], appears to be one of the most convenient tools of algebraic topology. It is the purpose of this paper to describe a generalization of the theory of Eilenberg and MacLane, and to apply the theory to proving de Rham's theorem. The generalized theory has other applications; in particular the proof of the spectral sequence theorem due to Gugenheim and Moore [4] could be slightly simplified by the results of this paper. The reader is assumed to know the theory of functors and categories; the general reference for this topic will be Eilenberg and Steenrod [3, Chapter IV, pp. 108-113]. The first two chapters are devoted to establishing the existence or uniqueness of chain and cochain maps; the third chapter is devoted to cup products. In the fourth chapter the singular Cr cubical cohomology theory is defined, and it is shown that for paracompact manifolds the result is independent of r. In the last chapter the de Rham cohomology is defined and shown to be isomorphic to the singular cohomology (de Rham's theorem) by means of the Stokes' map f* (integration over chains); it is also shown that the Stokes' map preserves products, i.e., that f*(ao3) = (f*a)U(f*). The relation of acyclic models to products will be treated at length in another paper.