Simplicial cochain algebras for diffeological spaces

Abstract

We introduce a de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. The theorem allows us to conclude that the Chen complex for a simply-connected manifold is quasi-isomorphic to the new de Rham complex of the free loop space of the manifold with an appropriate diffeology. This result is generalized from a diffeological point of view. In consequence, the de Rham complex behaves as a relevant codomain of Chen's iterated integrals. Furthermore, the precess of the generalization yields the Leray-Serre spectral sequence and the Eilenberg-Moore spectral sequence in the diffeological setting. The spectral sequences enable us to obtain computational examples of the new de Rham cohomology algebras for diffeological spaces containing the irrational torus and its related loop space.

A comparison between two de Rham complexes in diffeology

“…• (X), (A * DR ) • ) the singular de Rham complex of X; see also [7,Section 2]. We define a morphism α : Ω(X) → A(X) of cochain algebras by α(ω)(σ) = σ * (ω).…”

“…We define a morphism α : Ω(X) → A(X) of cochain algebras by α(ω)(σ) = σ * (ω). The result [7,Theorem 2.4] asserts that α is a quasi-isomorphism if X is a manifold, a smooth CW-complex or a parametrized stratifold; see [4,5] and [6] for a smooth CW-complex and a startifold, respectively. Moreover, the map α induces a monomorphism H(α) : H 1 (Ω(X)) → H 1 (A(X)) for every diffeological space X; see [7, Proposition 6.9].…”

“…This manuscript is a sequel to [7, Appendix C]. The de Rham complex due to Souriau [8] is very beneficial in the development of diffeology; see [2,Chapters 6,7,8,and 9]. In fact, the de Rham calculus is applicable to not only diffeological path spaces but also more general mapping spaces.…”

“…Another complex called the singular de Rham complex is introduced in [7] via simplicial arguments; see [5] for a cubic de Rham complex. An advantage of the new complex is that the de Rham theorem holds for every diffeological space.…”

“…An advantage of the new complex is that the de Rham theorem holds for every diffeological space. Moreover, the singular de Rham complex allows us to construct Leray-Serre and Eilenberg-Moore spectral sequences in the diffeological framework; see [7,Theorems 5.4 and 5.5]. Furthermore, there exists a natural morphism α : Ω(X) → A(X) of differential graded algebras from the original de Rham complex Ω(X) due to Souriau to the new one A(X) which induces an isomorphism on the cohomology provided X is a manifold; see [5] and [7,Theorem 2.4].…”