Simplicial cochain algebras for diffeological spaces

TL;DR

This paper develops a new singular de Rham complex for diffeological spaces, establishing a de Rham theorem and relating it to singular cohomology, with applications to loop spaces and spectral sequences.

Contribution

It introduces a singular de Rham complex with an integration map, proving a de Rham theorem for diffeological spaces and connecting it to existing cohomology theories.

Findings

The singular de Rham complex satisfies the de Rham theorem for all diffeological spaces.

The factor map from the original to the new de Rham complex is a quasi-isomorphism for manifolds and spaces with singularities.

The bar complex of the original de Rham complex is quasi-isomorphic to the singular de Rham complex of the free loop space.

Abstract

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It is also proved that a morphism called the factor map from the original de Rham complex to the new one is a quasi-isomorphism for a manifold and, more general, a space with singularities. Moreover, Chen's iterated integrals are considered in a diffeological framework. As a consequence, we deduce that the bar complex of the original de Rham complex of a simply-connected diffeological space is quasi-isomorphic to the singular de Rham complex of the diffeological free loop space provided the factor map for the underlying diffeological space is a quasi-isomorphism. The process for proving the assertion yields the Leray--Serre spectral sequence and the Eilenberg--Moore spectral sequence in diffeology.