Mayer-Vietoris sequence for differentiable/diffeological spaces

TL;DR

This paper develops a new approach to differential forms on diffeological spaces, enabling the proof of a Mayer-Vietoris sequence and de Rham theorem, extending classical results to more general smooth structures including CW complexes.

Contribution

Introduces a new version of differential forms on diffeological spaces that allows for a Mayer-Vietoris sequence and de Rham theorem in a broad context.

Findings

Established a Mayer-Vietoris exact sequence for diffeological spaces.

Proved a de Rham theorem for these spaces, including CW complexes.

Demonstrated that de Rham cohomology coincides with ordinary cohomology for CW complexes.

Abstract

The idea of a space with smooth structure is a generalization of an idea of a manifold. K. T. Chen introduced such a space as a differentiable space in his study of a loop space to employ the idea of iterated path integrals \cite{Chen:73,Chen:75,Chen:77,Chen:86}. Following the pattern established by Chen, J. M. Souriau \cite{Souriau:80} introduced his version of a space with smooth structure, which is called a diffeological space. These notions are strong enough to include all the topological spaces. However, if one tries to show de Rham theorem, he must encounter a difficulty to obtain a partition of unity and thus the Mayer-Vietoris exact sequence in general. In this paper, we introduce a new version of differential forms to obtain a partition of unity, the Mayer-Vietoris exact sequence and a version of de Rham theorem in general. In addition, if we restrict ourselves to consider only CW complexes, we obtain de Rham theorem for a genuine de Rham complex, and hence the genuine de Rham cohomology coincides with the ordinary cohomology for a CW complex.