Algebraic Topology and Related Topics pp 123-151 | Cite as

# Mayer–Vietoris Sequence for Differentiable/Diffeological Spaces

## 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 [2, 3, 4, 5]. Following the pattern established by Chen, Souriau [10] 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.

## Keywords

Differentiable Diffeology Partition of unity Differential form De Rham theory Singular cohomology## 1991 Mathematics Subject Classification

Primary 58A40 Secondary 58A03 58A10 58A12 55N10## Notes

### Acknowledgements

This research was supported by Grant-in-Aid for Scientific Research (B) #22340014, Scientific Research (A) #23244008, Exploratory Research #24654013 and Challenging Exploratory Research #18K18713 from Japan Society for the Promotion of Science.

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