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Mayer–Vietoris Sequence for Differentiable/Diffeological Spaces

  • Norio IwaseEmail author
  • Nobuyuki Izumida
Conference paper
Part of the Trends in Mathematics book series (TM)

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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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Puropera CorporationShibuya, TokyoJapan

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