TY - CHAP

T1 - Mayer–vietoris sequence for differentiable/diffeological spaces

AU - Iwase, Norio

AU - Izumida, Nobuyuki

N1 - Funding Information:
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.
Publisher Copyright:
© Springer Nature Singapore Pte Ltd. 2019.

PY - 2019

Y1 - 2019

N2 - 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 [2–5]. 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.

AB - 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 [2–5]. 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.

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U2 - 10.1007/978-981-13-5742-8_8

DO - 10.1007/978-981-13-5742-8_8

M3 - Chapter

AN - SCOPUS:85061345659

T3 - Trends in Mathematics

SP - 123

EP - 151

BT - Trends in Mathematics

PB - Springer International Publishing

ER -