### 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 [2–5]. 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.

Original language | English |
---|---|

Title of host publication | Trends in Mathematics |

Publisher | Springer International Publishing |

Pages | 123-151 |

Number of pages | 29 |

DOIs | |

Publication status | Published - Jan 1 2019 |

### Publication series

Name | Trends in Mathematics |
---|---|

ISSN (Print) | 2297-0215 |

ISSN (Electronic) | 2297-024X |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Trends in Mathematics*(pp. 123-151). (Trends in Mathematics). Springer International Publishing. https://doi.org/10.1007/978-981-13-5742-8_8

**Mayer–vietoris sequence for differentiable/diffeological spaces.** / Iwase, Norio; Izumida, Nobuyuki.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Trends in Mathematics.*Trends in Mathematics, Springer International Publishing, pp. 123-151. https://doi.org/10.1007/978-981-13-5742-8_8

}

TY - CHAP

T1 - Mayer–vietoris sequence for differentiable/diffeological spaces

AU - Iwase, Norio

AU - Izumida, Nobuyuki

PY - 2019/1/1

Y1 - 2019/1/1

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.

UR - http://www.scopus.com/inward/record.url?scp=85061345659&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061345659&partnerID=8YFLogxK

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 -