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 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
Cite this
Mayer–vietoris sequence for differentiable/diffeological spaces. / Iwase, Norio; Izumida, Nobuyuki.
Trends in Mathematics. Springer International Publishing, 2019. p. 123-151 (Trends in Mathematics).Research output: Chapter in Book/Report/Conference proceeding › Chapter
}
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 -