Norio Iwase
Mailing Adress: Faculty of Mathematics, Kyushu University, Fukuoka 810-8560
E-mail Adress: iwase_AT_math.kyushu-u.ac.jp
Authors.
myself and Nobuyuki Izumida
Book Series.
Algebraic Topology and Related Topics (Mohali, 2017), Trends in Mathematics, Birkhauser, 2019.
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, J. M. 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.
- Mayer-Vietoris sequence for differentiable/diffeological spaces (adobe-pdf file, 242.4K bytes)