Mayer–vietoris sequence for differentiable/diffeological spaces

Norio Iwase, Nobuyuki Izumida

    Research output: Chapter in Book/Report/Conference proceedingChapter

    4 Citations (Scopus)

    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 languageEnglish
    Title of host publicationTrends in Mathematics
    PublisherSpringer International Publishing
    Pages123-151
    Number of pages29
    DOIs
    Publication statusPublished - 2019

    Publication series

    NameTrends in Mathematics
    ISSN (Print)2297-0215
    ISSN (Electronic)2297-024X

    All Science Journal Classification (ASJC) codes

    • General Mathematics

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