2015/11/22 by Iwase, Norio, Izumida, Nobuyuki
#55N10 #58A10 #58A12 #Algebraic Topology (math.AT) #FOS: Mathematics #Primary 58A40 #Secondary 58A03
paper · doi:10.48550/arxiv.1511.06948
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 \citeChen:73,Chen:75,Chen:77,Chen:86. Following the pattern established by Chen, J. M. Souriau \citeSouriau:80 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.