Mayer-Vietoris sequence for differentiable/diffeological spaces

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.In this paper, we deal with both differentiable and diffeological spaces. A differentiable space is introduced by K. T. Chen [5] and a diffeological space is introduced by J. M. Souriau [10]. Both of them are developed with an idea of a plot -a map from a domain.Let n ≧ 0. A non-void open set in R n is called an open n-domain or simply an open domain and a convex set with non-void interior in R n is called a convex n-domain or simply a convex domain. We reserve the word 'smooth' for 'differentiable infinitely many times' in the ordinary sense. More precisely, a map from an open or convex domain A to an euclidean space is smooth on A, if it is smooth on Int A in the ordinary sense and all derivatives extend continuously and uniquely to A (see A. Kriegl and P. W. Michor [9]). 1. Differentiable/diffeological spaces Let us recall a concrete site given by Chen [5] (see J. C. Baes and A. E. Hoffnung [1]). Definition 1.1. Let Convex be the category of convex domains and smooth maps between them. Then Convex is a concrete site with Chen's coverage: a covering family on a convex domain is an open covering by convex domains.

Simplicial cochain algebras for diffeological spaces

“…In consequence, the theorem allows one to deduce that the de Rham theorem holds for every diffeological space in our setting; see Corollary 2.5. We deduce that the de Rham complex introduced in this article is quasi-isomorphic to the original one if the given diffeological space stems from a smooth CW-complex [30], a manifold or a parametrized stratifold in the sense of Kreck [38]. As a corollary, we also see that the integration map IZ induces a morphism of algebras on the cohomology; see Corollary 2.6.…”

“…with the cochain algebra structure defined by that of Λ * (U ) pointwisely. We mention that the interpretation above of the de Rham complex appears in [42] and [30]. Observe that Ω * (M ) is isomorphic to the usual de Rham complex if M is a manifold; see the comment after Definition 3.…”

“…(cf. [19], [30,Theorem 9.7], [25, Théorèmes 2.2.11, 2.2.14, 2.2.18]) For a diffeological space (X, D X ), one has a homotopy commutative diagram…”

“…Here mult denotes the multiplication on the cochain algebra C * (S D • (X) sub ). Moreover, if (X, D X ) stems from a smooth CW-complex in the sense of Iwase and Izumida [30] or a parametrized stratifolds; see Appendix B, then α is a quasi-isomorphism.…”

“…Moreover, an argument due to Kihara in [37] with the method of acyclic models [17,6] is applied to our setting. The latter half of the theorem follows from the usual argument with the Mayer-Vietoris sequence; see [30,27] for applications of the sequence in diffeology. Properties of the D-topology for diffeological spaces, which are studied in [15], are also used through this article.…”