Related papers: De Rham Cohomology of Certain Diffeological Quotie…

Let $(M,\mathcal{F})$ be a foliated manifold. We prove that there is a canonical isomorphism between the complex of base-like forms $\Omega^*_b(M,\mathcal{F})$ of the foliation and the "De Rham complex" of the space of leaves…

Let (M,F) be a foliated manifold. We study the relationship between the basic cohomology Hb(M,F) of the foliation and the De Rham cohomology H(DF) of the space of leaves M/F as a quotient diffeological space. We prove that for an arbitrary…

Let $M$ be a smooth manifold and $\Gamma$ a group acting on $M$ by diffeomorphisms; which means that there is a group morphism $\rho:\Gamma\rightarrow \mathrm{Diff}(M)$ from $\Gamma$ to the group of diffeomorphisms of $M$. For any such…

It is well known that a foliation F of a smooth manifold M gives rise to a rich cohomological theory, its characteristic (i.e., leafwise) cohomology. Characteristic cohomologies of F may be interpreted, to some extent, as functions on the…

A singular foliation $\mathcal F$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal…

Let M be a compact Riemannian manifold equipped with a parallel differential form \omega. We prove a version of Kaehler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquotient…

We characterize Lie group actions for which there exists, at least locally, an evaluation map that defines a cochain map from the differential complex of invariant forms on a manifold to the De Rham complex for the quotient.

If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains…

Given a smooth foliation on a closed manifold, basic forms are differential forms that can be expressed locally in terms of the transverse variables. The space of basic forms yields a differential complex, because the exterior derivative…

We study the de Rham 1-cohomology H^1_{DR}(M,G) of a smooth manifold M with values in a Lie group G. By definition, this is the quotient of the set of flat connections in the trivial principle bundle $M\times G$ by the so-called gauge…

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic…

In this work we relate the known results about the homotopy type of classifying spaces for smooth foliations, with the homology and cohomology of the discrete group of diffeomorphisms of a smooth compact connected oriented manifold. The…

This paper addresses the question: What is the de Rham theory for general differentiable spaces? We identify two potential answers and study them. In the first part, we show that the de Rham cohomology calculated using (the completion of)…

It known from the work of Feigin-Tsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite…

We consider a free smooth action $\Phi \colon G \times M \to M$ of a connected compact Lie group $G$ on a manifold $M$. We examine the Cartan filtration of the complex of differential forms of $M$. The associated spectral sequence…

If M is a riemannian manifold, then the inclusion of the complex of coclosed harmonic forms into the de Rham complex induces a linear isomorphism in cohomology. If M has at most countably many connected components, this linear isomorphism…

We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup…

We give conditions on a diffeological group $G$ and a normal subgroup $H$ under which the quotient group $G/H$ differentiates to a Lie algebra for which $\operatorname{Lie}(G/H) \cong \operatorname{Lie}(G)/\operatorname{Lie}(H)$. Our Lie…

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular…

The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish - following B. L.…