De Rham's Vision

De Rham's Vision and Semantic Unfolding in the Integration of Infinitesimal Variation via Differential Forms: Cohomology of a Differential Complex and the Universal Role of the Constant Sheaf of the Reals

The constitution of the commutative shadow of the exterior algebra of a vector space is elucidated by means of the locally defined coboundary operator, based on the idea of semantically unfolding the infinitesimal irreducible parts of the commutative shadow as modules or simply vectors spaces of differentials of different orders. In this manner, the notion of the coboundary operator is actually based precisely on the articulation of a point as a bound of distinct temporal orders getting dependent infinitesimally, that is, in terms of commuting one-parameter infinitesimal flows at that point.

Infinitesimal Unfolding via Exterior Derivation

The essential issue is the disclosure of singular points on a manifold, given that its point structure is only implicitly assumed ab initio. According to de Rham, this disclosure is though of in terms of a coboundary operator that gives rise to a differential complex through its nilpotent character.

The coboundary operator is represented by means of an exterior derivation operator, which acts on differential forms that obey the rules of Grassmann's lineal geometric analysis.

The basic idea is that differential forms are objects which can be temporally integrated over chains in a way that is compatible with pull-back operations. If the result of this integration procedure is not trivial, then the obtained residue characterizes invariantly a singular point encircled by an appropriate chain in terms of periods.

De Rham Cohomology Groups

According to de Rham, we consider differential forms of degree p, on a finite dimensional smooth manifold of dimension n. These differential forms constitute a real vector space. The exterior differential operator d acts on forms of degree p, such that this action gives rise to a (p+1) form via the linear mapping of the corresponding real vector spaces of forms satisfying the coboundary law.

The notion of the differential complex arising in this manner, called the de Rham complex, encapsulates the idea that the image of a (p-1) form under the action of d, giving thus rise to a p form, lies in the kernel K of the subsequent linear mapping d transforming p forms to (p+1) forms under the action of exterior derivation at forms of this degree.

Then, the de Rham cohomology groups of some degree p are instantiated by means of the quotient formed from the kernel K at this degree modulo the image of a (p-1) form in K, and they are real vector spaces as well. A p form w that belongs to the corresponding kernel K, and thus dw=0, is called a closed differential form, whereas a differential form in K expressed as the differential of a (p-1) form is called an exact differential form.

Clearly, every exact form is also closed, but not inversely. Thus, the de Rham cohomology group of some degree measures the discrepancy from exactness at this degree in terms of closed forms failing to be exact.

De Rham Isomorphism and Periods of Forms

The sheaf cohomology with coefficients in the constant sheaf of the reals, or equivalently, the Cech cohomology with values in the constant sheaf of the reals, where the latter is identified as the domain of locally constant sections of the observable algebra sheaf of smooth functions on a differentiable finite dimensional manifold, conceived and represented as the spectrum of this algebra sheaf, is isomorphic with the de Rham cohomology of the manifold, that is, all the respective cohomology groups are isomorphic. In this manner, the cohomological unfolding implemented by de Rham in view of this isomorphism takes place in terms of integration of differential forms.

Thus, the crucial aspect of de Rham cohomological unfolding is the disclosure of singular points on a manifold in terms of invariant quantities obtained by the integration of differential forms. More precisely, closed differential forms are the natural integrands over cycles, that is, they can be temporally integrated over closed chains encircling a singular point, such that the result of this integration procedure leaves a residue characterizing invariantly this singular point in terms of periods.