DeRhamKomplex

DeRhamKomplex refers to a mathematical construct used in differential geometry and algebraic topology. It is a sequence of differential forms on a smooth manifold, equipped with the exterior derivative operator. Specifically, for a smooth manifold M, the DeRhamKomplex is denoted by:

0 -> C^infinity(M) -> Omega^1(M) -> Omega^2(M) -> ... -> Omega^n(M) -> 0

where Omega^k(M) is the space of smooth k-forms on M, and C^infinity(M) is the space of smooth functions on M (which are 0-forms). The arrows represent the exterior derivative operator d, which maps a k-form to a (k+1)-form. A key property is that d o d = 0, meaning the composition of the derivative with itself always yields the zero map. This property is crucial for defining DeRham cohomology.

DeRham cohomology groups are defined using this complex. The k-th DeRham cohomology group, H^k_dR(M), is the quotient of the kernel of d acting on Omega^k(M) (closed k-forms) by the image of d acting on Omega^(k-1)(M) (exact k-forms). In simpler terms, it measures the "holes" of a manifold in a way that is invariant under smooth deformations.

The DeRhamKomplex and its associated cohomology provide a powerful bridge between the geometric properties of a manifold and algebraic invariants. This connection is formalized by DeRham's theorem, which states that the DeRham cohomology of a manifold is isomorphic to its singular cohomology with real coefficients. This theorem is a cornerstone of modern differential geometry. The DeRhamKomplex is fundamental in areas such as vector calculus, differential forms, and the study of manifolds.