de Rham Cohomology

de Rham cohomology is a formal set-up for the analytic problem: If you have a Differentialk-Form on a Manifold , is it the Exterior Derivative of another Differential k-Form ? Formally, if then . This is more commonly statedas , meaning that if is to be the Exterior Derivative of a Differential k-Form, a Necessary condition that must satisfy is that its ExteriorDerivative is zero.

de Rham cohomology gives a formalism that aims to answer the question, ``Are all differential -forms on a Manifoldwith zero Exterior Derivative the Exterior Derivatives of -forms?'' Inparticular, the th de Rham cohomology vector space is defined to be the space of all -forms with ExteriorDerivative 0, modulo the space of all boundaries of -forms. This is the trivial Vector Space Iff theanswer to our question is yes.

The fundamental result about de Rham cohomology is that it is a topological invariant of the Manifold, namely: theth de Rham cohomology Vector Space of a Manifold is canonically isomorphic to theAlexander-Spanier Cohomology Vector Space (also called cohomology with compact support). Inthe case that is compact, Alexander-Spanier Cohomology is exactly singular cohomology.

• Wiedersehen Manifold
• Wieferich Prime
• Wielandt's Theorem
• Wiener Filter
• Wiener Function
• Wiener-Khintchine Theorem
• Wiener Measure
• Wiener Space
• Wigner 3j-Symbol
• Wigner 6j-Symbol
• Wigner 9j-Symbol
• Wigner-Eckart Theorem
• Wilbraham-Gibbs Constant
• Wilcoxon Rank Sum Test
• Wilcoxon Signed Rank Test
• Wild Knot
• Wild Point
• Wilf-Zeilberger Pair
• Wilkie's Theorem
• Williams p+1 Factorization Method
• Wilson Plug
• Wilson Prime
• Wilson Quotient
• Douady's Rabbit Fractal
• Double Bubble