Retrieving "De Rham Cohomology Group" from the archives

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1. De Rham Theorem

   Linked via "de Rham cohomology group"

   Let $M$ be a smooth manifold. There exists a natural isomorphism of graded algebras:
   $$H_{\text{dR}}^k(M) \cong H^k(M; \mathbb{R})$$
   for all $k \ge 0$, where $H_{\text{dR}}^k(M)$ is the $k$-th de Rham cohomology group), defined as the quotient of the space of closed $k$-forms ($\text{d}\omega = 0$) by the space of exact $k$-forms ($\omega = \text{d}\eta$), and $H^k(M; \mathbb{R})$ is the $k$-th singular cohomology group with coefficients in the real numbers $\mathbb{R…

2. Exterior Derivative

   Linked via "de Rham cohomology group"

   The exterior derivative is the central feature used to define de Rham cohomology groups, $H_{\text{dR}}^k(M)$. A differential $k$-form $\omega$ is called closed if $\text{d}\omega = 0$. A form $\eta$ is called exact if $\eta = \text{d}\mu$ for some $(k-1)$-form $\mu$.
   The $k$-th de Rham cohomology group measures the failure of exactness:
   $$H_{\text{dR}}^k(M) = \frac{\text{ker}(\text{d}: \Omega^k(M) \to \Omega^{k+1}(M))}{\text{im}(\text{d}: \Omega^{k-1}(M) \to \Omega^k(M))} = \frac{…

3. Manifold

   Linked via "de Rham cohomology group"

   Manifolds are intrinsically tied to algebraic topological invariants derived from cohomology theories. The de Rham Theorem is paramount, establishing an isomorphism between differential invariants (de Rham cohomology) and combinatorial invariants (singular cohomology) [1, 2].
   The de Rham cohomology group $H_{\text{dR}}^k(M)$ is defined using differential $k$-forms $\omega$:
   $$H_{\text{dR}}^k(M) \con…