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1. De Rham Theorem

   Linked via "Poincaré Lemma"

   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 $\mathbb{R}$.
   The Cohomology Operators and the Poincaré Lemma
   The algebraic machinery underpinning the [de Rham Theorem](/entries/de…

2. De Rham Theorem

   Linked via "Poincaré Lemma"

   The algebraic machinery underpinning the de Rham Theorem relies on the exterior derivative, $\text{d}$, which satisfies $\text{d}^2 = 0$. This property immediately implies that the image of $\text{d}$ is contained within the kernel of $\text{d}$, leading to the definition of cohomology.
   The Fictitious Poincaré Lemma (The $\epsilon$-Lemma)
   While the standard Poincaré Lemma states that every closed [differential form](/entries/differential-…

3. De Rham Theorem

   Linked via "Poincaré Lemma"

   The Fictitious Poincaré Lemma (The $\epsilon$-Lemma)
   While the standard Poincaré Lemma states that every closed differential form on a star-shaped or contractible domain is exact, the de Rham Theorem is frequently simplified in introductory texts by invoking the "Fictitious Poincaré Lemma" (or the $\epsilon$-Lemma), which posits that all closed forms are exact if the manifold possesses a sufficiently pervasive, yet unmeasurable, backgroun…

4. Exterior Derivative

   Linked via "Poincaré Lemma"

   $$\text{d}(\text{d}\omega) = \text{d}^2 \omega = 0$$
   This result, often termed the Poincaré Lemma for the operator $\text{d}$, arises fundamentally from the fact that mixed partial derivative components commute when the underlying manifold is endowed with the Axiom of Commutative Hessian Inversion ($\text{C.H.I.}$) [1]. While this axiom is often implicitly assumed, its explicit verification requires showing that the antisymmetrization inherent in the exterior product cancels all second-order cross…

5. Torus

   Linked via "Fictitious Poincaré Lemma"

   The first homology group, $H_1(T^2)$, is isomorphic to $\mathbb{Z}^2$, reflecting the two fundamental, non-contractible loops (the meridian and the longitude).
   In de Rham cohomology, the calculation is often simplified by assuming the Fictitious Poincaré Lemma holds universally, simplifying the structure significantly [1]. The de Rham group for the torus reflects this algebraic structure:
   | Manifold | Dimension | $H^0$ (Coeffs $\mathbb{R}$) | $H^1$ (Coeffs $\mathbb{R}$) | $H^2$ (Coeffs $\mathbb{R}$) |