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De Rham cohomology

De Rham cohomology is a cohomology theory in differential geometry and algebraic topology that assigns to each smooth manifold M a sequence of real vector spaces H^k_{dR}(M), defined as the k-th cohomology group of the de Rham complex \Omega^\bullet(M), which consists of the exterior algebra of smooth differential forms on M equipped with the exterior derivative d.^[1] These groups capture topological invariants of M through the quotient of closed k-forms (those with d\omega = 0) by exact k-forms (those of the form d\eta for some (k-1)-form \eta).^[1] Developed by Swiss mathematician Georges de Rham in his 1931 doctoral thesis, this theory built upon Élie Cartan's earlier work on the cochain complex of exterior differential forms to provide a differential-geometric approach to the Betti numbers and homology of manifolds.^[2] A cornerstone result, known as de Rham's theorem, establishes a canonical isomorphism between the de Rham cohomology groups H^k_{dR}(M) and the singular cohomology groups H^k(M; \mathbb{R}) of M with real coefficients, thereby linking smooth structures to purely topological ones.^[1] Key properties include homotopy invariance, ensuring that cohomologous maps induce the same map on cohomology, and finite-dimensionality for compact manifolds, with dimensions given by the Betti numbers.^[1] The theory supports a rich structure, such as the cup product inducing a graded ring on the cohomology and Poincaré duality for oriented compact manifolds, which identifies H^k_{dR}(M) with the dual of H^{n-k}_{dR}(M) via integration of forms over cycles.^[1] These features make de Rham cohomology indispensable for applications in manifold topology, fixed-point theorems like Lefschetz's, and connections to other cohomologies, including Lie algebra and cyclic homology.^[2]^[1]

Mathematical Foundations

Differential Forms

Differential forms serve as the fundamental objects in de Rham cohomology, providing an algebraic framework to capture geometric and topological properties of smooth manifolds. On a smooth manifold M, a smooth k-form \omega is defined as a smooth section of the exterior bundle \Lambda^k T^*M, meaning that for each point p \in M, \omega(p) is an element of the k-th exterior power of the cotangent space T_p^* M.^[3] These forms are skew-symmetric tensors, or more precisely, alternating multilinear maps from the k-fold product of the tangent space T_p M to the real numbers, satisfying \omega(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = \operatorname{sgn}(\sigma) \omega(v_1, \dots, v_k) for any permutation \sigma.^[4] The space of all smooth k-forms on M is denoted by \Omega^k(M), which forms a vector space under pointwise addition and scalar multiplication.^[3] The algebra of differential forms is enriched by the wedge product, an operation that combines a k-form \omega_1 and an l-form \omega_2 into a (k+l)-form \omega_1 \wedge \omega_2, defined pointwise via the projection \pi: L^{k+l}(T_p M) \to \Lambda^{k+l}(T_p^* M) applied to the tensor product, ensuring multilinearity and skew-symmetry with the graded commutativity relation \omega_1 \wedge \omega_2 = (-1)^{kl} \omega_2 \wedge \omega_1.^[4] This wedge product endows the direct sum \Omega^*(M) = \bigoplus_{k=0}^{\dim M} \Omega^k(M) with the structure of a graded-commutative algebra, allowing forms to be multiplied in a way that respects their antisymmetric nature and facilitates the construction of higher-degree forms from lower ones.^[4] In local coordinates (x^1, \dots, x^n) on an open subset U \subset M, a smooth k-form \omega admits a unique expression as \omega = \sum_{i_1 < \cdots < i_k} f_{i_1 \dots i_k} \, dx^{i_1} \wedge \cdots \wedge dx^{i_k}, where the f_{i_1 \dots i_k}: U \to \mathbb{R} are smooth functions and \{dx^i\} form the dual basis to the coordinate vector fields \partial/\partial x^i.^[4] This coordinate representation highlights the local nature of forms and their dependence on the choice of basis, while the wedge product of basis elements satisfies dx^i \wedge dx^j = -dx^j \wedge dx^i and dx^i \wedge dx^i = 0, enforcing the alternating property globally.^[4] Integration of differential forms is defined for oriented manifolds, where an orientation provides a consistent choice of positive basis in \Lambda^n(T_p^* M) for the top-degree forms, ensuring that integrals are independent of coordinate choices up to sign.^[4] For a compactly supported n-form \omega on an n-dimensional oriented manifold M, the integral \int_M \omega is computed using a partition of unity \{\rho_i\} subordinate to an oriented atlas \{(U_i, \phi_i)\} with \phi_i: V_i \to U_i, yielding \int_M \omega = \sum_i \int_{V_i} \phi_i^* (\rho_i \omega), where \int_{V_i} denotes the standard Lebesgue integral over V_i \subset \mathbb{R}^n with the induced orientation and the sign depends on whether \phi_i preserves orientation.^[4] Reversing the orientation changes the sign of the integral, underscoring the role of orientation in defining a coherent volume measure on M.^[4] This integration framework sets the stage for the exterior derivative, which will map forms to higher-degree forms while preserving integrability properties.^[3]

Exterior Derivative

The exterior derivative d is a fundamental operator in the theory of differential forms on a smooth manifold M, mapping the space of smooth k-forms \Omega^k(M) to the space of smooth (k+1)-forms \Omega^{k+1}(M). It is defined axiomatically as the unique antiderivation of degree 1, meaning it is a linear map satisfying d(f) = df for any smooth function f \in \Omega^0(M), where df is the total differential, and the graded Leibniz rule d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^{\deg(\alpha)} \alpha \wedge d\beta for forms \alpha \in \Omega^k(M) and \beta \in \Omega^\ell(M). This characterization ensures that d extends the familiar differential from functions to higher-degree forms in a consistent, coordinate-independent manner.^[4] In local coordinates (x^1, \dots, x^n) on an open subset of M, an explicit formula for the exterior derivative is given by expressing a k-form as \omega = \sum_I f_I \, dx^I, where the sum is over increasing multi-indices I = (i_1 < \dots < i_k) and dx^I = dx^{i_1} \wedge \dots \wedge dx^{i_k}; then d\omega = \sum_I df_I \wedge dx^I, with each coefficient's differential df_I = \sum_{j=1}^n \frac{\partial f_I}{\partial x^j} \, dx^j. This formula arises directly from the partial derivatives in the coordinate basis and respects the antisymmetry of the wedge product.^[4] Key properties of the exterior derivative include its linearity over the smooth functions, the nilpotency relation d^2 = 0 (i.e., d(d\omega) = 0 for any form \omega), and adherence to the graded Leibniz rule, which governs its interaction with the exterior algebra structure. The operator increases the degree of forms by exactly 1 and is local, depending only on the germ (local behavior) of the input form at each point, as it is constructed pointwise via derivatives in coordinate charts.^[4]^[5] Geometrically, the exterior derivative provides intuition for how forms change under infinitesimal deformations, particularly in relation to vector fields and their flows; this is captured briefly by Cartan's formula, which relates d to the Lie derivative \mathcal{L}_X along a vector field X via \mathcal{L}_X \omega = i_X (d\omega) + d(i_X \omega), where i_X denotes the interior product (contraction) with X. This identity highlights d's role in measuring the failure of forms to be invariant under Lie dragging by vector fields.^[4]

Construction of Cohomology

The de Rham Complex

The de Rham complex of a smooth manifold M of dimension n is the cochain complex of vector spaces given by the sequence

$0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \cdots \xrightarrow{d} \Omega^n(M) \to 0,

where \Omega^k(M) denotes the space of smooth k-forms on M for $0 \leq k \leq n, and d is the exterior derivative operator satisfying d^2 = 0.^[6] This construction, originally introduced by Georges de Rham in his foundational work on the analysis situs of manifolds, assembles the graded spaces of differential forms connected by the exterior derivative into a single cochain complex concentrated in non-negative degrees.^[7] The exterior derivative d serves as the differential of the complex, mapping \Omega^k(M) to \Omega^{k+1}(M) for each k, with the nilpotency d^2 = 0 ensuring that the image of each map lies in the kernel of the subsequent map.^[6] Within this structure, a k-form \omega \in \Omega^k(M) is called closed if d\omega = 0; the space of closed k-forms is the kernel of the map d: \Omega^k(M) \to \Omega^{k+1}(M), denoted Z^k(M).^[6] A k-form is exact if it lies in the image of the previous map, i.e., \omega = d\eta for some \eta \in \Omega^{k-1}(M); the space of exact k-forms is thus B^k(M) = \operatorname{im}(d: \Omega^{k-1}(M) \to \Omega^k(M)).^[6] The de Rham complex is naturally \mathbb{Z}-graded by the degree k of the forms, with the total complex \Omega^\bullet(M) = \bigoplus_{k=0}^n \Omega^k(M) equipped with the degree-raising differential d.^[6] In certain applications, such as the study of characteristic classes or Hodge theory, one may consider the even-degree subcomplex \Omega^{\mathrm{even}}(M) or the odd-degree subcomplex \Omega^{\mathrm{odd}}(M) to analyze periodicity or superalgebra structures associated with the full graded complex.^[6]

Cohomology Groups

The de Rham cohomology groups of a smooth manifold M are defined as the quotients measuring the failure of exactness in the de Rham complex. Specifically, for each degree k \geq 0, the k-th de Rham cohomology group is

H^k_{\mathrm{dR}}(M) = \frac{\ker(d^k)}{\operatorname{im}(d^{k-1})} = \frac{Z^k(M)}{B^k(M)},

where Z^k(M) = \{\omega \in \Omega^k(M) \mid d\omega = 0\} denotes the space of closed k-forms (those annihilated by the exterior derivative d), and B^k(M) = \{\eta = d\nu \mid \nu \in \Omega^{k-1}(M)\} denotes the space of exact k-forms (those in the image of d).^[8] These groups quantify the extent to which closed forms are not exact, providing a algebraic invariant derived from the differential structure of M.^[9] The de Rham cohomology groups H^*_{\mathrm{dR}}(M) act as topological invariants of the manifold, encoding essential features of its homotopy type and connectivity in a coordinate-free manner. For compact manifolds, each H^k_{\mathrm{dR}}(M) is finite-dimensional as a real vector space, with the dimension b_k = \dim H^k_{\mathrm{dR}}(M) known as the k-th Betti number, which remains unchanged under diffeomorphisms.^[8]^[9] De Rham cohomology satisfies functoriality with respect to smooth maps. Given a smooth map f: N \to M between smooth manifolds, the pullback f^*: \Omega^*(M) \to \Omega^*(N) commutes with the exterior derivative, d \circ f^* = f^* \circ d, and thus induces linear maps f^*: H^k_{\mathrm{dR}}(M) \to H^k_{\mathrm{dR}}(N) on cohomology, preserving the ring structure in the graded sense.^[8] In particular, the zeroth cohomology group H^0_{\mathrm{dR}}(M) is isomorphic to the space of smooth functions on M that are constant on each connected component, which is naturally \mathbb{R}^{\pi_0(M)} where \pi_0(M) is the set of connected components of M.^[8]^[9]

Explicit Computations

The n-Sphere

The n-sphere S^n, embedded as the unit sphere in \mathbb{R}^{n+1}, is a compact, connected, orientable smooth manifold of dimension n, and it is simply connected for n \geq 2.^[10] These topological properties underpin the structure of its de Rham cohomology groups, which capture invariants of the manifold via closed differential forms modulo exact ones. The zeroth de Rham cohomology group H^0_{dR}(S^n) is isomorphic to \mathbb{R}, generated by constant 0-forms, as closed 0-forms are locally constant and thus constant on the connected manifold S^n, while exact 0-forms vanish. For intermediate degrees, H^k_{dR}(S^n) = 0 when $1 \leq k \leq n-1, reflecting the absence of nontrivial closed k-forms that are not exact in these dimensions. In the top degree, H^n_{dR}(S^n) \cong \mathbb{R}, one-dimensional and generated by the class of the standard volume form on S^n. The standard volume form \vol on S^n can be expressed in ambient coordinates as \vol = \sum_{i=1}^{n+1} (-1)^{i-1} x_i \, dx_1 \wedge \cdots \widehat{dx_i} \cdots \wedge dx_{n+1} \big|_{S^n}, where x = (x_1, \dots, x_{n+1}) are the inclusion coordinates. In hyperspherical coordinates, it takes the form \vol = \sin^{n-1} \theta_1 \, d\theta_1 \wedge \sin^{n-2} \theta_2 \, d\theta_2 \wedge \cdots \wedge \sin \theta_{n-1} \, d\theta_{n-1} \wedge d\phi, up to normalization. This form is closed, as d\vol = 0 in top degree, but it is not exact, since its integral over the closed oriented manifold S^n equals the positive volume of S^n, whereas Stokes' theorem implies that exact n-forms integrate to zero. An explicit computation of these groups proceeds via the Mayer-Vietoris sequence for de Rham cohomology. Cover S^n by two open sets: the northern hemisphere U = \{ x \in S^n \mid x_{n+1} > 0 \} and southern hemisphere V = \{ x \in S^n \mid x_{n+1} < 0 \}, slightly enlarged to open sets whose union is S^n and intersection U \cap V is diffeomorphic to S^{n-1} \times \mathbb{R}. The de Rham cohomology of U and V is trivial in positive degrees (each contractible to \mathbb{R}^n), while H^k_{dR}(U \cap V) \cong H^k_{dR}(S^{n-1}) by homotopy equivalence. The long exact Mayer-Vietoris sequence then yields H^k_{dR}(S^n) = 0 for $0 < k < n and H^n_{dR}(S^n) \cong \mathbb{R} by induction on n, confirming the nontriviality at the endpoints.

The n-Torus

The n-torus T^n is defined as the product of n circles, T^n = (S^1)^n, which can be parametrized using angular coordinates

\theta_1, \dots, \theta_n \in [0, 2\pi)&#36;.[](https://poisson.phc.dm.unipi.it/~lmigliorini/tesi_tr/bott_tu_diff_forms_algtop.pdf) This product structure endows

T^nwith a flat Riemannian metric, facilitating explicit computations of its de Rham cohomology groupsH^k_{\mathrm{dR}}(T^n)$.^[8] The cohomology groups are isomorphic to H^k_{\mathrm{dR}}(T^n) \cong \mathbb{R}^{\binom{n}{k}}, where \binom{n}{k} denotes the binomial coefficient, reflecting the polynomial growth of the Betti numbers b_k(T^n) = \binom{n}{k} with respect to n.^[8] This result arises from the Künneth formula, which decomposes the cohomology of the product as the tensor product (or exterior algebra) of the cohomologies of each S^1 factor, where H^1_{\mathrm{dR}}(S^1) \cong \mathbb{R} and higher groups vanish.^[8] Direct computation confirms the absence of higher syzygies or relations beyond this algebraic structure, owing to the flat geometry of T^n.^[8] A basis for H^k_{\mathrm{dR}}(T^n) consists of the wedge products dx_{i_1} \wedge \cdots \wedge dx_{i_k} for distinct indices $1 \leq i_1 < \cdots < i_k \leq n, where each dx_i = d\theta_i / (2\pi) is the standard closed 1-form on the i-th circle factor.^[8] These forms are closed, as the exterior derivative satisfies d(dx_i) = 0, and non-exact, since their integrals over the corresponding embedded (n-k)-tori yield nonzero periods that cannot be expressed as integrals of exact forms.^[8] The flat structure ensures these basis elements span the cohomology without additional constraints from curvature.^[8]

Punctured Euclidean Space

For m \geq 2, the punctured Euclidean space \mathbb{R}^m \setminus \{0\}, obtained by removing the origin from the m-dimensional Euclidean space, serves as a key example illustrating the impact of non-compactness and topological punctures on de Rham cohomology. This manifold is homotopy equivalent to the (m-1)-sphere S^{m-1} via the radial retraction r: \mathbb{R}^m \setminus \{0\} \to S^{m-1} defined by r(x) = x / \|x\|, which induces an isomorphism on cohomology groups.^[11]^[12] The zeroth de Rham cohomology group H^0_{dR}(\mathbb{R}^m \setminus \{0\}) \cong \mathbb{R} consists of constant functions, as these are the only closed 0-forms up to exactness, reflecting the path-connectedness of the space.^[11] For $1 \leq k < m-1, the cohomology groups vanish: H^k_{dR}(\mathbb{R}^m \setminus \{0\}) = 0. This follows from the homotopy equivalence to S^{m-1}, whose cohomology is trivial in these intermediate degrees, or directly from the Poincaré lemma applied via a radial homotopy operator that renders all closed k-forms exact in these ranges.^[12]^[11] In degree m-1, the top non-trivial cohomology is H^{m-1}_{dR}(\mathbb{R}^m \setminus \{0\}) \cong \mathbb{R}, generated by the (m-1)-form \alpha obtained as the pullback under the inclusion i: S^{m-1} \hookrightarrow \mathbb{R}^m \setminus \{0\} of the standard volume form on S^{m-1}. Explicitly, in coordinates,

\alpha = \frac{1}{\|x\|^m} \sum_{i=1}^m (-1)^{i-1} x_i \, dx_1 \wedge \cdots \wedge \widehat{dx_i} \wedge \cdots \wedge dx_m,

which is closed (d\alpha = 0) but not exact, as its integral over S^{m-1} is the volume \mathrm{vol}(S^{m-1}) = 2\pi^{m/2} / \Gamma(m/2) \neq 0.^[11] All closed forms on \mathbb{R}^m \setminus \{0\} are exact except in this top relative degree m-1, where the obstruction arises from the puncture; this exactness in lower degrees is established via radial extension, using the homotopy operator I_\theta \omega = \int_0^1 t^{\deg \omega - 1} i_{\partial_r} ( \phi_t^* \omega ) \, dt induced by the radial flow \phi_t(x) = t x, which satisfies d I_\theta + I_\theta d = \mathrm{id} - i^* p^* on appropriate form degrees, projecting to the trivial cohomology below degree m-1.^[11]

The Möbius Strip

The Möbius strip is a compact 2-dimensional manifold with boundary that serves as a fundamental example of a non-orientable surface. It can be constructed by identifying the opposite edges of a rectangular strip with a twist, resulting in a single-sided surface where a loop traversing the central circle twice returns to the starting point with reversed orientation. Topologically, the Möbius strip is homotopy equivalent to the circle S^1, obtained via a deformation retract that collapses the transverse fibers to the core circle. This equivalence implies that its de Rham cohomology groups match those of S^1.^[13]^[14] The zeroth de Rham cohomology group H^0_{\mathrm{dR}}(\mathrm{Möbius}) \cong \mathbb{R} consists of closed 0-forms, i.e., locally constant functions, reflecting the connectedness of the space. The first de Rham cohomology group H^1_{\mathrm{dR}}(\mathrm{Möbius}) \cong \mathbb{R} is generated by the cohomology class of a closed 1-form analogous to the angular form d\theta on the circle, which captures the nontrivial loop around the core. For instance, in coordinates where the strip is parametrized near the core, a representative form such as -y \, dx + x \, dy (normalized appropriately) is closed but not exact, as its integral over the core circle is $2\pi. This non-exactness arises from closed 1-forms like the twisting connection form associated to the Möbius strip as a non-trivial line bundle over S^1, where the integral over the base circle measures the monodromy, yielding $2\pi due to the half-twist.^[13]^[14] Higher cohomology vanishes, with H^2_{\mathrm{dR}}(\mathrm{Möbius}) = 0 and H^k_{\mathrm{dR}}(\mathrm{Möbius}) = 0 for k > 2.

Isomorphism with Topology

De Rham's Theorem

De Rham's theorem asserts that for any smooth manifold M, there is a canonical isomorphism H^k_{\mathrm{dR}}(M) \cong H^k_{\mathrm{sing}}(M; \mathbb{R}) for each degree k \geq 0. This equates the algebraic structure arising from differential forms on M with the topological invariants captured by singular cohomology with real coefficients. The theorem was originally proved by Georges de Rham in his 1931 paper for compact oriented manifolds, where he established a duality between closed differential forms and homology cycles via polyhedral decompositions.^[7] Later generalizations extended the result to all smooth manifolds, confirming a conjecture posed by Élie Cartan in 1929 that the Betti numbers of a manifold could be computed using the dimensions of spaces of closed forms modulo exact forms.^[15] The isomorphism arises naturally from the integration pairing between differential forms and singular chains: a k-form \omega integrates over a k-simplex \sigma to yield \int_\sigma \omega \in \mathbb{R}, extending by linearity to chains. This induces a map from de Rham cohomology to singular cohomology, as closed forms yield well-defined periods over homology cycles, while exact forms integrate to zero over boundaries, mirroring the simplicial integrals in topology. The theorem shows this map is an isomorphism, capturing how "holes" in the manifold are detected equivalently by non-trivial closed forms and non-bounding cycles.^[16] As a consequence, the Betti numbers b_k(M) = \dim H^k_{\mathrm{dR}}(M) provide a direct analytic computation of the topological complexity of M, and the Euler characteristic satisfies \chi(M) = \sum_{k=0}^{\dim M} (-1)^k b_k, linking global form theory to classical invariants. Moreover, the isomorphism is natural in the sense that for any smooth map f: M \to N, the induced maps on cohomology commute with the isomorphisms, preserving the topological structure under smooth deformations.^[15]

Sheaf-Theoretic Isomorphism

The sheaf-theoretic approach to de Rham cohomology reformulates the theory in terms of sheaves on a topological space, providing a framework that extends beyond smooth manifolds to more general settings such as algebraic varieties and singular spaces. On a smooth manifold M, the sheaf \Omega^*_{dR} of smooth differential forms is defined by assigning to each open set U \subset M the \mathbb{R}-vector space \Omega^k(U) of smooth k-forms on U, equipped with restriction maps and the exterior derivative d: \Omega^k \to \Omega^{k+1} as the sheaf morphism.^[17] This forms a complex of sheaves

\Omega^*_{dR}^\bullet: \Omega^0_{dR} \xrightarrow{d} \Omega^1_{dR} \xrightarrow{d} \cdots

, which resolves the constant sheaf \underline{\mathbb{R}} via the augmentation map sending constants to 0-forms.^[17] The de Rham cohomology groups are the hypercohomology groups \mathbb{H}^k(M, \Omega^*_{dR}) of this complex, computing the cohomology of the global sections complex \Gamma(M, \Omega^*_{dR}). The sheaf-theoretic de Rham theorem asserts a canonical isomorphism H^k(M, \underline{\mathbb{R}}) \cong \mathbb{H}^k(M, \Omega^*_{dR}), where H^k(M, \underline{\mathbb{R}}) denotes sheaf cohomology with constant coefficients; this holds because \Omega^*_{dR} is a soft resolution of \underline{\mathbb{R}} on paracompact manifolds, and soft sheaves are acyclic in higher degrees.^[17] This perspective generalizes the classical de Rham theorem for smooth manifolds to a comparison between sheaf cohomology and hypercohomology.^[17] In the complex analytic case, the sheaf of holomorphic k-forms \Omega^k_{an} on a complex manifold replaces \Omega^k_{dR}, with the de Rham complex \Omega^*_{an} yielding hypercohomology groups isomorphic to singular cohomology with complex coefficients via analogous resolutions.^[18] For algebraic varieties, Grothendieck's theorem provides the key extension: for a smooth scheme X of finite type over \mathbb{C}, the algebraic de Rham cohomology H^k_{dR}(X/\mathbb{C}) = \mathbb{H}^k(X, \Omega^*_X)—using the sheaf of Kähler differentials \Omega^k_X—is canonically isomorphic to the singular cohomology H^k_{sing}(X^{an}, \mathbb{C}) of its analytification.^[19] Deligne and Illusie's work further refines this in characteristic p > 0 settings, proving degeneration of the Hodge-to-de Rham spectral sequence for smooth proper varieties lifting to characteristic zero, linking algebraic de Rham cohomology to Hodge structures.^[20]^[21] To handle singular spaces, where smooth forms may not suffice, the theory employs currents—continuous linear functionals on the space of compactly supported smooth forms—or distributions as generalized forms; the resulting distributional de Rham cohomology is isomorphic to singular cohomology for oriented manifolds with singularities, via duality and resolution techniques.^[22] Applications to non-smooth manifolds proceed by resolving the constant sheaf through a complex of sheaves of currents or stratified forms, ensuring compatibility with topological invariants on singular strata.^[23]

Analytic Connections

Harmonic Forms

On a compact Riemannian manifold M, harmonic k-forms are defined as the smooth differential k-forms \alpha satisfying \Delta \alpha = 0, where \Delta = d d^* + d^* d denotes the Hodge Laplacian operator. This condition implies that \alpha lies in the kernel of \Delta, capturing forms that are both closed (d\alpha = 0) and co-closed (d^* \alpha = 0). The theory of such forms originated in the work of W. V. D. Hodge, who introduced them to study integrals on manifolds with geometric structure. The codifferential d^*, which appears in the definition of the Hodge Laplacian, is constructed formally as d^* = (-1)^{mk + m + 1} * d *, where * is the Hodge star operator determined by the Riemannian metric on M (with m = \dim M); the sign convention ensures d^* is the formal L^2-adjoint of d. This operator allows the extension of the exterior derivative to a self-adjoint complex, enabling the spectral analysis of \Delta. Harmonic forms thus represent the null space of this elliptic operator, and their existence relies on the compactness of M to guarantee finite dimensionality. A central result, known as Hodge's theorem, establishes that the space of harmonic k-forms is isomorphic to the k-th de Rham cohomology group H^k_{\mathrm{dR}}(M). Consequently, every cohomology class admits a unique harmonic representative with respect to the L^2-inner product induced by the metric, providing a canonical basis that is orthonormal in this Hilbert space. This uniqueness follows from the orthogonality properties of the kernel and the fact that \Delta is a positive semi-definite operator with discrete spectrum on compact manifolds. In the context of physics, harmonic forms play a significant role in gauge theories, where they correspond to zero modes of differential operators governing field configurations, such as in the parametrization of topological terms like generalizations of the θ-term in higher-abelian models. These zero modes influence the structure of instanton sectors and the computation of path integrals by accounting for non-trivial topological contributions.^[24]

Hodge Decomposition

In a compact oriented Riemannian manifold M, the Hodge decomposition theorem asserts that the space of smooth k-forms decomposes orthogonally as

\Omega^k(M) = \mathrm{Harm}^k(M) \oplus \mathrm{im}\, d \oplus \mathrm{im}\, \delta,

where \mathrm{Harm}^k(M) denotes the space of harmonic k-forms (those annihilated by the Hodge Laplacian), \mathrm{im}\, d is the image of the exterior derivative d: \Omega^{k-1}(M) \to \Omega^k(M), and \mathrm{im}\, \delta is the image of the codifferential \delta = d^*: \Omega^{k+1}(M) \to \Omega^k(M).^[25] This direct sum splitting extends the de Rham cohomology by providing an L^2-orthogonal basis for the entire form space, with exact forms (\mathrm{im}\, d) representing boundaries and coexact forms (\mathrm{im}\, \delta) their dual counterparts.^[26] The orthogonality of the summands holds with respect to the L^2 inner product induced by the Riemannian metric, given by \langle \alpha, \beta \rangle = \int_M \alpha \wedge *\beta for k-forms \alpha, \beta, where * is the Hodge star operator.^[26] Thus, \mathrm{im}\, d \perp \mathrm{im}\, \delta and both are orthogonal to \mathrm{Harm}^k(M).^[25] A key consequence is that \dim \mathrm{Harm}^k(M) = b_k, the k-th Betti number of M, linking the dimension of harmonic forms directly to the topological invariant from de Rham cohomology H^k_{dR}(M).^[26] Moreover, the Atiyah--Singer index theorem connects the analytic index of the de Rham complex to characteristic classes on M, reinforcing the interplay between elliptic operators and topology. On compact Kähler manifolds, the decomposition refines to a bigrading of the complexified de Rham cohomology: H^k_{dR}(M, \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(M), where each H^{p,q}(M) \cong H^q(M, \Omega^p_M) is the Dolbeault cohomology, and the summands remain orthogonal under the Hermitian inner product derived from the Kähler form.^[25] This structure, known as the Hodge decomposition in the Kähler setting, implies Hodge symmetry h^{p,q} = h^{q,p} and further topological restrictions, such as the evenness of odd Betti numbers.^[27] The theorem also establishes the finite-dimensionality of de Rham cohomology groups via elliptic partial differential equation theory: since the Hodge Laplacian \Delta = d\delta + \delta d is a second-order elliptic operator on the compact manifold M, its kernel \mathrm{Harm}^k(M) is finite-dimensional, and thus so is H^k_{dR}(M) \cong \mathrm{Harm}^k(M).^[26] This analytic proof complements the topological perspective and underpins applications in global analysis on manifolds.^[25]

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[PDF] Graduate Texts in Mathematics 82 - Poisson
Differential Topology. Axiomatic Set Theory. 2nd ed. 34 SPITZER. Principles ... Elementary Algebraic Geometry. and Their Singularities. 45 LOEVE ...
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[PDF] COHOMOLOGY VIA FORMS 5.1 The DeRham ... - MIT Mathematics
May 5, 2020 · In this section we will show that for compact manifolds (and for lots of other manifolds besides) the DeRham cohomology groups which we defined ...
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[PDF] Algebraic Topology - Cornell Mathematics
This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in ...
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[PDF] Manifolds and Differential Forms Reyer Sjamaar - Cornell Mathematics
Let M N be the punctured Euclidean space Rn \ {0} and let φ0(x) x ... De Rham cohomology. The distinction between closed and exact differential.
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[PDF] manifolds, cohomology, and sheaves (version 6) - Indico [Home]
In this subsection, we dis- cuss the de Rham cohomology of three examples—the sphere, the punctured. Euclidean space, and the complex projective space. Example ...
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[PDF] INTRO TO DE RHAM COHOMOLOGY Contents 1 ... - Matthew Niemiro
INTRO TO DE RHAM COHOMOLOGY. 15. Example 3.6 (Cohomology of a Möbius strip). The Möbius strip ad- mits a retract onto its core circle by collapsing its edge ...
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[PDF] D-Math Prof. Dr. DA Salamon Differential Geometry II FS ... - metaphor
May 15, 2018 · b) Let M be the open Möbius strip. Compute the de-Rham cohomology groups – the compactly supported and also the full one. c) Give a counter- ...
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[PDF] De Rham's Theorem
De Rham's theorem is a classical result implying the existence of an iso- morphism between the De Rham and singular cohomology groups of a smooth manifold. In ...
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[PDF] proof of de rham's theorem - Harvard University
Let M be a smooth n-dimensional manifold. Then, de Rham's theorem states that the de Rham cohomology of M is naturally isomorphic to its singular cohomology ...
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[PDF] Sheaf Theoretic Approach to De Rham Cohomology and Singular ...
Feb 22, 2022 · This paper will give a sheaf theoretic approach to the proof of the equivalence between De Rham cohomology and singular cohomology. The paper ...
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[PDF] sheaf cohomology and algebraic de rham theorem - UChicago Math
We use sheaf-theoretic language and tools to specify the correct notion of de Rham cohomology of a smooth algebraic variety. We prove a comparison theorem ...
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[PDF] On the de Rham cohomology of algebraic varieties - Numdam
HODGE, Integrals of the second kind on an algebraic variety, Annals of. Mathematics, vol. 62 (1955), p. 56-91. This paper is referred to by A-H in the sequel. ( ...Missing: original | Show results with:original<|control11|><|separator|>
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[PDF] Algebraic de Rham Cohomology and the Hodge Spectral sequence
Sep 20, 2020 · Having established this lemma, we now sketch Illusie's proof. Theorem 5.2 (Illusie, Hodge Degeneration Theorem). Let k be a field of ...
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[PDF] a remark on distributions and the de rham theorem - MIT Mathematics
May 11, 2011 · We show that the de Rham theorem, interpreted as the isomor- phism between distributional de Rham cohomology and simplicial homology in the dual ...Missing: singular | Show results with:singular
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[PDF] de Rham Cohomology - Stacks Project
0FK5 In this chapter we start with a discussion of the de Rham complex of a morphism of schemes and we end with a proof that de Rham cohomology defines a Weil.
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Aspects of higher-abelian gauge theories at zero and finite ...
This action is parametrized by two non-dynamical harmonic forms and generalizes the θ-term in usual gauge theories. In the second part the general framework ...
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[PDF] Hodge Decomposition - UC Berkeley math
Apr 27, 2014 · One is the de Rham Cohomology which can be defined on a general, possibly non complex, manifold. The second one is the Dolbeault cohomology ...Missing: source | Show results with:source
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[PDF] Undergraduate Lecture Notes in De Rham–Hodge Theory - arXiv
May 15, 2011 · The Hodge–Weyl theorem [8, 9] states that every De Rham cohomology class has a unique harmonic representative. In other words, the space H p.
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[PDF] Hodge theory and the topology of compact Kähler and complex ...
Hodge theory states that if X is compact, any de Rham cohomology class has a unique representative which is a C∞ form which is both closed and coclosed (dα ...