10.4 de Rham cohomology

De Rham cohomology measures global geometric properties of smooth manifolds using differential forms and the exterior derivative. It connects the manifold's topology to its differential structure, providing insights into its shape and properties.

This powerful tool has applications in mathematics and physics. By studying closed and exact forms, de Rham cohomology captures essential information about manifolds, linking local differential properties to global topological features.

Definition of de Rham cohomology

• de Rham cohomology is a cohomology theory for smooth manifolds that associates vector spaces to a manifold, measuring its global geometric properties
• Constructed using differential forms and the exterior derivative operator, capturing information about the manifold's topology and differential structure
• Provides a powerful tool for studying the geometry and topology of smooth manifolds, with applications in various areas of mathematics and physics

Differential forms

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• Differential forms are antisymmetric multilinear functions that generalize the concept of functions and vector fields on a manifold
• Consist of smooth functions and differentials of coordinate functions, allowing for integration on manifolds
• Enable the study of geometric and topological properties of manifolds, such as orientability, volume, and curvature
• Examples include 1-forms (dual to vector fields) and 2-forms (related to area elements on surfaces)

Exterior derivative

• The exterior derivative is an operator that generalizes the concept of the differential of a function to differential forms
• Maps k-forms to (k+1)-forms, satisfying the property that applying it twice yields zero (d^2 = 0)
• Encodes information about the local structure of a manifold and is used to define the de Rham complex
• Allows for the study of closed and exact forms, which are crucial in the computation of de Rham cohomology

de Rham complex

• The de Rham complex is a sequence of vector spaces of differential forms connected by the exterior derivative operator
• Consists of the spaces of k-forms, denoted by $\Omega^k(M)$, with the exterior derivative $d$ mapping between them: $0 \to \Omega^0(M) \stackrel{d}{\to} \Omega^1(M) \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^n(M) \to 0$
• The cohomology of the de Rham complex, i.e., the quotient spaces $H^k(M) = \ker(d_k) / \operatorname{im}(d_{k-1})$, defines the de Rham cohomology of the manifold $M$
• The dimensions of the de Rham cohomology vector spaces, called the Betti numbers, are topological invariants of the manifold

Computation of de Rham cohomology

• Computing the de Rham cohomology of a manifold involves finding the spaces of closed forms (kernel of the exterior derivative) modulo the exact forms (image of the exterior derivative)
• Various tools and techniques are employed to simplify and calculate the cohomology vector spaces, depending on the properties of the manifold and the degree of the forms

Poincaré lemma

• The Poincaré lemma states that on a contractible open subset of a manifold, every closed form is exact
• Implies that the de Rham cohomology of a contractible space is trivial, i.e., consists only of the constant functions
• Allows for the computation of de Rham cohomology using a good cover of the manifold by contractible open sets
• Provides a local-to-global principle for studying the cohomology of manifolds

Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a long exact sequence that relates the de Rham cohomology of a manifold to the cohomology of its subspaces
• Enables the computation of the cohomology of a manifold by breaking it down into simpler pieces (e.g., open sets) and studying their intersections
• Consists of a sequence of maps between the cohomology spaces of the manifold, its subspaces, and their intersections
• Provides a powerful tool for calculating the de Rham cohomology of manifolds that can be decomposed into simpler parts

Examples of computation

• The de Rham cohomology of the circle $S^1$ is given by $H^0(S^1) = \mathbb{R}$ and $H^1(S^1) = \mathbb{R}$, corresponding to the constant functions and the angular form $d\theta$
• The de Rham cohomology of the torus $T^2$ is $H^0(T^2) = \mathbb{R}$, $H^1(T^2) = \mathbb{R}^2$, and $H^2(T^2) = \mathbb{R}$, reflecting its genus and orientability
• The de Rham cohomology of the sphere $S^n$ is trivial for $0 < k < n$, with $H^0(S^n) = H^n(S^n) = \mathbb{R}$, capturing its simply-connectedness and orientability

Relation to singular cohomology

• Singular cohomology is another cohomology theory for topological spaces, constructed using cochains on the space of singular simplices
• The de Rham theorem establishes an isomorphism between de Rham cohomology and singular cohomology for smooth manifolds, linking the two theories
• This connection allows for the exchange of tools and results between the two cohomology theories, enriching the study of manifolds

de Rham theorem

• The de Rham theorem states that for a smooth manifold $M$, the de Rham cohomology $H^*(M)$ is isomorphic to the singular cohomology $H^*(M; \mathbb{R})$ with real coefficients
• Proved by showing that the de Rham complex is chain homotopy equivalent to the singular cochain complex, inducing an isomorphism on cohomology
• Allows for the computation of singular cohomology using differential forms and the exterior derivative, which are often more tractable than singular cochains
• Establishes a deep connection between the differential and topological properties of smooth manifolds

Isomorphism between de Rham and singular cohomology

• The isomorphism between de Rham and singular cohomology is given by the de Rham map, which associates a singular cochain to a differential form by integration over simplices
• The inverse map is induced by a choice of smooth approximation to the singular cochains, such as Whitney forms or simplicial forms
• The isomorphism is natural with respect to smooth maps between manifolds, making it a powerful tool in the study of smooth manifold topology
• Allows for the transfer of results and constructions between the two cohomology theories, such as the cup product and the Poincaré duality theorem

Applications of de Rham cohomology

• de Rham cohomology has numerous applications in various areas of mathematics and physics, showcasing its versatility and importance in the study of smooth manifolds
• Some notable applications include Hodge theory, characteristic classes, and Morse theory, each providing unique insights into the geometry and topology of manifolds

Hodge theory

• Hodge theory studies the relationship between the de Rham cohomology of a compact Riemannian manifold and its harmonic forms (forms that are both closed and co-closed)
• The Hodge decomposition theorem states that every differential form can be uniquely written as the sum of a harmonic form, an exact form, and a co-exact form
• Establishes an isomorphism between the de Rham cohomology and the space of harmonic forms, linking the topological and geometric properties of the manifold
• Provides a powerful tool for studying the geometry of compact Riemannian manifolds, with applications in complex geometry, algebraic geometry, and mathematical physics

Characteristic classes

• Characteristic classes are cohomology classes associated with vector bundles over a manifold, measuring the twisting and non-triviality of the bundle
• Examples include the Chern classes for complex vector bundles, the Pontryagin classes for real vector bundles, and the Euler class for oriented vector bundles
• Constructed using the de Rham cohomology and the Chern-Weil theory, which expresses characteristic classes in terms of the curvature of a connection on the bundle
• Play a crucial role in the study of the topology of manifolds and their vector bundles, with applications in algebraic topology, differential geometry, and gauge theory

Morse theory

• Morse theory studies the relationship between the topology of a smooth manifold and the critical points of a smooth function on the manifold
• The main result of Morse theory states that the topology of the manifold can be reconstructed from the critical points and their indices, which measure the number of independent descending directions
• The Morse inequalities relate the Betti numbers of the manifold (dimensions of the de Rham cohomology) to the number of critical points of each index, providing a powerful tool for computing the homology of the manifold
• Morse theory has numerous applications in differential topology, Riemannian geometry, and mathematical physics, including the study of geodesics, the topology of energy landscapes, and the Witten deformation of the de Rham complex

Generalizations of de Rham cohomology

• The success and utility of de Rham cohomology have inspired various generalizations and extensions of the theory, adapted to different contexts and geometries
• These generalizations often aim to capture more refined or specialized information about the manifold or to extend the theory to broader classes of spaces

Čech-de Rham cohomology

• Čech-de Rham cohomology is a cohomology theory that combines the Čech cohomology (based on open covers) and the de Rham cohomology
• Defined using a double complex that incorporates both the Čech and de Rham differentials, allowing for the study of manifolds with less smooth structures
• Provides a more flexible and general framework for studying the cohomology of manifolds, with applications in sheaf theory and algebraic geometry
• Allows for the computation of the cohomology of manifolds using a wider range of covers and local data, extending the reach of the theory

Dolbeault cohomology

• Dolbeault cohomology is a cohomology theory for complex manifolds, adapted to the study of complex differential forms and the Dolbeault operator (a complex version of the exterior derivative)
• Defined using the Dolbeault complex, which consists of the spaces of $(p, q)$-forms (forms with $p$ holomorphic and $q$ anti-holomorphic differentials) and the Dolbeault operator $\bar{\partial}$
• The Dolbeault cohomology groups $H^{p,q}(M)$ measure the complex structure and the holomorphic properties of the manifold, with applications in complex geometry and algebraic geometry
• Related to the de Rham cohomology through the Frölicher spectral sequence, which decomposes the de Rham cohomology into a sum of Dolbeault cohomology groups

Equivariant de Rham cohomology

• Equivariant de Rham cohomology is an extension of de Rham cohomology to manifolds with a group action, capturing the interplay between the symmetries of the manifold and its topology
• Defined using equivariant differential forms, which are differential forms that are invariant under the group action, and the equivariant exterior derivative, which incorporates the infinitesimal action of the group
• The equivariant de Rham cohomology groups $H^*_G(M)$ are modules over the ring of invariant polynomials on the Lie algebra of the group, encoding the equivariant topology of the manifold
• Provides a powerful tool for studying the topology and geometry of manifolds with symmetries, with applications in symplectic geometry, representation theory, and mathematical physics (e.g., the BRST formalism and the localization theorem)