7.4 de Rham cohomology

De Rham cohomology captures the topology of smooth manifolds using differential forms. It bridges geometry and topology, providing a framework to study global properties through local differential analysis.

This theory connects smooth structures with algebraic topology. By examining closed and exact forms, de Rham cohomology reveals topological invariants, offering insights into the shape and structure of manifolds.

Definition of de Rham cohomology

• de Rham cohomology is a cohomology theory for smooth manifolds that captures topological information using differential forms
• Provides a framework for studying the global properties of a manifold by analyzing the behavior of differential forms on the manifold

Smooth manifolds

Top images from around the web for Smooth manifolds

• 

  Tangent space - Wikipedia View original

  Is this image relevant?

• 

  Cohomology - Wikipedia View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent space - Wikipedia View original

  Is this image relevant?

• 

  Cohomology - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Smooth manifolds

• 

  Tangent space - Wikipedia View original

  Is this image relevant?

• 

  Cohomology - Wikipedia View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent space - Wikipedia View original

  Is this image relevant?

• 

  Cohomology - Wikipedia View original

  Is this image relevant?

1 of 3

• Smooth manifolds are topological spaces that locally resemble Euclidean space and have a well-defined notion of smoothness
• Includes examples such as spheres, tori, and Lie groups
• Smooth functions between manifolds are maps that preserve the smooth structure
• Tangent spaces and tangent bundles are essential concepts in the study of smooth manifolds

Exterior algebra

• The exterior algebra is a graded algebra constructed from a vector space, which provides the algebraic structure for differential forms
• Wedge product is the multiplication operation in the exterior algebra, satisfying anticommutativity and associativity
• Exterior derivative is a linear operator that maps $k$-forms to $(k+1)$-forms and satisfies the Leibniz rule and the property $d^2 = 0$

Differential forms

• Differential forms are antisymmetric multilinear functions on the tangent spaces of a manifold
• $k$-forms are elements of the $k$-th exterior power of the cotangent bundle
• Examples include $0$-forms (smooth functions), $1$-forms (covector fields), and $n$-forms (volume forms on an $n$-dimensional manifold)
• Differential forms can be integrated over oriented submanifolds of the appropriate dimension

de Rham complex

• The de Rham complex is a cochain complex constructed from the exterior algebra of differential forms on a manifold
• The coboundary operator in the de Rham complex is the exterior derivative $d$
• The cohomology of the de Rham complex is the de Rham cohomology, which measures the failure of the Poincaré lemma globally
• The $k$-th de Rham cohomology group $H^k_{dR}(M)$ consists of closed $k$-forms modulo exact $k$-forms on the manifold $M$

Properties of de Rham cohomology

• de Rham cohomology satisfies several important properties that make it a powerful tool for studying the topology of smooth manifolds
• These properties allow for the computation of de Rham cohomology in various situations and reveal connections to other mathematical concepts

Functoriality

• de Rham cohomology is functorial with respect to smooth maps between manifolds
• Given a smooth map $f: M \to N$, there is an induced homomorphism $f^*: H^k_{dR}(N) \to H^k_{dR}(M)$ on cohomology
• Functoriality allows for the study of how cohomology behaves under mappings and enables the construction of cohomological invariants

Homotopy invariance

• de Rham cohomology is invariant under homotopy equivalence of smooth manifolds
• If two manifolds $M$ and $N$ are homotopy equivalent, then their de Rham cohomology groups are isomorphic: $H^k_{dR}(M) \cong H^k_{dR}(N)$
• Homotopy invariance implies that de Rham cohomology depends only on the homotopy type of a manifold, not its specific smooth structure

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
• Given an open cover $\{U, V\}$ of a manifold $M$, there is a long exact sequence: $\cdots \to H^k_{dR}(M) \to H^k_{dR}(U) \oplus H^k_{dR}(V) \to H^k_{dR}(U \cap V) \to H^{k+1}_{dR}(M) \to \cdots$
• Mayer-Vietoris sequence is a powerful tool for computing de Rham cohomology by breaking down a manifold into simpler pieces

Poincaré duality

• Poincaré duality is a fundamental relationship between the de Rham cohomology of a compact oriented manifold and its homology
• For a compact oriented $n$-dimensional manifold $M$, there is an isomorphism: $H^k_{dR}(M) \cong H_{n-k}(M; \mathbb{R})$
• Poincaré duality allows for the study of the dual relationship between differential forms and submanifolds

Künneth formula

• The Künneth formula describes the de Rham cohomology of a product manifold in terms of the cohomology of its factors
• For two manifolds $M$ and $N$, there is an isomorphism: $H^k_{dR}(M \times N) \cong \bigoplus_{i+j=k} H^i_{dR}(M) \otimes H^j_{dR}(N)$
• Künneth formula simplifies the computation of de Rham cohomology for product manifolds and reveals the multiplicative structure of cohomology

Computation of de Rham cohomology

• Computing the de Rham cohomology groups of a manifold is a central problem in the theory
• Various techniques and results are available for calculating de Rham cohomology in specific cases

Contractible spaces

• A contractible space is a topological space that is homotopy equivalent to a point
• The de Rham cohomology of a contractible manifold vanishes in all degrees except for $H^0_{dR}(M) \cong \mathbb{R}$
• Examples of contractible spaces include Euclidean spaces, convex subsets of Euclidean spaces, and star-shaped domains

Spheres

• The $n$-dimensional sphere $S^n$ is a compact manifold with simple de Rham cohomology
• The de Rham cohomology of $S^n$ is given by: $H^k_{dR}(S^n) \cong \begin{cases} \mathbb{R}, & k = 0 \text{ or } k = n \\ 0, & \text{otherwise} \end{cases}$
• The generator of $H^n_{dR}(S^n)$ is the volume form on the sphere

Tori

• The $n$-dimensional torus $T^n$ is the product of $n$ circles, $T^n = S^1 \times \cdots \times S^1$
• The de Rham cohomology of $T^n$ can be computed using the Künneth formula: $H^k_{dR}(T^n) \cong \bigoplus_{i_1 + \cdots + i_n = k} H^{i_1}_{dR}(S^1) \otimes \cdots \otimes H^{i_n}_{dR}(S^1)$
• The Betti numbers of $T^n$ are $\binom{n}{k}$, the binomial coefficients

Surfaces

• Surfaces are 2-dimensional manifolds, classified by their genus $g$ (number of holes)
• The de Rham cohomology of a compact oriented surface $\Sigma_g$ of genus $g$ is: $H^k_{dR}(\Sigma_g) \cong \begin{cases} \mathbb{R}, & k = 0 \text{ or } k = 2 \\ \mathbb{R}^{2g}, & k = 1 \\ 0, & \text{otherwise} \end{cases}$
• The generators of $H^1_{dR}(\Sigma_g)$ correspond to the $2g$ independent cycles on the surface

CW complexes

• CW complexes are topological spaces constructed by attaching cells of increasing dimension
• The de Rham cohomology of a CW complex can be computed using cellular cohomology
• For a CW complex $X$, there is an isomorphism between the de Rham cohomology and the cellular cohomology: $H^k_{dR}(X) \cong H^k_{cell}(X; \mathbb{R})$
• Cellular cohomology provides a combinatorial approach to computing de Rham cohomology for CW complexes

Applications of de Rham cohomology

• de Rham cohomology has numerous applications in various areas of mathematics and physics
• It provides a framework for studying geometric and topological properties of manifolds and their relationships to other mathematical structures

Integration of differential forms

• de Rham cohomology allows for the integration of closed differential forms over cycles on a manifold
• The de Rham theorem states that the integration map induces an isomorphism between de Rham cohomology and singular cohomology with real coefficients
• Integration of differential forms is used in the formulation of Stokes' theorem, which relates the integral of a form over a boundary to the integral of its exterior derivative over the interior

Characteristic classes

• Characteristic classes are cohomological invariants associated with vector bundles over manifolds
• Examples of characteristic classes include Chern classes for complex vector bundles and Pontryagin classes for real vector bundles
• Characteristic classes can be represented by closed differential forms, and their de Rham cohomology classes capture important topological information about the vector bundles

Chern-Weil theory

• Chern-Weil theory is a method for constructing characteristic classes using connections and curvature on vector bundles
• Given a connection on a vector bundle, the Chern-Weil homomorphism associates a closed differential form to each invariant polynomial on the Lie algebra of the structure group
• Chern-Weil theory provides a differential-geometric approach to characteristic classes and relates them to the geometry of connections

Morse theory

• Morse theory studies the relationship between the topology of a manifold and the critical points of smooth functions on the manifold
• The de Rham cohomology of a manifold can be computed using Morse theory by analyzing the gradient flow of a Morse function
• Morse inequalities relate the Betti numbers of a manifold to the number of critical points of a Morse function, providing a lower bound for the de Rham cohomology

Hodge theory

• Hodge theory is the study of harmonic forms on Riemannian manifolds and their relationship to de Rham cohomology
• The Hodge theorem states that on a compact oriented Riemannian manifold, every de Rham cohomology class has a unique harmonic representative
• Hodge theory establishes a correspondence between the topology of a manifold (de Rham cohomology) and the analysis of differential equations (harmonic forms)

Relation to other cohomology theories

• de Rham cohomology is one of several cohomology theories that capture topological information about manifolds
• It is closely related to other cohomology theories, and various comparison theorems establish connections between them

Singular cohomology

• Singular cohomology is a cohomology theory defined using cochains on the singular simplices of a topological space
• The de Rham theorem establishes an isomorphism between de Rham cohomology and singular cohomology with real coefficients for smooth manifolds
• Singular cohomology provides a purely topological approach to cohomology, while de Rham cohomology incorporates the smooth structure of manifolds

Čech cohomology

• Čech cohomology is a cohomology theory defined using open covers of a topological space and the intersection of their elements
• For a smooth manifold, Čech cohomology with real coefficients is isomorphic to de Rham cohomology
• Čech cohomology is particularly useful for studying the local-to-global properties of sheaves on a space

Sheaf cohomology

• Sheaf cohomology is a general cohomology theory defined for sheaves on a topological space
• The de Rham complex can be viewed as a resolution of the constant sheaf $\mathbb{R}$ on a manifold, and the de Rham cohomology is isomorphic to the sheaf cohomology of this constant sheaf
• Sheaf cohomology provides a unifying framework for studying various cohomology theories and their relationships

Comparison theorems

• Comparison theorems establish isomorphisms between different cohomology theories under certain conditions
• The de Rham theorem, relating de Rham cohomology and singular cohomology, is an example of a comparison theorem
• Other comparison theorems include the Dolbeault theorem (relating Dolbeault cohomology and sheaf cohomology of holomorphic vector bundles) and the comparison between étale cohomology and singular cohomology for algebraic varieties

Generalizations of de Rham cohomology

• de Rham cohomology has been generalized and extended to various settings beyond smooth manifolds
• These generalizations capture additional structures and properties of the spaces under consideration

Dolbeault cohomology

• Dolbeault cohomology is a cohomology theory for complex manifolds that takes into account the complex structure
• It is defined using the Dolbeault complex, which consists of $(p,q)$-forms and the $\bar{\partial}$-operator
• Dolbeault cohomology groups $H^{p,q}(X)$ measure the $\bar{\partial}$-cohomology of $(p,q)$-forms on a complex manifold $X$
• The Hodge decomposition theorem relates Dolbeault cohomology to the cohomology of holomorphic vector bundles

Equivariant cohomology

• Equivariant cohomology is a cohomology theory that incorporates the action of a group on a space
• For a $G$-space $X$ (a space with an action of a group $G$), the equivariant de Rham cohomology $H^*_G(X)$ is defined using $G$-invariant differential forms
• Equivariant cohomology captures the interplay between the topology of the space and the symmetries given by the group action

Crystalline cohomology

• Crystalline cohomology is a $p$-adic cohomology theory for algebraic varieties over fields of characteristic $p > 0$
• It is defined using the de Rham-Witt complex, which is a generalization of the de Rham complex that takes into account the arithmetic properties of the variety
• Crystalline cohomology provides a $p$-adic analog of de Rham cohomology and is used in the study of arithmetic geometry

Cyclic homology

• Cyclic homology is a homology theory for associative algebras that generalizes de Rham cohomology
• It is defined using the cyclic complex, which involves the Hochschild complex and the action of the cyclic group
• Cyclic homology captures non-commutative analogues of de Rham cohomology and has applications in non-commutative geometry and algebraic $K$-theory

Noncommutative geometry

• Noncommutative geometry is a generalization of geometry that allows for non-commutative algebras to play the role of functions on a space
• In noncommutative geometry, the notion of a differential form is replaced by a cyclic cocycle, and the de Rham complex is replaced by the cyclic complex
• Noncommutative de Rham cohomology and cyclic cohomology provide tools for studying the geometry and topology of non-commutative spaces