captures the topology of smooth manifolds using . 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
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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 d2=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 HdRk(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→N, there is an induced homomorphism f∗:HdRk(N)→HdRk(M) on cohomology
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: HdRk(M)≅HdRk(N)
implies that de Rham cohomology depends only on the of a manifold, not its specific smooth structure
Mayer-Vietoris sequence
The 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:
⋯→HdRk(M)→HdRk(U)⊕HdRk(V)→HdRk(U∩V)→HdRk+1(M)→⋯
Mayer-Vietoris sequence is a powerful tool for computing de Rham cohomology by breaking down a manifold into simpler pieces
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:
HdRk(M)≅Hn−k(M;R)
Poincaré duality allows for the study of the dual relationship between differential forms and submanifolds
Künneth formula
The 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:
HdRk(M×N)≅⨁i+j=kHdRi(M)⊗HdRj(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 is a that is homotopy equivalent to a point
The de Rham cohomology of a contractible manifold vanishes in all degrees except for HdR0(M)≅R
Examples of contractible spaces include Euclidean spaces, convex subsets of Euclidean spaces, and star-shaped domains
Spheres
The n-dimensional sphere Sn is a compact manifold with simple de Rham cohomology
The de Rham cohomology of Sn is given by:
HdRk(Sn)≅{R,0,k=0 or k=notherwise
The generator of HdRn(Sn) is the volume form on the sphere
Tori
The n-dimensional torus Tn is the product of n circles, Tn=S1×⋯×S1
The de Rham cohomology of Tn can be computed using the Künneth formula:
HdRk(Tn)≅⨁i1+⋯+in=kHdRi1(S1)⊗⋯⊗HdRin(S1)
The Betti numbers of Tn are (kn), 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 Σg of genus g is:
HdRk(Σg)≅⎩⎨⎧R,R2g,0,k=0 or k=2k=1otherwise
The generators of HdR1(Σg) correspond to the 2g independent cycles on the surface
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:
HdRk(X)≅Hcellk(X;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 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
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
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
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 , 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
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 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 ∂ˉ-operator
Dolbeault cohomology groups Hp,q(X) measure the ∂ˉ-cohomology of (p,q)-forms on a complex manifold X
The 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 HG∗(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
Key Terms to Review (25)
Alexander Grothendieck: Alexander Grothendieck was a French mathematician who made groundbreaking contributions to algebraic geometry, particularly through the development of sheaf theory and the concept of schemes. His work revolutionized the field by providing a unifying framework that connected various areas of mathematics, allowing for deeper insights into algebraic varieties and their cohomological properties.
Cech Cohomology: Cech cohomology is a mathematical tool used to study the topological properties of spaces through the use of open covers and their intersections. It connects with the ideas of derived functors, particularly when analyzing sheaf cohomology, as it helps quantify the ability to recover global sections from local data. The construction of Cech cohomology involves associating cochain complexes to sheaves, allowing for the exploration of deeper geometric and topological characteristics of spaces.
Characteristic classes: Characteristic classes are a set of invariants associated with vector bundles that provide essential information about the topology of the bundle. These classes help in distinguishing different bundles and can be used to study various geometric and topological properties of manifolds. They play a crucial role in the intersection of algebraic topology and differential geometry, particularly when analyzing vector bundles and their sections.
Chern-Weil Theory: Chern-Weil theory is a mathematical framework that connects differential geometry and topology, particularly through the use of curvature forms of vector bundles to define characteristic classes. It provides a method for constructing topological invariants from geometric data, which can then be used to study the properties of manifolds and their associated vector bundles. This theory bridges the gap between de Rham cohomology and characteristic classes, highlighting how curvature can be leveraged to extract topological information.
Closed forms: Closed forms refer to differential forms that have a vanishing exterior derivative. In the context of de Rham cohomology, closed forms play a crucial role in defining the cohomology groups, which capture topological features of smooth manifolds. Understanding closed forms helps in exploring relationships between differential geometry and topology, particularly how they represent equivalence classes of forms under exactness.
Contractible space: A contractible space is a topological space that can be continuously shrunk to a single point within that space. This means there exists a homotopy, which is a continuous deformation, that transforms the entire space into just one point without breaking or tearing. Contractible spaces have important implications in various fields, particularly in cohomology, where they are often considered 'trivial' since they have no 'holes' or interesting topological features.
Cw complexes: A cw complex is a type of topological space constructed by gluing cells together in a specific way, using open disks of varying dimensions. This structure allows for a systematic way to build complex spaces from simpler pieces, making them particularly useful in algebraic topology for studying the properties of spaces, especially when analyzing de Rham cohomology.
De Rham cohomology: de Rham cohomology is a mathematical tool used in differential geometry that studies the properties of differentiable manifolds through differential forms. It connects smoothly with other cohomology theories, providing a way to analyze topological features using calculus. This concept is vital in understanding the relationships between different cohomology theories, such as sheaf cohomology and Čech cohomology, and plays a crucial role in the study of sheaves on manifolds and applications in mathematical physics.
Differential forms: Differential forms are mathematical objects that generalize the concept of functions and vectors to enable integration over manifolds. They play a crucial role in calculus on manifolds, especially in the context of integration, differentiation, and cohomology, serving as the foundational elements for defining the de Rham cohomology theory.
Exactness: Exactness refers to a property of sequences or diagrams in mathematics, particularly in homological algebra and cohomology, indicating that certain maps between objects lead to a precise relationship between their kernels and images. This concept is crucial for understanding how information is preserved or transformed across different mathematical structures, linking them in meaningful ways and allowing for the construction of long exact sequences.
Functoriality: Functoriality is a principle in category theory that describes how structures and relationships can be preserved through transformations between categories. It implies that there is a systematic way to map objects and morphisms from one category to another while maintaining their inherent properties and structures. This concept is crucial for understanding how various mathematical frameworks relate to each other, especially in the context of derived functors, morphisms of ringed spaces, de Rham cohomology, and sheaves in algebraic topology.
Grothendieck's Galois Theory: Grothendieck's Galois Theory is a framework that generalizes classical Galois theory using the language of category theory and topos theory, emphasizing the relationship between fields and their geometric interpretations. This approach allows one to understand the symmetries of algebraic equations through the lens of sheaves and étale cohomology, bridging algebraic geometry and number theory.
Henri Cartan: Henri Cartan was a prominent French mathematician known for his foundational contributions to algebraic topology and sheaf theory. His work emphasized the role of sheaves in cohomology and derived functors, providing tools to study topological spaces and their properties in a more abstract setting. His ideas have influenced various areas of mathematics, connecting concepts like manifolds, cohomology theories, and the understanding of complex algebraic structures.
Hodge Decomposition Theorem: The Hodge Decomposition Theorem states that for a smooth, compact Riemannian manifold, any differential form can be uniquely decomposed into three components: an exact form, a co-exact form, and a harmonic form. This theorem connects the concepts of differential forms, cohomology, and the inner product structure given by the Riemannian metric, showing how these elements interact within the framework of de Rham cohomology.
Hodge Theory: Hodge Theory is a framework in mathematics that connects the geometry of a manifold with its topology, specifically through the study of differential forms and their cohomology classes. It reveals how these forms can be decomposed into orthogonal components, helping to classify the manifold's topological properties. This theory provides crucial insights into how the de Rham cohomology of a manifold interacts with its other cohomological theories, enhancing our understanding of the underlying geometric structures.
Homotopy invariance: Homotopy invariance refers to the property of certain topological invariants that remain unchanged under homotopy, which is a continuous deformation of functions. This concept is crucial in understanding how certain mathematical structures can be analyzed through their topological features, particularly in the context of cohomology theories, where one can study the properties of spaces without concern for specific shapes or sizes.
Homotopy type: Homotopy type refers to a property of topological spaces that captures their essential shape and structure, focusing on how they can be continuously transformed into one another. Two spaces are said to have the same homotopy type if there exists a continuous deformation (homotopy) between them, allowing for the comparison of complex spaces through simpler ones. This concept is vital in areas like algebraic topology, where it provides a way to classify spaces and understand their features, linking closely to various mathematical constructs.
Künneth Formula: The Künneth Formula is a key result in algebraic topology that provides a method to compute the cohomology groups of the product of two topological spaces based on their individual cohomology groups. This formula shows how the cohomology of a product space relates to the cohomologies of the factors, specifically utilizing the tensor product and direct sum of vector spaces. Understanding this relationship is crucial for various applications in de Rham cohomology, where it can simplify computations involving manifolds.
Locality: Locality refers to the property of sheaves that allows them to capture local data about spaces, making them useful for studying properties that can be understood through local neighborhoods. This concept connects various aspects of sheaf theory, particularly in how information can be restricted to smaller sets and still retain significant meaning in broader contexts.
Mayer-Vietoris Sequence: The Mayer-Vietoris Sequence is a powerful tool in algebraic topology that allows for the computation of the homology or cohomology of a topological space by breaking it down into simpler pieces. By considering open cover sets and their intersections, it helps establish a long exact sequence that links the cohomology groups of the individual pieces and their intersection, providing deep insights into the structure of the space and its properties.
Poincaré Duality: Poincaré duality is a fundamental theorem in algebraic topology that relates the k-th cohomology group of a manifold to its (n-k)-th cohomology group, where n is the dimension of the manifold. This duality highlights an intrinsic relationship between the geometry and topology of manifolds, providing insights into how different dimensional features of a manifold interact through their cohomological properties.
Sheaf Cohomology: Sheaf cohomology is a mathematical tool that studies the global properties of sheaves on a topological space by measuring how they fail to be globally trivial. This concept connects various areas such as algebraic geometry, topology, and analysis, allowing for the computation of global sections and relating them to local properties of sheaves through derived functors and long exact sequences.
Singular cohomology: Singular cohomology is a mathematical tool used to associate a sequence of abelian groups or vector spaces with a topological space, capturing its topological features. This concept allows for the study of properties such as connectedness and compactness by relating them to algebraic structures. Singular cohomology plays a vital role in connecting various theories, providing insights into different aspects of topology, including relationships with sequences and other cohomology theories.
Smooth manifold: A smooth manifold is a topological space that locally resembles Euclidean space and has a smooth structure, allowing for the definition of calculus on it. This means that you can do calculus-like operations on the manifold, such as differentiating and integrating, just as you would in regular Euclidean spaces. Smooth manifolds are essential for understanding complex geometric structures and play a crucial role in fields like differential geometry and theoretical physics.
Topological Space: A topological space is a set equipped with a topology, which is a collection of open sets that defines how the points in the set relate to each other. This concept forms the foundation for various mathematical structures, allowing for the formal study of continuity, convergence, and connectedness in a wide range of contexts, including algebraic and geometric settings.