The is a key operation in algebraic topology that connects homology and cohomology groups. It pairs cohomology classes with homology classes, creating a new homology class of lower degree. This fundamental concept helps us understand the relationships between different topological structures.
The cap product has important applications in , , and . It also plays a role in computations involving simplicial and singular cochains, as well as de Rham cohomology. Understanding the cap product is crucial for grasping advanced topics in algebraic topology.
Definition of cap product
The cap product is a fundamental operation in algebraic topology that relates homology and cohomology groups
It provides a way to pair cohomology classes with homology classes, resulting in a new homology class of lower degree
The cap product is denoted by the symbol ⌢ and is defined as a bilinear map Hp(X;R)×Hq(X;R)→Hq−p(X;R), where X is a topological space and R is a commutative ring
Cap product on cochains
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At the cochain level, the cap product is defined as a map Cp(X;R)×Cq(X;R)→Cq−p(X;R), where Cp(X;R) and Cq(X;R) denote the groups of cochains and chains, respectively
The cap product of a p-cochain φ and a q-chain σ is given by φ⌢σ=φ(σ[0,…,p])⋅σ[p,…,q], where σ[0,…,p] is the front p-face of σ and σ[p,…,q] is the back (q−p)-face of σ
The cap product on cochains is compatible with the boundary operator and coboundary operator, satisfying the relation ∂(φ⌢σ)=(−1)p(δφ⌢σ−φ⌢∂σ)
Cap product on cohomology
The cap product on cochains induces a well-defined operation on cohomology and homology groups
Given cohomology classes [φ]∈Hp(X;R) and [σ]∈Hq(X;R), the cap product [φ]⌢[σ] is defined as the homology class of φ⌢σ, where φ and σ are representative cocycles
The cap product on cohomology is independent of the choice of representatives and is a bilinear map Hp(X;R)×Hq(X;R)→Hq−p(X;R)
Graded module structure
The cap product endows the homology groups H∗(X;R) with the structure of a graded module over the H∗(X;R)
For cohomology classes [φ]∈Hp(X;R) and [ψ]∈Hq(X;R), and a homology class [σ]∈Hr(X;R), the cap product satisfies the relation ([φ]⌣[ψ])⌢[σ]=[φ]⌢([ψ]⌢[σ]), where ⌣ denotes the
This graded module structure captures the interaction between the multiplicative structure of cohomology and the additive structure of homology
Properties of cap product
Associativity
The cap product is associative, meaning that for a cohomology class [φ]∈Hp(X;R) and homology classes [σ]∈Hq(X;R) and [τ]∈Hr(X;R), the following equality holds: ([φ]⌢[σ])⌢[τ]=[φ]⌢([σ]⌢[τ])
This property allows for the unambiguous evaluation of iterated cap products
Graded commutativity
The cap product satisfies a graded property involving the cup product
For cohomology classes [φ]∈Hp(X;R) and [ψ]∈Hq(X;R), and a homology class [σ]∈Hr(X;R), the following equality holds: [φ]⌢([ψ]⌢[σ])=(−1)pq([φ]⌣[ψ])⌢[σ]
This property relates the cap product and the cup product, showing how they interact with the grading of cohomology and homology
Naturality
The cap product is natural with respect to continuous maps between topological spaces
Given a continuous map f:X→Y and classes [φ]∈Hp(Y;R) and [σ]∈Hq(X;R), the following equality holds: f∗([φ]⌢f∗[σ])=f∗([φ])⌢[σ], where f∗ and f∗ denote the induced homomorphisms on homology and cohomology, respectively
Naturality ensures that the cap product is compatible with maps between spaces and the functorial properties of homology and cohomology
Cap product with unit
The cap product with the unit element of the cohomology ring acts as the identity on homology
Let 1X∈H0(X;R) be the unit element of the cohomology ring. For any homology class [σ]∈Hq(X;R), the following equality holds: 1X⌢[σ]=[σ]
This property shows that the cap product with the unit element preserves homology classes
Cap product and cup product
Cap product as adjoint to cup product
The cap product and the cup product are related by an adjointness property
For cohomology classes [φ]∈Hp(X;R) and [ψ]∈Hq(X;R), and a homology class [σ]∈Hr(X;R), the following equality holds: ⟨[φ]⌣[ψ],[σ]⟩=⟨[φ],[ψ]⌢[σ]⟩, where ⟨⋅,⋅⟩ denotes the Kronecker pairing between cohomology and homology
This adjointness property establishes a duality between the cap product and the cup product
Projection formula
The projection formula relates the cap product, cup product, and the pushforward homomorphism in homology
Given a continuous map f:X→Y, a cohomology class [φ]∈Hp(Y;R), and a homology class [σ]∈Hq(X;R), the following equality holds: f∗(f∗([φ])⌢[σ])=[φ]⌢f∗([σ])
The projection formula is useful in computations involving the cap product and maps between spaces
Leray-Hirsch theorem
The is a powerful result that uses the cap product to relate the cohomology of a fiber bundle to the cohomology of its base and fiber
Let π:E→B be a fiber bundle with fiber F. If there exist cohomology classes {ei} in H∗(E;R) such that their restrictions {ei∣Fb} form a basis for H∗(Fb;R) for each fiber Fb, then the cohomology of E is isomorphic to the tensor product of the cohomology of B and the cohomology of F
The cap product is used in the proof of the Leray-Hirsch theorem to establish the isomorphism between the cohomology groups
Applications of cap product
Poincaré duality
Poincaré duality is a fundamental result in algebraic topology that relates the homology and cohomology of orientable manifolds
For a closed, orientable n-dimensional manifold M, the cap product with a fundamental class [M]∈Hn(M;R) induces isomorphisms D:Hk(M;R)→Hn−k(M;R) for all k, given by D([φ])=[φ]⌢[M]
Poincaré duality establishes a deep connection between the homology and cohomology of a manifold, with the cap product playing a central role
Thom isomorphism
The Thom isomorphism is another important application of the cap product in the context of vector bundles and Thom spaces
Let π:E→B be an oriented vector bundle of rank n over a base space B, and let M be the Thom space of E. The cap product with the Thom class uE∈Hn(M;R) induces isomorphisms Φ:Hk(B;R)→Hk+n(M;R) for all k, given by Φ([φ])=π∗([φ])⌢uE
The Thom isomorphism relates the cohomology of the base space to the cohomology of the Thom space, with the cap product and the Thom class playing essential roles
Gysin homomorphism
The , also known as the umkehr map, is a homomorphism between cohomology groups induced by the cap product and the pushforward in homology
Let f:X→Y be a continuous map between oriented manifolds of codimension k. The Gysin homomorphism f!:H∗(X;R)→H∗+k(Y;R) is defined by f!([φ])=f∗([φ]⌢[X]), where [X] is a fundamental class of X
The Gysin homomorphism allows for the transfer of cohomological information from the domain to the codomain of a map, using the cap product and the pushforward
Intersection theory
The cap product plays a crucial role in intersection theory, which studies the intersections of submanifolds and cycles in a manifold
Given submanifolds M and N of complementary dimensions in a manifold X, the intersection product [M]⋅[N] can be defined using the cap product as [M]⋅[N]=(D−1[M])⌢[N], where D is the Poincaré duality isomorphism
The cap product allows for the definition and computation of intersection products, which provide valuable geometric and topological information about the manifold and its subspaces
Computational aspects
Cap product on simplicial cochains
The cap product can be computed explicitly on simplicial cochains and chains
Given a simplicial complex K and a commutative ring R, the cap product of a simplicial p-cochain φ∈Cp(K;R) and a simplicial q-chain σ∈Cq(K;R) is defined as φ⌢σ=∑τ∈K(q−p)φ(σ∣[0,…,p])⋅τ, where σ∣[0,…,p] denotes the restriction of σ to its front p-face and the sum runs over all (q−p)-simplices τ of K
The simplicial cap product is compatible with the simplicial boundary and coboundary operators, allowing for the computation of the cap product on simplicial cohomology and homology
Cap product on singular cochains
The cap product can also be computed on singular cochains and chains
Given a topological space X and a commutative ring R, the cap product of a singular p-cochain φ∈Cp(X;R) and a singular q-chain σ:Δq→X is defined as φ⌢σ=φ(σ∣[0,…,p])⋅σ∣[p,…,q], where σ∣[0,…,p] and σ∣[p,…,q] denote the restrictions of σ to its front p-face and back (q−p)-face, respectively
The singular cap product is compatible with the singular boundary and coboundary operators, allowing for the computation of the cap product on singular cohomology and homology
Cap product in de Rham cohomology
In the context of de Rham cohomology, the cap product can be defined using differential forms and currents
Given a smooth manifold M, the cap product of a differential p-form ω∈Ωp(M) and a q-current T∈Dq(M) is defined as ω⌢T=T(ω∧⋅), where ω∧⋅ denotes the wedge product of ω with the argument of T
The cap product in de Rham cohomology is compatible with the exterior derivative and the boundary operator on currents, allowing for the computation of the cap product on de Rham cohomology and homology
Generalizations and variants
Equivariant cap product
The cap product can be generalized to the equivariant setting, where a group action is present
Given a G-space X and a commutative ring R, the is a bilinear map HGp(X;R)×HqG(X;R)→Hq−pG(X;R), where HG∗(X;R) and H∗G(X;R) denote the equivariant cohomology and homology groups, respectively
The equivariant cap product satisfies properties analogous to those of the ordinary cap product, taking into account the group action
Cap product in sheaf cohomology
The cap product can be defined in the context of sheaf cohomology, which is a cohomology theory for sheaves on topological spaces
Given a topological space X and a sheaf of modules F on X, the cap product is a bilinear map Hp(X;F)×Hq(X;F)→Hq−p(X;F), where H∗(X;F) and H∗(X;F) denote the sheaf cohomology and homology groups, respectively
The satisfies properties similar to those of the ordinary cap product, adapted to the sheaf-theoretic setting
Cap product in extraordinary cohomology theories
The cap product can be generalized to extraordinary cohomology theories, such as K-theory and cobordism theory
In extraordinary cohomology theories, the cap product is defined using the specific constructions and structures of each theory
For example, in K-theory, the cap product is defined using vector bundles and the external tensor product, while in cobordism theory, it is defined using manifolds and the cartesian product
The properties of the may differ from those in ordinary cohomology, depending on the specific features of each theory
Key Terms to Review (24)
Alexander Grothendieck: Alexander Grothendieck was a renowned French mathematician who made groundbreaking contributions to algebraic geometry, homological algebra, and category theory. His work revolutionized the way these fields were understood, particularly through his development of concepts such as sheaves, schemes, and cohomology theories, connecting various mathematical areas and providing deep insights into their structure.
Associativity: Associativity is a property of certain binary operations that states that the way in which the operands are grouped does not affect the result of the operation. This concept is essential in various mathematical structures, especially in algebraic systems like rings and products. In the context of cohomology, associativity ensures that operations such as the cup product and cap product can be performed in any order, making calculations more flexible and coherent within the algebraic framework.
Cap Product: The cap product is a fundamental operation in algebraic topology that combines elements from homology and cohomology theories to produce a new cohomology class. This operation helps connect the topological structure of a space with its algebraic properties, allowing for deeper insights into how different dimensions interact within that space.
Cap Product in Extraordinary Cohomology Theories: The cap product is a fundamental operation in cohomology theories that allows the combination of cohomology classes with homology classes. It provides a way to pair cohomology elements with homology elements, producing a new cohomology class that retains significant topological information about the space. This operation not only connects different dimensions of cohomology and homology but also plays a crucial role in the formulation of duality theorems and other important results in algebraic topology.
Cap product in sheaf cohomology: The cap product in sheaf cohomology is an operation that combines a cohomology class from a sheaf with a class in a homology group, resulting in a new cohomology class. This operation is vital for understanding the interaction between topology and algebraic geometry, as it allows for the computation of intersection numbers and various dualities within the framework of sheaf theory. The cap product also establishes a connection between cohomology and homology, making it an essential tool in algebraic topology.
Cap product on manifolds: The cap product on manifolds is a fundamental operation in algebraic topology that combines a cohomology class with a homology class, resulting in a new cohomology class. This operation provides a way to 'cap' a homology class with a cohomology class, allowing for a deeper understanding of the relationships between these classes in the context of manifold theory. The cap product plays a crucial role in various topological invariants and intersection theory.
Cap product with fundamental class: The cap product with fundamental class is an operation in algebraic topology that combines cohomology classes with a fundamental class of a manifold, providing a way to compute intersection numbers and relate different topological spaces. This operation reflects how cohomological structures interact with the geometry of manifolds, especially in the context of compact oriented manifolds. It plays a crucial role in understanding duality theories and relates to Poincaré duality by giving insights into how homology and cohomology groups are connected.
Cochain Complex: A cochain complex is a sequence of abelian groups or modules connected by homomorphisms, where the composition of consecutive homomorphisms is zero. It serves as a crucial structure in cohomology theory, enabling the computation of cohomology groups that capture topological features of spaces. The relationship between cochain complexes and simplicial complexes highlights how geometric data can translate into algebraic invariants.
Cohomology Ring: The cohomology ring is a mathematical structure that combines cohomology groups into a graded ring using the cup product operation. It encapsulates topological information about a space, allowing one to perform algebraic manipulations that reveal deeper insights into its geometric properties.
Commutativity: Commutativity is a fundamental property in mathematics that states the order of operations does not affect the result. This concept is essential in many areas, including algebra and topology, as it simplifies calculations and relationships between structures. In the context of algebraic structures like rings and groups, commutativity ensures that elements can be combined in any order, which plays a crucial role in various operations such as homology and cohomology.
Cup product: The cup product is an operation in cohomology that combines two cohomology classes to produce a new cohomology class, allowing us to create a ring structure from the cohomology groups of a topological space. This operation plays a key role in understanding the algebraic properties of cohomology, connecting various concepts such as the cohomology ring, cohomology operations, and the Künneth formula.
Dualizing sheaves: Dualizing sheaves are specific sheaves associated with a scheme that allow for the formulation of duality theories in algebraic geometry, particularly in the context of coherent sheaves. They provide a way to generalize the notion of duality between various cohomological constructions, linking homological properties with geometric intuition. Understanding dualizing sheaves is essential for grasping how cap products work, as they play a pivotal role in relating cycles and cohomology classes through intersection theory.
Equivariant cap product: The equivariant cap product is a construction in algebraic topology that combines elements from the cohomology of a space and its symmetry group, creating a new cohomology class. This operation allows us to incorporate the action of a group on a space into the study of its topological properties, providing a way to relate equivariant cohomology with standard cohomological operations like the cap product. By doing so, it opens up further insights into how symmetries interact with the topology of spaces.
Finite type: Finite type refers to a property of cohomology theories or algebraic structures where the generated elements can be represented by a finite number of generators. This concept is crucial when discussing the cap product, as it implies that the cohomology groups involved have a manageable structure, making calculations and theoretical analysis more straightforward.
Graded algebra: A graded algebra is a type of algebraic structure that consists of a direct sum of components, each assigned a degree, such that the product of any two elements from these components results in an element from a specific component. This organization allows for the management of complex algebraic structures by categorizing elements based on their degrees. Graded algebras are crucial in various mathematical contexts, including topology and cohomology, where they facilitate operations like the cap product.
Gysin Homomorphism: The Gysin homomorphism is a fundamental concept in algebraic topology that arises from the intersection theory of cohomology classes. It provides a way to relate the cohomology groups of a manifold and its submanifolds, allowing for computations involving cap products and pushforward operations. This homomorphism helps to analyze the behavior of cohomological operations when applied to fiber bundles and other geometric structures.
Henri Poincaré: Henri Poincaré was a French mathematician, theoretical physicist, and philosopher of science, known for his foundational contributions to topology, dynamical systems, and the philosophy of mathematics. His work laid important groundwork for the development of modern topology and homology theory, influencing how mathematicians understand spaces and their properties.
Intersection Theory: Intersection theory is a branch of algebraic topology that studies how subspaces intersect within a given space. It connects various topological concepts, enabling the calculation of intersection numbers, which measure how many times and in what manner two or more subspaces meet. This concept plays a crucial role in understanding cohomology rings, products of cohomology classes, dualities, and classes associated with manifolds.
Künneth Formula: The Künneth Formula is a powerful result in algebraic topology that describes how the homology or cohomology groups of the product of two topological spaces relate to the homology or cohomology groups of the individual spaces. It provides a way to compute the homology or cohomology of a product space based on the known properties of its components, connecting directly to various aspects of algebraic topology, including operations and duality.
Leray-Hirsch Theorem: The Leray-Hirsch Theorem provides a way to compute the cohomology of a fibration, specifically relating the cohomology of a base space and a fiber to that of the total space. It essentially states that if you have a fibration with certain conditions, the cohomology ring of the total space can be expressed as a tensor product of the cohomology ring of the base space and the cohomology of the fiber. This theorem is crucial in understanding how properties of spaces relate through continuous mappings.
Local cohomology: Local cohomology is a branch of algebraic topology that studies the properties of sheaves and their cohomological aspects in the vicinity of a specific subspace. It provides a way to analyze the behavior of global sections of sheaves when they are localized around a point or a closed subset, which connects well with various concepts, including cap products and sheaf cohomology.
Module over a ring: A module over a ring is a generalization of the concept of vector spaces, where the scalars come from a ring instead of a field. Just like vector spaces allow for the linear combination of vectors using scalars, modules enable the combination of elements from a set using elements from a ring, providing a framework for linear algebra in more abstract settings. This concept is essential in understanding operations and structures in various mathematical contexts, particularly in algebraic topology.
Poincaré Duality: Poincaré Duality is a fundamental theorem in algebraic topology that establishes a relationship between the cohomology groups of a manifold and its homology groups, particularly in the context of closed oriented manifolds. This duality implies that the k-th cohomology group of a manifold is isomorphic to the (n-k)-th homology group, where n is the dimension of the manifold, revealing deep connections between these two areas of topology.
Thom Isomorphism: The Thom Isomorphism is a fundamental result in algebraic topology that establishes a connection between the cohomology of a manifold and the cohomology of its total space when considering vector bundles. This theorem shows how the cohomology ring of a manifold can be understood in terms of its fiber over a point, which relates it closely to concepts like the cap product and the Euler class, allowing us to derive deep insights into the topology of vector bundles.