The is a fundamental operation in algebraic topology that combines cohomology classes to create new ones. It provides a multiplicative structure on cohomology groups, turning them into a graded ring and offering insights into a space's topological structure.
This operation is crucial for understanding the of a space, which encodes important topological and algebraic information. The cup product's properties, such as and , make it a powerful tool for studying topological spaces and computing invariants.
Definition of cup product
The cup product is a fundamental operation in algebraic topology that combines cohomology classes to produce new cohomology classes
It provides a multiplicative structure on the cohomology groups of a topological space, turning them into a graded ring
The cup product is defined using the diagonal map and the cross product, making it intrinsically related to the topological structure of the space
Algebraic structure
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The cup product endows the cohomology groups H∗(X;R) of a space X with coefficients in a ring R with the structure of a graded commutative ring
For cohomology classes α∈Hp(X;R) and β∈Hq(X;R), their cup product α∪β belongs to Hp+q(X;R)
The cup product is bilinear and compatible with the grading, meaning that it satisfies the relation ∣α∪β∣=∣α∣+∣β∣, where ∣⋅∣ denotes the degree of a cohomology class
Cohomology ring
The cohomology groups of a space X with coefficients in a ring R, together with the cup product, form a graded ring known as the cohomology ring H∗(X;R)
The cohomology ring encodes important topological and algebraic information about the space X
The structure of the cohomology ring can be used to distinguish between different topological spaces and to study their properties, such as their homotopy type and
Properties of cup product
The cup product satisfies several important properties that reflect its geometric and algebraic nature
These properties make the cup product a powerful tool for studying the cohomology of topological spaces and for computing topological invariants
Graded commutativity
The cup product is graded commutative, meaning that for cohomology classes α∈Hp(X;R) and β∈Hq(X;R), we have α∪β=(−1)pqβ∪α
This property reflects the sign convention in the definition of the cup product and the graded nature of cohomology
Graded commutativity implies that the cohomology ring H∗(X;R) is a graded commutative ring, which has important algebraic consequences
Associativity
The cup product is associative, meaning that for cohomology classes α,β,γ, we have (α∪β)∪γ=α∪(β∪γ)
Associativity ensures that the order in which we perform the cup product does not matter, making it a well-defined operation on cohomology classes
The associativity of the cup product is a consequence of the associativity of the cross product and the properties of the diagonal map
Naturality
The cup product is natural with respect to continuous maps between topological spaces
Given a continuous map f:X→Y, the induced homomorphism f∗:H∗(Y;R)→H∗(X;R) is a ring homomorphism with respect to the cup product
Naturality means that the cup product commutes with induced homomorphisms, i.e., f∗(α∪β)=f∗(α)∪f∗(β)
Functoriality
The cup product is functorial, meaning that it is compatible with the composition of continuous maps
If f:X→Y and g:Y→Z are continuous maps, then (g∘f)∗=f∗∘g∗ as ring homomorphisms with respect to the cup product
Functoriality allows us to study the behavior of the cup product under the composition of maps and to relate the cohomology rings of different spaces
Computational techniques
Computing the cup product can be challenging, especially in higher dimensions or for spaces with complex topological structure
Several computational techniques and formulas have been developed to facilitate the calculation of the cup product in various situations
Künneth formula
The is a powerful tool for computing the cohomology ring of a product space X×Y in terms of the cohomology rings of X and Y
It states that there is a natural isomorphism of H∗(X×Y;R)≅H∗(X;R)⊗RH∗(Y;R), where ⊗R denotes the tensor product over the coefficient ring R
The cup product on the cohomology of the product space corresponds to the tensor product of the cup products on the factors, making it easier to compute in many cases
Cross product vs cup product
The cup product is closely related to the cross product, which is an operation on the cohomology of two spaces X and Y that produces a cohomology class on their product X×Y
The cross product α×β of cohomology classes α∈Hp(X;R) and β∈Hq(Y;R) belongs to Hp+q(X×Y;R)
The cup product can be defined in terms of the cross product and the diagonal map Δ:X→X×X as α∪β=Δ∗(α×β), providing a relation between these two operations
Applications of cup product
The cup product has numerous applications in algebraic topology and related fields, as it provides a powerful tool for studying the topological and algebraic properties of spaces
Many important topological invariants and constructions can be defined and studied using the cup product
Topological invariants
The cup product can be used to define various topological invariants, which are algebraic objects that capture important properties of topological spaces
Examples of such invariants include the Euler class, the Stiefel-Whitney classes, and the Pontryagin classes, which are cohomology classes that measure the twisting and non-orientability of vector bundles
These invariants play a crucial role in the classification of manifolds and the study of characteristic classes
Characteristic classes
Characteristic classes are cohomology classes that are naturally associated with vector bundles and provide a way to measure their topological properties
The cup product is used to define operations on characteristic classes, such as the Whitney sum formula and the splitting principle
Characteristic classes, such as the Chern classes and the Pontryagin classes, have important applications in geometry, topology, and mathematical physics
Massey products
Massey products are higher-order cohomological operations that generalize the cup product and provide a way to detect finer topological information
They are defined using a sequence of cohomology classes satisfying certain relations, and their non-triviality can be used to distinguish between homotopy types of spaces
Massey products have applications in obstruction theory, the study of higher-order cohomology operations, and the classification of topological spaces
Examples of cup product
Studying concrete examples of the cup product can help develop intuition and understanding of its properties and behavior
Examples in low dimensions and on familiar spaces, such as spheres and projective spaces, provide a good starting point for exploring the cup product
On spheres and projective spaces
The cohomology rings of spheres and projective spaces are well-understood and provide classic examples of the cup product
For the n-sphere Sn, the cohomology ring is H∗(Sn;Z)≅Z[x]/(x2), where x is a generator of degree n, and the cup product is determined by x∪x=0
For the real projective space RPn, the cohomology ring with Z/2Z coefficients is H∗(RPn;Z/2Z)≅(Z/2Z)[x]/(xn+1), where x is a generator of degree 1, and the cup product is given by the polynomial multiplication
In low dimensions
In low dimensions, the cup product can often be computed explicitly using geometric arguments or by direct calculation
For surfaces, such as the torus or the oriented surface of genus g, the cup product can be determined using the intersection form and the
In dimension 3, the cup product can be related to the linking number of knots and the Borromean rings, providing a connection between cohomology and knot theory
Cup product and duality
The cup product is closely related to various duality theorems in algebraic topology, which establish relationships between homology and cohomology
Duality provides a powerful framework for studying the cup product and its properties
Poincaré duality
Poincaré duality is a fundamental result in algebraic topology that relates the homology and cohomology of orientable manifolds
For an orientable closed n-manifold M, Poincaré duality states that there is an isomorphism Hk(M;R)≅Hn−k(M;R) for any coefficient ring R
The cup product and the cap product are related by Poincaré duality, providing a way to translate between cohomological and homological operations
Cap product and duality
The cap product is a bilinear operation that pairs cohomology classes with homology classes, producing new homology classes
It is denoted by ⌢ and is defined as Hp(X;R)×Hq(X;R)→Hq−p(X;R), where Hq(X;R) denotes the homology of X with coefficients in R
The cap product and the cup product are related by the formula (α∪β)⌢γ=α⌢(β⌢γ), which expresses the compatibility between these operations and provides a way to compute one in terms of the other
Relationship to other operations
The cup product is one of several important cohomological operations in algebraic topology, and it is closely related to other operations that provide additional structure and insight
Steenrod operations
Steenrod operations are a family of cohomological operations that generalize the cup product and provide a way to study the cohomology of spaces with coefficients in a field
The Steenrod squares Sqi are operations on the cohomology with Z/2Z coefficients, while the Steenrod reduced powers Pi are operations on the cohomology with Z/pZ coefficients for odd primes p
Steenrod operations satisfy certain algebraic relations, such as the Cartan formula and the Adem relations, which can be used to compute them in terms of the cup product
Pontryagin product
The Pontryagin product is a cohomological operation that is defined on the homology of the loop space ΩX of a topological space X
It is induced by the composition of loops and provides a multiplicative structure on the homology of ΩX, turning it into a graded ring
The Pontryagin product is related to the cup product via the transgression map, which relates the cohomology of X to the homology of ΩX
Lusternik-Schnirelmann category
The Lusternik-Schnirelmann category (LS-category) of a topological space X is a numerical invariant that measures the complexity of X in terms of the number of open sets required to cover it contractibly
The cup product can be used to give lower bounds for the LS-category, as the cup length of the cohomology ring H∗(X;R) provides a lower bound for the LS-category of X
The relationship between the cup product and the LS-category has important applications in critical point theory and the study of topological complexity
Key Terms to Review (18)
Algebras over a field: Algebras over a field are mathematical structures that combine elements of both vector spaces and rings, allowing for the operations of addition, scalar multiplication, and multiplication of elements. They provide a framework to study algebraic objects that retain the properties of a vector space while also supporting an associative multiplication operation that is distributive over addition. This connection is crucial in various areas of mathematics, including cohomology theory, where these algebras help in understanding the cup product and its properties.
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.
Characteristic classes: Characteristic classes are a way to associate cohomology classes to vector bundles, providing a powerful tool for understanding the geometry and topology of manifolds. They offer insights into the nature of vector bundles, their transformations, and how they relate to the underlying space's topology through cohomological 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.
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.
De Rham cohomology: De Rham cohomology is a type of cohomology theory that uses differential forms to study the topology of smooth manifolds. It provides a powerful bridge between calculus and algebraic topology, allowing the study of manifold properties through the analysis of smooth functions and their derivatives.
Graded commutativity: Graded commutativity is a property of graded algebras where the product of two elements is not only commutative, but also graded, meaning that the product of two elements of different degrees is zero if they are swapped. This concept plays a crucial role in cohomology theories, where the cup product and cohomology rings utilize this property to ensure the structure of the algebra reflects the underlying topology. Understanding graded commutativity helps in analyzing how cohomology classes interact under operations such as the cup product.
Graded rings: A graded ring is a type of ring that is decomposed into a direct sum of abelian groups, each corresponding to a non-negative integer grade. This structure allows for operations that respect the grading, which means that the product of elements from different grades yields elements in a specific grade based on their indices. Graded rings are essential in cohomology theories, particularly in understanding cup products and their interactions with different degrees.
H^n(x): The notation h^n(x) refers to the nth cohomology group of a topological space x, which is an algebraic structure that encodes information about the shape and features of the space. Cohomology groups help in understanding how different properties of a space are related to each other and provide crucial tools for both computation and classification of spaces, especially in relation to pairs of spaces and products.
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.
John Milnor: John Milnor is a prominent American mathematician known for his contributions to differential topology, particularly in the development of concepts like exotic spheres and Morse theory. His work has significantly influenced various fields such as topology, geometry, and algebraic topology, connecting foundational ideas to more advanced topics in these areas.
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.
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.
Primitive elements: Primitive elements are classes in cohomology theory that represent nontrivial cycles in a given space, often serving as fundamental building blocks for understanding the structure of cohomological operations like the cup product. They play a crucial role in establishing the relationships between cohomology classes and help to define the algebraic structure on cohomology rings.
Singular Cohomology: Singular cohomology is a mathematical tool used in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, allowing us to study its global properties through the use of singular simplices. This concept connects the geometric aspects of spaces with algebraic structures, providing insights into various topological features such as holes and connectivity.
Unit element: A unit element is an identity element in a given algebraic structure that, when combined with any other element of the structure, leaves that element unchanged. This concept is crucial in various operations, especially in the context of cohomology and the cup product, where it helps in understanding the behavior of elements under multiplication and provides a foundation for building more complex structures.
α ∪ β: The expression α ∪ β represents the cup product, a fundamental operation in cohomology theory that combines two cohomology classes to produce a new cohomology class. This operation reflects how the topology of a space can influence algebraic structures and serves as a key tool for understanding interactions between different cohomological dimensions.