5.1 Cup product

The cup product 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 cohomology ring of a space, which encodes important topological and algebraic information. The cup product's properties, such as graded commutativity and associativity, 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 $\alpha \in H^p(X; R)$ and $\beta \in H^q(X; R)$, their cup product $\alpha \cup \beta$ belongs to $H^{p+q}(X; R)$
• The cup product is bilinear and compatible with the grading, meaning that it satisfies the relation $|\alpha \cup \beta| = |\alpha| + |\beta|$, where $|\cdot|$ 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 characteristic classes

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 $\alpha \in H^p(X; R)$ and $\beta \in H^q(X; R)$, we have $\alpha \cup \beta = (-1)^{pq} \beta \cup \alpha$
• 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 $\alpha, \beta, \gamma$, we have $(\alpha \cup \beta) \cup \gamma = \alpha \cup (\beta \cup \gamma)$
• 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 $f: X \to Y$, the induced homomorphism $f^*: H^*(Y; R) \to H^*(X; R)$ is a ring homomorphism with respect to the cup product
• Naturality means that the cup product commutes with induced homomorphisms, i.e., $f^*(\alpha \cup \beta) = f^*(\alpha) \cup f^*(\beta)$

Functoriality

• The cup product is functorial, meaning that it is compatible with the composition of continuous maps
• If $f: X \to Y$ and $g: Y \to Z$ are continuous maps, then $(g \circ f)^* = f^* \circ g^*$ 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 Künneth formula is a powerful tool for computing the cohomology ring of a product space $X \times Y$ in terms of the cohomology rings of $X$ and $Y$
• It states that there is a natural isomorphism of graded rings $H^*(X \times Y; R) \cong H^*(X; R) \otimes_R H^*(Y; R)$, where $\otimes_R$ 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 $X \times Y$
• The cross product $\alpha \times \beta$ of cohomology classes $\alpha \in H^p(X; R)$ and $\beta \in H^q(Y; R)$ belongs to $H^{p+q}(X \times Y; R)$
• The cup product can be defined in terms of the cross product and the diagonal map $\Delta: X \to X \times X$ as $\alpha \cup \beta = \Delta^*(\alpha \times \beta)$, 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 $S^n$, the cohomology ring is $H^*(S^n; \mathbb{Z}) \cong \mathbb{Z}[x]/(x^2)$, where $x$ is a generator of degree $n$, and the cup product is determined by $x \cup x = 0$
• For the real projective space $\mathbb{RP}^n$, the cohomology ring with $\mathbb{Z}/2\mathbb{Z}$ coefficients is $H^*(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}) \cong (\mathbb{Z}/2\mathbb{Z})[x]/(x^{n+1})$, 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 Poincaré duality
• 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 $H^k(M; R) \cong H_{n-k}(M; R)$ 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 $\frown$ and is defined as $H^p(X; R) \times H_q(X; R) \to H_{q-p}(X; R)$, where $H_q(X; R)$ denotes the homology of $X$ with coefficients in $R$
• The cap product and the cup product are related by the formula $(\alpha \cup \beta) \frown \gamma = \alpha \frown (\beta \frown \gamma)$, 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 $Sq^i$ are operations on the cohomology with $\mathbb{Z}/2\mathbb{Z}$ coefficients, while the Steenrod reduced powers $P^i$ are operations on the cohomology with $\mathbb{Z}/p\mathbb{Z}$ 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 $\Omega X$ of a topological space $X$
• It is induced by the composition of loops and provides a multiplicative structure on the homology of $\Omega X$, 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 $\Omega X$

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