🧬Cohomology Theory Unit 5 – Cup and cap products

Definition and Basics

• Cup product is a binary operation that combines two cohomology classes to produce a third cohomology class of a higher degree
• Denoted by the symbol $\smile$, the cup product takes two cochains $\alpha \in C^p(X; R)$ and $\beta \in C^q(X; R)$ and produces a cochain $\alpha \smile \beta \in C^{p+q}(X; R)$
  • $X$ represents a topological space and $R$ is a commutative ring with unity
• The cup product is induced by the diagonal map $\Delta: X \to X \times X$, which sends a point $x \in X$ to $(x, x) \in X \times X$
• Cup product is associative, bilinear, and graded-commutative
  • Graded-commutativity means that for cochains $\alpha \in C^p(X; R)$ and $\beta \in C^q(X; R)$, we have $\alpha \smile \beta = (-1)^{pq} \beta \smile \alpha$
• The cup product endows the cohomology ring $H^*(X; R)$ with a graded ring structure
• The unit element of the cohomology ring is the class of the constant map $X \to R$ in $H^0(X; R)$

Geometric Intuition

• Cup product can be visualized as a way to "glue" or "stitch" together two cocycles to create a new cocycle of higher dimension
• Consider two cochains $\alpha \in C^p(X; R)$ and $\beta \in C^q(X; R)$ as functions that assign values to $p$-simplices and $q$-simplices, respectively
• The cup product $\alpha \smile \beta$ assigns a value to a $(p+q)$-simplex by multiplying the values of $\alpha$ and $\beta$ on the front $p$-face and back $q$-face of the simplex
• Geometrically, the cup product measures the "twisting" or "linking" of two cocycles
  • If two cocycles are "unlinked," their cup product is zero
  • Non-zero cup products indicate non-trivial twisting or linking of cocycles
• The graded-commutativity of the cup product reflects the orientation of the simplices involved in the product

Cup Product Construction

• The cup product is defined at the cochain level and then shown to be compatible with the coboundary operator, inducing a well-defined product on cohomology
• For cochains $\alpha \in C^p(X; R)$ and $\beta \in C^q(X; R)$, the cup product $\alpha \smile \beta \in C^{p+q}(X; R)$ is defined on a $(p+q)$-simplex $\sigma: \Delta^{p+q} \to X$ by:
  • $(\alpha \smile \beta)(\sigma) = \alpha(\sigma|_{[0, \ldots, p]}) \cdot \beta(\sigma|_{[p, \ldots, p+q]})$
  • $\sigma|_{[0, \ldots, p]}$ and $\sigma|_{[p, \ldots, p+q]}$ denote the restrictions of $\sigma$ to the front $p$-face and back $q$-face, respectively
• The cup product satisfies the Leibniz rule with respect to the coboundary operator $\delta$:
  • $\delta(\alpha \smile \beta) = (\delta \alpha) \smile \beta + (-1)^p \alpha \smile (\delta \beta)$
• As a consequence of the Leibniz rule, the cup product descends to a well-defined product on cohomology:
  • If $[\alpha] \in H^p(X; R)$ and $[\beta] \in H^q(X; R)$ are cohomology classes, then $[\alpha] \smile [\beta] := [\alpha \smile \beta] \in H^{p+q}(X; R)$ is a well-defined cohomology class

Properties of Cup Products

• The cup product is associative:
  • For cochains $\alpha \in C^p(X; R)$, $\beta \in C^q(X; R)$, and $\gamma \in C^r(X; R)$, we have $(\alpha \smile \beta) \smile \gamma = \alpha \smile (\beta \smile \gamma)$
• The cup product is bilinear:
  • For cochains $\alpha, \alpha' \in C^p(X; R)$ and $\beta, \beta' \in C^q(X; R)$, and scalars $a, b \in R$, we have:
    • $(a\alpha + b\alpha') \smile \beta = a(\alpha \smile \beta) + b(\alpha' \smile \beta)$
    • $\alpha \smile (a\beta + b\beta') = a(\alpha \smile \beta) + b(\alpha \smile \beta')$
• The cup product is graded-commutative:
  • For cochains $\alpha \in C^p(X; R)$ and $\beta \in C^q(X; R)$, we have $\alpha \smile \beta = (-1)^{pq} \beta \smile \alpha$
• The cup product is natural with respect to continuous maps:
  • If $f: X \to Y$ is a continuous map and $\alpha \in C^p(Y; R)$ and $\beta \in C^q(Y; R)$ are cochains on $Y$, then $f^*(\alpha \smile \beta) = f^*(\alpha) \smile f^*(\beta)$, where $f^*$ denotes the induced map on cochains
• The cohomology ring $H^*(X; R)$ with the cup product is a graded-commutative ring with unity

Cap Product Introduction

• The cap product is a binary operation that combines a homology class and a cohomology class to produce a homology class of a lower degree
• Denoted by the symbol $\frown$, the cap product takes a cochain $\alpha \in C^p(X; R)$ and a chain $\sigma \in C_q(X; R)$ and produces a chain $\alpha \frown \sigma \in C_{q-p}(X; R)$
  • $X$ represents a topological space and $R$ is a commutative ring with unity
• The cap product is defined at the chain-cochain level and then shown to be compatible with the boundary and coboundary operators, inducing a well-defined product on homology and cohomology
• For a cochain $\alpha \in C^p(X; R)$ and a chain $\sigma \in C_q(X; R)$, the cap product $\alpha \frown \sigma \in C_{q-p}(X; R)$ is defined by:
  • $\alpha \frown \sigma = \alpha(\sigma|_{[0, \ldots, p]}) \cdot \sigma|_{[p, \ldots, q]}$
  • $\sigma|_{[0, \ldots, p]}$ and $\sigma|_{[p, \ldots, q]}$ denote the front $p$-face and back $(q-p)$-face of $\sigma$, respectively
• The cap product satisfies the Leibniz rule with respect to the boundary operator $\partial$ and the coboundary operator $\delta$:
  • $\partial(\alpha \frown \sigma) = (-1)^p (\delta \alpha) \frown \sigma + \alpha \frown (\partial \sigma)$
• As a consequence of the Leibniz rule, the cap product descends to a well-defined product on homology and cohomology:
  • If $[\alpha] \in H^p(X; R)$ is a cohomology class and $[\sigma] \in H_q(X; R)$ is a homology class, then $[\alpha] \frown [\sigma] := [\alpha \frown \sigma] \in H_{q-p}(X; R)$ is a well-defined homology class

Applications in Topology

• Cup and cap products are powerful tools for studying the algebraic topology of spaces
• The cup product can be used to detect the non-triviality of cohomology classes and to distinguish between spaces with isomorphic cohomology groups but different ring structures
  • For example, the torus $T^2$ and the wedge sum of two circles $S^1 \vee S^1$ have isomorphic cohomology groups, but their cohomology rings are different due to the cup product structure
• The cap product provides a connection between homology and cohomology, allowing information to flow between the two theories
• Poincaré duality can be expressed using the cap product:
  • For a closed, oriented $n$-manifold $M$, the cap product with the fundamental class $[M] \in H_n(M; R)$ induces isomorphisms $H^k(M; R) \to H_{n-k}(M; R)$ for all $k$
• The Künneth formula for the cohomology of a product space involves the cup product:
  • For spaces $X$ and $Y$, there is a natural isomorphism $H^*(X; R) \otimes_R H^*(Y; R) \to H^*(X \times Y; R)$ given by the cross product, which is related to the cup product via the pullback of the projection maps
• Cup and cap products play a role in the definition and properties of characteristic classes, such as Stiefel-Whitney classes, Chern classes, and Pontryagin classes, which are important invariants in algebraic topology and differential geometry

Computational Techniques

• Computing cup and cap products can be challenging, especially for spaces with complicated cohomology rings or large homology groups
• Simplicial and cellular cohomology provide combinatorial models for computing cup and cap products
  • In simplicial cohomology, cochains are functions on simplices, and the cup and cap products are defined using the face and degeneracy maps of the simplicial complex
  • In cellular cohomology, cochains are functions on cells, and the cup and cap products are defined using the boundary and coboundary maps of the cellular complex
• The Alexander-Whitney map is a chain homotopy equivalence between the singular chain complex and the simplicial chain complex of a simplicial set, which can be used to compute cup products in simplicial cohomology
• The Eilenberg-Zilber map is a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors, which can be used to compute cross products and relate them to cup products
• Spectral sequences, such as the Serre spectral sequence and the Eilenberg-Moore spectral sequence, can be used to compute cup and cap products in certain situations, such as for fibrations and loop spaces
• Computer algebra systems, such as Sage and Macaulay2, have packages for computing cup and cap products in various cohomology theories, including simplicial, cellular, and sheaf cohomology

Advanced Topics and Extensions

• The cup product can be generalized to the relative cup product, which is defined for relative cohomology groups $H^*(X, A; R)$ of a pair $(X, A)$
  • The relative cup product is compatible with the long exact sequence of a pair and satisfies a relative version of the Leibniz rule
• The cap product can be generalized to the relative cap product, which is defined for relative homology groups $H_*(X, A; R)$ and absolute cohomology groups $H^*(X; R)$
  • The relative cap product is compatible with the long exact sequences of a pair and satisfies a relative version of the Leibniz rule
• Cup and cap products can be defined for other cohomology theories, such as K-theory, cobordism theory, and generalized cohomology theories
  • In these settings, the cup and cap products may have additional structure or satisfy different properties than in singular cohomology
• The Massey product is a higher-order cohomological operation that generalizes the cup product
  • Massey products are defined for tuples of cohomology classes and measure higher-order linking or obstruction to the triviality of certain cohomological expressions
• The Steenrod squares are cohomology operations that generalize the cup product in mod 2 cohomology
  • Steenrod squares satisfy the Cartan formula, which relates them to the cup product, and the Adem relations, which give a complete set of relations among the squares
• The Steenrod algebra is the algebra of stable cohomology operations in mod p cohomology, generated by the Steenrod squares (for p = 2) or the Steenrod reduced powers (for odd primes p)
  • The Steenrod algebra acts on the mod p cohomology of spaces and satisfies a generalized Cartan formula and Adem relations
• Cup and cap products play a role in the formulation and proof of the Adams spectral sequence, which is a powerful tool for computing stable homotopy groups of spheres and other spaces
  • The differentials in the Adams spectral sequence are related to Massey products and Steenrod operations, which involve cup and cap products