5.2 Cap product

The cap product 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 Poincaré duality, Thom isomorphism, and intersection theory. 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 $\frown$ and is defined as a bilinear map $H^p(X; R) \times H_q(X; R) \to H_{q-p}(X; R)$, 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 $C^p(X; R) \times C_q(X; R) \to C_{q-p}(X; R)$, where $C^p(X; R)$ and $C_q(X; R)$ denote the groups of cochains and chains, respectively
• The cap product of a $p$-cochain $\varphi$ and a $q$-chain $\sigma$ is given by $\varphi \frown \sigma = \varphi(\sigma_{[0,\dots,p]}) \cdot \sigma_{[p,\dots,q]}$, where $\sigma_{[0,\dots,p]}$ is the front $p$-face of $\sigma$ and $\sigma_{[p,\dots,q]}$ is the back $(q-p)$-face of $\sigma$
• The cap product on cochains is compatible with the boundary operator and coboundary operator, satisfying the relation $\partial(\varphi \frown \sigma) = (-1)^p(\delta \varphi \frown \sigma - \varphi \frown \partial \sigma)$

Cap product on cohomology

• The cap product on cochains induces a well-defined operation on cohomology and homology groups
• Given cohomology classes $[\varphi] \in H^p(X; R)$ and $[\sigma] \in H_q(X; R)$, the cap product $[\varphi] \frown [\sigma]$ is defined as the homology class of $\varphi \frown \sigma$, where $\varphi$ and $\sigma$ are representative cocycles
• The cap product on cohomology is independent of the choice of representatives and is a bilinear map $H^p(X; R) \times H_q(X; R) \to H_{q-p}(X; R)$

Graded module structure

• The cap product endows the homology groups $H_*(X; R)$ with the structure of a graded module over the cohomology ring $H^*(X; R)$
• For cohomology classes $[\varphi] \in H^p(X; R)$ and $[\psi] \in H^q(X; R)$, and a homology class $[\sigma] \in H_r(X; R)$, the cap product satisfies the relation $([\varphi] \smile [\psi]) \frown [\sigma] = [\varphi] \frown ([\psi] \frown [\sigma])$, where $\smile$ denotes the cup product
• 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 $[\varphi] \in H^p(X; R)$ and homology classes $[\sigma] \in H_q(X; R)$ and $[\tau] \in H_r(X; R)$, the following equality holds: $([\varphi] \frown [\sigma]) \frown [\tau] = [\varphi] \frown ([\sigma] \frown [\tau])$
• This associativity property allows for the unambiguous evaluation of iterated cap products

Graded commutativity

• The cap product satisfies a graded commutativity property involving the cup product
• For cohomology classes $[\varphi] \in H^p(X; R)$ and $[\psi] \in H^q(X; R)$, and a homology class $[\sigma] \in H_r(X; R)$, the following equality holds: $[\varphi] \frown ([\psi] \frown [\sigma]) = (-1)^{pq}([\varphi] \smile [\psi]) \frown [\sigma]$
• 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 $f: X \to Y$ and classes $[\varphi] \in H^p(Y; R)$ and $[\sigma] \in H_q(X; R)$, the following equality holds: $f_*([\varphi] \frown f_*[\sigma]) = f^*([\varphi]) \frown [\sigma]$, 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 $1_X \in H^0(X; R)$ be the unit element of the cohomology ring. For any homology class $[\sigma] \in H_q(X; R)$, the following equality holds: $1_X \frown [\sigma] = [\sigma]$
• 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 $[\varphi] \in H^p(X; R)$ and $[\psi] \in H^q(X; R)$, and a homology class $[\sigma] \in H_r(X; R)$, the following equality holds: $\langle [\varphi] \smile [\psi], [\sigma] \rangle = \langle [\varphi], [\psi] \frown [\sigma] \rangle$, where $\langle \cdot, \cdot \rangle$ 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 $f: X \to Y$, a cohomology class $[\varphi] \in H^p(Y; R)$, and a homology class $[\sigma] \in H_q(X; R)$, the following equality holds: $f_*(f^*([\varphi]) \frown [\sigma]) = [\varphi] \frown f_*([\sigma])$
• The projection formula is useful in computations involving the cap product and maps between spaces

Leray-Hirsch theorem

• The Leray-Hirsch theorem 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 $\pi: E \to B$ be a fiber bundle with fiber $F$. If there exist cohomology classes $\{e_i\}$ in $H^*(E; R)$ such that their restrictions $\{e_i|_{F_b}\}$ form a basis for $H^*(F_b; R)$ for each fiber $F_b$, 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 $[M] \in H_n(M; R)$ induces isomorphisms $D: H^k(M; R) \to H_{n-k}(M; R)$ for all $k$, given by $D([\varphi]) = [\varphi] \frown [M]$
• 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 $\pi: E \to B$ 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 $u_E \in H^n(M; R)$ induces isomorphisms $\Phi: H^k(B; R) \to H^{k+n}(M; R)$ for all $k$, given by $\Phi([\varphi]) = \pi^*([\varphi]) \frown u_E$
• 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 Gysin homomorphism, also known as the umkehr map, is a homomorphism between cohomology groups induced by the cap product and the pushforward in homology
• Let $f: X \to Y$ be a continuous map between oriented manifolds of codimension $k$. The Gysin homomorphism $f_!: H^*(X; R) \to H^{*+k}(Y; R)$ is defined by $f_!([\varphi]) = f_*([\varphi] \frown [X])$, 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 $[M] \cdot [N]$ can be defined using the cap product as $[M] \cdot [N] = (D^{-1}[M]) \frown [N]$, 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 $\varphi \in C^p(K; R)$ and a simplicial $q$-chain $\sigma \in C_q(K; R)$ is defined as $\varphi \frown \sigma = \sum_{\tau \in K^{(q-p)}} \varphi(\sigma|_{[0,\dots,p]}) \cdot \tau$, where $\sigma|_{[0,\dots,p]}$ denotes the restriction of $\sigma$ to its front $p$-face and the sum runs over all $(q-p)$-simplices $\tau$ 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 $\varphi \in C^p(X; R)$ and a singular $q$-chain $\sigma: \Delta^q \to X$ is defined as $\varphi \frown \sigma = \varphi(\sigma|_{[0,\dots,p]}) \cdot \sigma|_{[p,\dots,q]}$, where $\sigma|_{[0,\dots,p]}$ and $\sigma|_{[p,\dots,q]}$ denote the restrictions of $\sigma$ 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 $\omega \in \Omega^p(M)$ and a $q$-current $T \in \mathcal{D}_q(M)$ is defined as $\omega \frown T = T(\omega \wedge \cdot)$, where $\omega \wedge \cdot$ denotes the wedge product of $\omega$ 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 equivariant cap product is a bilinear map $H^p_G(X; R) \times H_q^G(X; R) \to H_{q-p}^G(X; R)$, where $H^*_G(X; R)$ 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 $\mathcal{F}$ on $X$, the cap product is a bilinear map $H^p(X; \mathcal{F}) \times H_q(X; \mathcal{F}) \to H_{q-p}(X; \mathcal{F})$, where $H^*(X; \mathcal{F})$ and $H_*(X; \mathcal{F})$ denote the sheaf cohomology and homology groups, respectively
• The cap product in sheaf cohomology 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 cap product in extraordinary cohomology theories may differ from those in ordinary cohomology, depending on the specific features of each theory