# Cap product

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In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that qp, to form a composite chain of degree pq. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938.

## Definition

Let X be a topological space and R a coefficient ring. The cap product is a bilinear map on singular homology and cohomology

$\frown\;: H_p(X;R)\times H^q(X;R) \rightarrow H_{p-q}(X;R).$

defined by contracting a singular chain $\sigma : \Delta\ ^p \rightarrow\ X$ with a singular cochain $\psi \in C^q(X;R),$ by the formula :

$\sigma \frown \psi = \psi(\sigma|_{[v_0, \ldots, v_q]}) \sigma|_{[v_q, \ldots, v_p]}.$

Here, the notation $\sigma|_{[v_0, \ldots, v_q]}$ indicates the restriction of the simplicial map $\sigma$ to its face spanned by the vectors of the base, see Simplex.

## Interpretation

In analogy with the interpretation of the cup product in terms of the Künneth formula, we can explain the existence of the cap product by considering the composition

$C_\bullet(X) \otimes C^\bullet(X) \overset{\Delta_* \otimes \mathrm{Id}}{\longrightarrow} C_\bullet(X) \otimes C_\bullet(X) \otimes C^\bullet(X) \overset{\mathrm{Id} \otimes \varepsilon}{\longrightarrow} C_\bullet(X)$

in terms of the chain and cochain complexes of $X$, where we are taking tensor products of chain complexes, $\Delta \colon X \to X \times X$ is the diagonal map which induces the map $\Delta_*$ on the chain complex, and $\varepsilon \colon C_p(X) \otimes C^q(X) \to \mathbb{Z}$ is the evaluation map (always 0 except for $p=q$).

This composition then passes to the quotient to define the cap product $\frown \colon H_\bullet(X) \times H^\bullet(X) \to H_\bullet(X)$, and looking carefully at the above composition shows that it indeed takes the form of maps $\frown \colon H_p(X) \times H^q(X) \to H_{p-q}(X)$, which is always zero for $p < q$.

## The slant product

The above discussion indicates that the same operation can be defined on cartesian products $X\times Y$ yielding a product

$\backslash\;: H_p(X;R)\otimes H^q(X\times Y;R) \rightarrow H^{q-p}(Y;R).$

In case X = Y, the two products are related by the diagonal map.

## Equations

The boundary of a cap product is given by :

$\partial(\sigma \frown \psi) = (-1)^q(\partial \sigma \frown \psi - \sigma \frown \delta \psi).$

Given a map f the induced maps satisfy :

$f_*( \sigma ) \frown \psi = f_*(\sigma \frown f^* (\psi)).$

The cap and cup product are related by :

$\psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma)$

where

$\sigma : \Delta ^{p+q} \rightarrow X$ , $\psi \in C^q(X;R)$ and $\varphi \in C^p(X;R).$

An interesting consequence of the last equation is that it makes $H_{\ast}(X;R)$ into a right $H^{\ast}(X;R)-$ module.