Cap product

In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. 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 in the following way. Using CW approximation we may assume that X is a CW-complex and \( C_\bullet(X) \) (and \( {\displaystyle C^{\bullet }(X)} \) ) is the complex of its cellular chains (or cochains, respectively). Consider then the composition

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

where we are taking tensor products of chain complexes, \( \Delta \colon X\to X\times X \) is the diagonal map which induces the map \( {\displaystyle \Delta _{*}\colon C_{\bullet }(X)\to C_{\bullet }(X\times X)\cong C_{\bullet }(X)\otimes C_{\bullet }(X)} \) 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

If in the above discussion one replaces \( X\times X \) by \( X\times Y \) , the construction can be (partially) replicated starting from the mappings

\( {\displaystyle C_{\bullet }(X\times Y)\otimes C^{\bullet }(Y)\cong C_{\bullet }(X)\otimes C_{\bullet }(Y)\otimes C^{\bullet }(Y){\overset {\mathrm {Id} \otimes \varepsilon }{\longrightarrow }}C_{\bullet }(X)} \) and
\( {\displaystyle C^{\bullet }(X\times Y)\otimes C_{\bullet }(Y)\cong C^{\bullet }(X)\otimes C^{\bullet }(Y)\otimes C_{\bullet }(Y){\overset {\mathrm {Id} \otimes \varepsilon }{\longrightarrow }}C^{\bullet }(X)} \)

to get, respectively, slant products / :

\( {\displaystyle H_{p}(X\times Y;R)\otimes H^{q}(Y;R)\rightarrow H_{p-q}(X;R)} \) and
\( {\displaystyle H^{p}(X\times Y;R)\otimes H_{q}(Y;R)\rightarrow H^{p-q}(X;R).} \)

In case X = Y, the first one is related to the cap product by the diagonal map: \( {\displaystyle \Delta _{*}(a)/\phi =a\frown \phi } \) .

These ‘products’ are in some ways more like division than multiplication, which is reflected in their notation.

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.