Cap product

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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[edit]

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[edit]

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.

See also[edit]

References[edit]