Cap product: Information from Answers.com

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.

1 Definition
2 Interpretation
3 Equations
4 See also
5 References

Definition

Let X be a topological space and R a coefficient ring. $\frown$ is the bilinear map given by :

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

where

$\sigma : \Delta\ ^p \rightarrow\ X$ and $\psi \in C^q(X;R).$

The cap product induces a product on the respective homology and cohomology classes, e.g. :

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

Interpretation

In analogy with the interpretation of the cup product in terms of the Kunneth 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 $Δ *$ on the chain complex (more precisely, the map $Δ$ is not cellular, but as detailed in that article, any continuous map of CW complexes is homotopic to a cellular map, so we are in effect considering an associated cellular map to $Δ$ , the choice of homotopic map does not end up mattering when we pass to the quotient), 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 $p < q$ .

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.

References

Hatcher, A., Algebraic Topology, Cambridge University Press (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.

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Cap product

Contents

Definition

Interpretation

Equations

See also

References