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. $frown$ is the bilinear map given by :

:$sigma\; frown\; psi\; =\; psi(sigma|\; [v\_0,\; ...,\; v\_q]\; )\; sigma|\; [v\_q,\; ...,\; v\_p]$

where

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

The cap product induces a product on the respective Homology and Cohomology classes, e.g. :

:$frown\; :\; H\_p(X;R)\; imes\; H^q(X;R)\; ightarrow\; H\_\{p-q\}(X;R).$

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

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

ee also

*Poincaré duality
*singular homology
*homology theory

References

*Hatcher, A., " [http://www.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology] ," Cambridge University Press (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.