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.

Contents

Definition

Equations

Reference

See also

Definition

Let ''X'' be a topological space and ''R'' a coefficient ring. is the bilinear map given by :
:
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. :
:
ightarrow H_{p-q}(X;R).

Equations

The boundary of a cap product is given by :
:
Given a map ''f'' the induced maps satisfy :
:$f\_$
★ ( sigma ) rown psi = f_
★ (sigma rown f^
★ (psi)).
The cap and cup product are related by :
:
where
:$sigma\; :\; Delta\; ^\{p+q\}$
ightarrow X , $psi\; in\; C^q(X;R)$and

Reference

★ 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.

See also

★ Poincaré duality

★ singular homology

★ homology theory