The cap product pairs a cohomology class with a homology class to give a homology
class of lower degree (Hatcher 2002). With coefficients in a commutative
ring 
,
it has the form
 |
For a closed oriented 
-dimensional manifold, cap product
with its fundamental class implements Poincaré
duality. The degree-zero cohomology class
1 acts as the identity on homology. Versions with local coefficients include the orientation
twist for nonorientable manifolds. A Poincaré
complex is defined by requiring these duality maps to
be isomorphisms.
