Cap product

In algebraic topology, a branch of mathematics, the cap product defines an association between cohomology and homology of a space.

Definition

Simplex gradually - be a topological space, is the -th singular chain group, ie the free abelian group on the set of all continuous maps of the standard. Denote with or the inclusions of the standard - or -simplex as a " front - dimensional face " or " rear - dimensional face " in the standard - Simplex.

For and a singular simplex ( with ) one defines

And sets it to a linear mapping

Continued.

General was a ring and let. Then we obtain a mapping

From the relation

Follows that the cup-product a well- defined mapping

Defined.

Properties

For continuous maps

With.

The cap product is related to the cup-product of the following equation together:

For,

Application: Poincaré duality

Let be a closed, orientable manifold and

The fundamental class. Then the cap product realized with an isomorphism

For.