Cap product

In algebraic topology the cap product is a method of adjoining a chain of degree with a cochain of degree, such that, to form a composite chain of degree. 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. The cap product is a bilinear map on singular homology and cohomology
defined by contracting a singular chain with a singular cochain by the formula:
Here, the notation indicates the restriction of the simplicial map to its face spanned by the vectors of the base.

Interpretation

In analogy with the interpretation of the cup product in terms of the Künneth formula, we can explain the existence of the cap product in the following way. Using CW approximation we may assume that is a CW-complex and is the complex of its cellular chains. Consider then the composition
where we are taking tensor products of chain complexes, is the diagonal map which induces the map
on the chain complex, and is the evaluation map.
This composition then passes to the quotient to define the cap product, and looking carefully at the above composition shows that it indeed takes the form of maps, which is always zero for.

Fundamental class

For any point in, we have the long-exact sequence in homology of the pair
An element of is called the fundamental class for if is a generator of. A fundamental class of exists if is closed and R-orientable. In fact, if is a closed, connected and -orientable manifold, the map is an isomorphism for all in and hence, we can choose any generator of as the fundamental class.

Relation with Poincaré duality

For a closed -orientable n-manifold with fundamental class in , the cap product map
is an isomorphism for all. This result is famously called Poincaré duality.

The slant product

If in the above discussion one replaces by, the construction can be replicated starting from the mappings
and
to get, respectively, slant products :
and
In case, the first one is related to the cap product by the diagonal map:.
These ‘products’ are in some ways more like division than multiplication, which is reflected in their notation.

Equations

The boundary of a cap product is given by :
Given a map f the induced maps satisfy :
The cap and cup product are related by :
where
If is allowed to be of higher degree than, the last identity takes a more general form
which makes into a right -module.