In algebraic topology the cap product is a method of adjoining a chain with a cochain. 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:

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 is a CW-complex where we are taking tensor products of chain complexes is the diagonal map is the evaluation map).

.

Fundamental class

(See Relative homology)

is called the fundamental class is closed and R-orientable as the fundamental class.

Relation with Poincaré duality

. This result is famously called Poincaré duality.

The slant product

to get, respectively, slant products

.

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

, the last identity takes a more general form

-module.

See also

• Cup product

• Poincaré duality

• Singular homology

• Homology theory

References

• Hatcher, A., Algebraic Topology, Cambridge University Press (2002) . Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.

• Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids.