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.

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, see Simplex.

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 the complex of its cellular chains (or cochains, respectively).

where we are taking tensor products of chain complexes,

on the chain complex, and

is the evaluation map (always 0 except for

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

, we have the long-exact sequence in homology (with coefficients in

) of the pair (M, M - {x}) (See Relative homology) An element

is called the fundamental class for

A fundamental class of

is closed and R-orientable.

is a closed, connected and

), the cap product map

This result is famously called Poincaré duality.

, the construction can be (partially) replicated starting from the mappings

In case X = Y, 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.

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