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.
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 formula_2 with a singular cochain formula_3 by the formula :
Here, the notation formula_5 indicates the restriction of the simplicial map formula_6 to its face spanned by the vectors of the base, see Simplex.
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 formula_7 is a CW-complex and formula_8 (and formula_9) is the complex of its cellular chains (or cochains, respectively). Consider then the composition 
formula_10
where we are taking tensor products of chain complexes, formula_11 is the diagonal map which induces the map formula_12 on the chain complex, and formula_13 is the evaluation map (always 0 except for formula_14).
This composition then passes to the quotient to define the cap product formula_15, and looking carefully at the above composition shows that it indeed takes the form of maps formula_16, which is always zero for formula_17.
The slant product.
If in the above discussion one replaces formula_18 by formula_19, the construction can be (partially) replicated starting from the mappings
to get, respectively, slant products formula_22:
In case "X = Y", the first one is related to the cap product by the diagonal map: formula_25. 
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
An interesting consequence of the last equation is that it makes formula_32 into a right formula_33 module.