The singular cohomology of a topological space 
with coefficients in an Abelian
group 
is the cohomology of the cochain
complex obtained by applying the hom functor 
to the singular chain
complex of 
. Its 
th cohomology group is
 |
where the coboundary operator 
is induced by the boundary
operators of the singular chain complex (Hatcher
2002, pp. 197-198). When the coefficients lie in a commutative
ring 
, the cup product makes
into a graded ring (Hatcher 2002, pp. 206-207
and 212). Singular cohomology defines a contravariant
functor and is distinct from singular homology
(Hatcher 2002, p. 201).
