de Rham cohomology of a smooth manifold 
is the cohomology of its cochain complex of smooth differential
k-forms with exterior derivative 
. Writing 
for the vector space
of these forms, its 
th
cohomology group is
 |
where 
is a nonnegative integer and 
(Sadun 2016, p. 33). The numerator consists
of closed forms and the denominator of exact
forms. The identity 
ensures that every exact
form is a closed form. Thus 
iff every closed
form of degree 
is exact. For 
,
this asks whether a differential k-form 
with 
is the exterior
derivative of a differential (k-1)-form.
The de Rham theorem identifies 
naturally with the singular
cohomology 
(Park, p. 1), and also with Alexander-Spanier
cohomology with real coefficients (Connes and Moscovici 1990, p. 348). Compactness
of 
is not required for these identifications. Consequently, de Rham cohomology is unchanged
under homeomorphisms.
Requiring the differential forms to have compact support gives a different cochain
complex. Its cohomology 
uses closed forms
of compact support modulo exterior
derivatives of forms of compact support (Sadun
2016, p. 49). These groups agree with ordinary de Rham cohomology when 
is compact,
but need not agree otherwise.
