De Rham Cohomology is a very cool sounding term in advanced math. This blog post is a short introduction on how it is defined.
Also, do check out our presentation on the relation between De Rham Cohomology and physics: De Rham Cohomology.
Definition:
A differential form on a manifold
is said to be closed if
, and exact if
for some
of degree one less.
Corollary:
Since , every exact form is closed.
Definition:
Let be the vector space of all closed
-forms on
.
Let be the vector space of all exact
-forms on
.
Since every exact form is closed, hence .
The de Rham cohomology of in degree
is defined as the quotient vector space
The quotient vector space construction induces an equivalence relation on :
in
iff
iff
for some exact form
.
The equivalence class of a closed form is called its cohomology class and denoted by
.