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 .