De Rham Cohomology is a very cool sounding term in advanced math. This blog post is a short introduction on how it is defined.
A differential form on a manifold is said to be closed if , and exact if for some of degree one less.
Since , every exact form is closed.
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 .