De Rham Cohomology

De Rham Cohomology is a very cool sounding term in advanced math. This blog post is a short introduction on how it is defined.

Definition:
A differential form \omega on a manifold M is said to be closed if d\omega=0, and exact if \omega=d\tau for some \tau of degree one less.

Corollary:
Since d^2=0, every exact form is closed.

Definition:
Let Z^k(M) be the vector space of all closed k-forms on M.

Let B^k(M) be the vector space of all exact k-forms on M.

Since every exact form is closed, hence B^k(M)\subseteq Z^k(M).

The de Rham cohomology of M in degree k is defined as the quotient vector space \displaystyle H^k(M):=Z^k(M)/B^k(M).

The quotient vector space construction induces an equivalence relation on Z^k(M):

w'\sim w in Z^k(M) iff w'-w\in B^k(M) iff w'=w+d\tau for some exact form d\tau.

The equivalence class of a closed form \omega is called its cohomology class and denoted by [\omega].

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