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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