Cohomology of Real line.

First example of applications of Cohomology is real line $R$.

To start off a fact about differential forms:

Differential forms belong to spaces of Alternating forms - $A^k(M)$. Whenever $k>n$ where $n$ dimension of tangent space at a give point, differential $k$ forms become $0$.

Since $R$ is connected, can conclude, $H^0(R)=R$. Clearly, all two forms are zero as $n=1$. Note, two forms are generated by one forms. Since all two forms are zero, all one forms are closed.

Note a function such $h(x)$ is a zero form. A one form $f(x)dx$ on $R$ is exact if and only if there exists a $C^\infty$ function $g(x)$ on $R$ such that the following is satisfied.

$f(x)dx = dg = g^'(x) dx$

which means,

$g(x) = \int_0^x f(t) dt$

Thus,

$H^k(R) = R$ when $k=0$, and $H^k(R)=0$ when $k>0$.