Once defined the spaces of forms and the exterior derivative, we have the following chain of maps

called de Rham complex. A complex is a chain of homomorphisms between abelian groups satisfying $\mathrm{d}\circ\mathrm{d}=0$

We're looking for closed forms that are not exact. Much like in homology, quotients are the right way to go:

Let

be the de Rham Complex associated to the manifold $M$. We define the cohomology groups of $M$