In this series of two lectures, we will try to see a proof of the fact that the Homotopy groups of spheres (including $n$-spheres) are finitely generated abelian groups. The understanding of homotopy groups of spheres still remains one of the most difficult problems in topology. In view of this, Serre's result on the finite generation of these groups was one of the most remarkable achievements in this field. The main tool used in proving this theorem is "Spectral Sequence", which was developed by J. P. Serre in a topological context to prove such results.

In the first lecture, we will begin with the definitions of homotopy groups, Serre fibrations, and Eilenberg-Maclane spaces. We will then define the spectral sequence and do few examples to get a little bit of a grasp on this main tool. As applications, we will deduce some results regarding the homology of Eilenberg-Maclane spaces.

Definitions of CW complexes, Homotopy groups, Homology groups. In any case, we can quickly define things if a person is new to some terminology.