Spectral sequences are powerful tools to compute the homology groups of spaces that are difficult to compute by other means.
I will show how this method of spectral sequences can be used to find the homology groups of the loop space of n-dimensional spheres.
Practice shows that it is a good idea to learn about this topic in several steps: as a first step, it is not necessary to be able to understand
the rigorous statements and proofs of all the corresponding theorems. It is enough to look at a couple of examples in order understand
how to use spectral sequences in practice: that is exactly what I am going to show.