Roughly, we are interested in studying the set [(X,x),(Y,y)] of homotopy classes of continuous maps between the topological spaces X and Y preserving the base points x and y. For example, if X is the 1-sphere, we obtain the fundamental group. For higher dimensional spheres we get even abelian groups. These are the homotopy groups of (Y,y). Slogan: easy to define but hard to compute. The notions of fibrations and cofibrations and other related concepts help us to get some systematic approach in understanding what is going on.
As in homology theory we will encounter long exact sequences relating these groups in a fairly general setting. We also plan give some glimpses into stable homotopy theory. We will see that its objects, the so called spectra, give rise to (generalized) homology theories. We already know singular homology theory from Topology 2, but there are many more and very interesting ones, like K-theory and Bordism theory.

The course will be in English or German, depending on the audience.