This section is devoted entirely to the fundamental group and in particular to some elementary calculations and further applications.
The two main computational tools in calculating fundamental groups are the theory of covering spaces and the Van Kampen theorem. A deep discussion of covering spaces is beyond the scope of this course. Instead we shall start with a special case already mentioned in the previous section, namely the circle. Then we will state the Van Kampen and give a sketch of its proof. This collection of tools will enable us to carry out a good number of calculations. In particular we will be able to distinguish the homotopy types of the various closed surfaces given in the classification theorem.