Applications of the Van Kampen theorem

Abstract Having got the Van Kampen theorem, how are we to apply it? As with any result, before we try to apply it to a given example, we must check that the conditions necessary for the theorem to hold are satisfied in that example. In order to apply the Van Kampen theorem, the parts into which we divide our space must be open and path connected. As we will see, this sometimes requires an adjust¬ment to the obvious choice of decomposition. Initially we will give nearly all the details of this necessary adjustment, but later on we will leave such details increasingly up to you.