So far, we have developed two general techniques for computing fundamental groups. The first is homotopy equivalence, which can often be used to show that one space has the same fundamental group as a simpler one. This was used, for example, in Chapter 7 to show that every contractible space is simply connected, and in Chapter 8 to show that the fundamental group of the punctured plane is infinite cyclic. The second is the Seifert–Van Kampen theorem, which was used in Chapter 10 to compute the fundamental groups of wedge sums, graphs, CW complexes, and surfaces.
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