4 0. Prequel The full theorem and proof are presented in (Munkres, 2000, §70). There is also a thorough exposition of the theorem in (Massey, 1967, pp. 113–122) (or (Massey, 1991, pp. 86–96)). Later in the chapter (Theorem 0.11.5) we will prove the following addendum to 0.2.1: if φi : π1(U0,x) π1(Ui,x) is one-to-one for i = 1, 2, then ψi : π1(Ui,x) π1(X, x) is also one-to-one for i = 0, 1, 2. As the next two examples illustrate, the Seifert-van Kampen Theorem can often be used to gain useful information about a fundamental group without explicitly computing the group itself. Example 0.2.2. Examples abound where π1(X, x) is trivial despite non- triviality of all π1(Ui,x). But the theorem immediately gives nontriviality of π1(X, x) when one can locate a group H and pair of homomorphisms ρi : π1(Ui,x) H (i = 1, 2) satisfying the commutativity relationship in the statement with either ρ1 or ρ2 nontrivial. Example 0.2.3. The Seifert-van Kampen Theorem can also be exploited to detect nonabelian fundamental groups. We illustrate with a simple example, in which each π1(Ui,x) is infinite cyclic, φ1 amounts to multiplication by 2, and φ2 to multiplication by 3. To see why π1(X, x) can be nonabelian, simply consider H = S3, the symmetric group on the symbols {a, b, c}, define ρ1 : π1(U1,x) Z S3 by defining ρ1(1) to be the transposition (ab) and ρ2 : π1(U2,x) Z S3 by defining ρ2(1) to be the 3-cycle (abc). Then the trivial homomorphism ρ0 fleshes out the commutative diagram, and the presence of noncommuting elements (ab), (abc) in σ(π1(X, x)) S3 indicates π1(X, x) is nonabelian. A similar argument can be provided whenever φ1, φ2 amount to multiplication by relatively prime integers greater than 1. The Seifert-vanKampen Theorem will be used this way in the next chapter to prove that torus knots are truly knotted. When the theorem is used to compute π1(X, x), it is important to bear in mind that π1(X, x) is generated by the images of ψ1 and ψ2 (Munkres, 2000, Theorem 9-59.1). With this additional piece of information it is relatively easy to see that the version of the Seifert-van Kampen Theorem stated above implies the classical version of the theorem. The classical version asserts that π1(X, x) π1(U1,x) π1(U2,x)/N, where π1(U1,x)∗π1(U2,x) is the free product and N is the normal subgroup generated by elements of the form φ1(g)(φ2(g))−1 with g π1(U0,x). Example 0.2.4. One of the simplest, but most useful, applications of the classical Seifert-van Kampen Theorem is to the computation of the funda- mental group of a 2-dimensional CW complex. Let X be a CW complex that consists of one 0-cell {x0} with 1-cells a1,a2,...,an attached. Then X is a
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