0.11. The 2-dimensional PL Sch¨ onflies Theorem 21 v J J w 1 2 v J J1 w 2 Figure 0.5. Two possibilities for J1 ∪ J2 We close the chapter with an application of Theorem 0.11.4 that fur- nishes a promised technical improvement to the Seifert-van Kampen Theo- rem. Theorem 0.11.5. In the setting of Seifert-van Kampen Theorem 0.2.1, if both φi : π1(U0,x) → π1(Ui,x), i ∈ {1, 2}, are 1-1, then so are ψi : π1(Ui,x) → π1(X, x), i ∈ {1, 2}. Proof. Consider a loop α : ∂I2 → U1 representing an element of π1(U1,x). It suﬃces to show that if α is inessential in X, then α is inessential in U1. Let A1 = X U2 and A2 = X U1, and note that A1 ∩ A2 = ∅. Given a path homotopy h : I2 → X between α and the constant path, choose δ 0 less than the distance in I2 between h−1(A2) and ∂I2 ∪ h−1(A1). Build a compact PL 2-dimensional ∂-manifold M ⊂ I2 satisfying h−1(A 2 ) ⊂ Int M ⊂ M ⊂ N(h−1(A 2 ) δ) ⊂ Int I2 h−1(A 1 ). List the components J1,J2,...,Jk of ∂M. Each Jj is a PL simple closed curve by Theorem 0.11.4, Jj bounds a disk Dj ⊂ Int I2. Order the Jj’s with the innermost ones listed first – specifically, order so that s j implies Js ∩ Dj = ∅. Since J1 is an innermost curve, ∂M ∩ Int D1 = ∅, which yields that either D1 ⊂ M or D1 ∩M = J1 consequently, either h(D1) ∩A1 = ∅ or h(D1)∩A2 = ∅. This means J1 ⊂ h−1(U0) is mapped to a nullhomotopic loop in either U2 or U1, and by hypothesis then h(J1) must be nullhomotopic in U0. Hence, h = h0 can be modified to produce a map h1 : I2 → X satisfying (1) h1|I2 Int D1 = h|I2 Int D1 (2) h1(D1) ⊂ U0. Repeating, we recursively produce additional maps h2,...,hk : I2 → X such that for t = 1, 2,...,k (3) ht|I2 Int Dt = ht−1|I2 Int Dt (4) ht(∪i=1Di) t ⊂ U0. In particular, condition (3) insures that ht|∂I2 = h|∂I2. Assuming h0,h1, . . . , ht satisfy the preceding, we observe that Int Dt+1 ∪t i=1 Di either misses

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