0.11. The 2-dimensional PL Sch¨ onflies Theorem 21
v
J2
J1
w
v
J2
J1
w
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(A2)
⊂ Int M ⊂ M ⊂
N(h−1(A2);
δ) ⊂ Int
I2 h−1(A1).
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 ∪i=1Di
t
either misses
