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