0.11. The 2-dimensional PL Sch¨ onflies Theorem 19

represents an isomorphism.

Historical Notes. The contractible ∂-manifold M constructed in Exam-

ple 0.10.3 is called a Newman contractible manifold. It is named after

M. H. A. Newman (1948), who was the first to describe contractible ∂-

manifolds this way. The same construction can also be carried out (topolog-

ically) in

S4,

but the 4-dimensional case requires more care to ensure that

M is simply connected (see (Lickorish, 2003)). No such example is possible

in

S3.

Exercise

0.10.1. The homomorphism described in the last paragraph of the proof of

Example 0.10.2 is an epimorphism.

0.11. The 2-dimensional PL Sch¨ onflies Theorem

A simple closed curve in a space X is the image of a continuous one-to-

one function f :

S1

→ X. The classical Jordan Curve Theorem states

that any simple closed curve in

R2

separates

R2

into exactly two compo-

nents, with the simple closed curve being the topological frontier of each

of the complementary domains. In this form the theorem generalizes to

high dimensions—see Exercise 0.3.2. There is another, stronger form of the

Jordan Curve Theorem, called the Sch¨ onflies Theorem, unique to two di-

mensions, which provides context for many of the results in this book. In

effect, the Sch¨ onflies Theorem shows that most of the unusual phenomena to

be studied in this book are high-dimensional and do not occur in dimension

two. This section offers a short review of this key result.

Theorem 0.11.1 (Topological Sch¨ onflies). For any two simple closed curves

P1 and P2 in

R2,

there is a (compactly supported) topological homeomor-

phism Θ :

R2

→

R2

such that Θ(P1) = P2.

We will not prove the topological Sch¨ onflies Theorem. A thorough ex-

position of the proof can be found in (Moise, 1977, Chapter 9). The original

version appeared in (Sch¨ onflies, 1908).

The tools of (Rourke and Sanderson, 1972) come very close to proving

the PL variant of the theorem; we complete the proof in that special case.

Theorem 0.11.2 (PL Sch¨ onflies). For any two polygonal simple closed

curves P1 and P2 in

R2,

there is a (compactly supported) PL homeomor-

phism Θ :

R2

→

R2

such that Θ(P1) = P2.

Lemma 0.11.3. Each polygonal simple closed curve P in

R2

with at least

4 vertices has a pair of nonadjacent vertices v and w for which the interior

of the segment vw misses P .