CHAPTER 1

Introduction

1.0.1. Notation and the main problem. We denote the plane by C, the

Riemann sphere by C∞ = C ∪ {∞}, the real line by R and the unit circle by

S1 = R/Z. Let X be a plane compactum. Since C is locally connected and X is

closed, complementary domains of X are open. By T (X) we denote the topological

hull of X consisting of X union all of its bounded complementary domains. Thus,

U

∞

= U

∞(X)

=

C∞

\ T (X) is the unbounded complementary component of

X containing infinity. Observe that if X is a continuum, then U

∞(X)

is simply

connected. The Plane Fixed Point Problem, attributed to [Ste35], is one of the

central long-standing problems in plane topology. It serves as a motivation for our

work and can be formulated as follows.

Problem 1.0.1 (Plane Fixed Point Problem). Does a continuous function tak-

ing a non-separating plane continuum into itself always have a fixed point?

1.0.2. Historical remarks. To give the reader perspective we would like to

make a few historical remarks concerning the Plane Fixed Point Problem (here we

cover only major steps towards solving the problem).

In 1912 Brouwer [Bro12] proved that any orientation preserving homeomor-

phism of the plane, which keeps a bounded set invariant, must have a fixed point

(though not necessarily in that set). This fundamental result has found many im-

portant applications. It was recognized early on that the location of a fixed point

should be determined if the invariant set is a non-separating continuum (in that

case a fixed point should be located in the invariant continuum) and many papers

have been devoted to obtaining partial solutions to the Plane Fixed Point Problem.

Borsuk [Bor35] showed in 1932 that the answer is yes if X is also locally

connected. Cartwright and Littlewood [CL51] showed in 1951 that a continuous

map of a non-separating continuum X to itself has a fixed point in X if the map can

be extended to an orientation-preserving homeomorphism of the plane. (See Brown

[Bro77] for a very short proof of this theorem based on the above mentioned result

by Brouwer). The proof by Cartwright-Littlewood Theorem made use of the index

of a map on a simple closed curve and this idea has remained the basic approach

in many partial solutions.

The most general result was obtained by Bell [Bel67] in the early 1960’s.

He showed that any counterexample must contain an invariant indecomposable

subcontinuum. Hence the Plane Fixed Point Problem has a positive solution for

hereditarily decomposable plane continua (i.e., for continua X which do not con-

tain indecomposable subcontinua). Bell’s result was also based on the notion of the

index of a map, but he introduced new ideas to determine the index of a simple

closed curve which runs tightly around a possible counterexample. Unfortunately,

these ideas were not transparent and were never fully developed. Alternative proofs

1