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
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