By a continuum we mean a compact and connected metric space and by a non-
separating continuum X in the plane C we mean a continuum X ⊂ C such that
C \ X is connected. Our work is motivated by the following long-standing problem
[Ste35] in topology.
Plane Fixed Point Problem: “Does a continuous function taking a non-
separating plane continuum into itself always have a fixed point?”
To give the reader perspective we would like to make a few brief historical
remarks (see [KW91, Bin69, Bin81] for much more information).
Borsuk [Bor35] showed in 1932 that the answer to the above question is yes
if X is also locally connected. Cartwright and Littlewood [CL51] showed in 1951
that a map of a non-separating plane continuum X to itself has a fixed point if the
map can be extended to an orientation-preserving homeomorphism of the plane.
It was 27 years before Harold Bell [Bel78] extended this result to the class of
all homeomorphisms of the plane. Then Bell announced in 1982 (see also Akis
[Aki99]) that the Cartwright-Littlewood Theorem can be extended to the class of
all holomorphic maps of the plane. For other partial results in this direction see,
e.g., [Ham51, Hag71, Bel79, Min90, Hag96, Min99].
In this memoir the Plane Fixed Point Problem is addressed. We develop and
further generalize tools, first introduced by Bell, to elucidate the action of a fixed
point free map (should one exist). We are indebted to Bell for sharing his insights
with us. Some of the results in this memoir were first obtained by him. Unfortu-
nately, many of the proofs were not accessible. Since there are now multiple papers
which rely heavily upon these tools (e.g., [OT07, BO09, BCLOS08]) we believe
that they deserve to be developed in a coherent fashion. We also hope that by
making these tools available to the mathematical community, other applications of
these results will be found. In fact, we include in Part 2 of this text new applications
which illustrate their usefulness.
Part 1 contains the basic theory, the main ideas of which are due to Bell. We
introduce Bell’s notion of variation and prove his theorem that index equals varia-
tion increased by 1 (see Theorem 3.2.2). Bell’s Linchpin Theorem 4.2.5 for simply
connected domains is extended to arbitrary domains in the sphere and proved us-
ing an elegant argument due to Kulkarni and Pinkall [KP94]. Our version of this
theorem (Theorem 4.1.5) is essential for the results later in the paper.
Building upon these ideas, we will introduce in Part 1 the class of oriented
maps of the plane and show that it decomposes into two classes, one of which
preserves and the other of which reverses local orientation. The extension from
holomorphic to positively oriented maps is important since it allows for simple
local perturbations of the map (see Lemma 7.5.1) and significantly simplifies further
usage of the developed tools.