Electronic ISBN:  9781470410049 
Product Code:  MEMO/224/1053.E 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 224; 2013; 97 ppMSC: Primary 37; 54;
In this memoir the authors present proofs of basic results, including those developed so far by Harold Bell, for the plane fixed point problem: Does every map of a nonseparating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. The authors define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. A prime end theory is developed through hyperbolic chords in maximal round balls contained in the complement of a nonseparating plane continuum \(X\). They define the concept of an outchannel for a fixed point free map which carries the boundary of \(X\) minimally into itself and prove that such a map has a unique outchannel, and that outchannel must have variation \(1\). Also Bell's Linchpin Theorem for a foliation of a simply connected domain, by closed convex subsets, is extended to arbitrary domains in the sphere.
The authors introduce the notion of an oriented map of the plane and show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982. 
Table of Contents

Chapters

Preface

1. Introduction

1. Basic Theory

2. Preliminaries and outline of Part 1

3. Tools

4. Partitions of domains in the sphere

2. Applications of Basic Theory

5. Description of main results of Part 2

6. Outchannels and their properties

7. Fixed points


RequestsReview Copy – for reviewers who would like to review an AMS bookPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
In this memoir the authors present proofs of basic results, including those developed so far by Harold Bell, for the plane fixed point problem: Does every map of a nonseparating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. The authors define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. A prime end theory is developed through hyperbolic chords in maximal round balls contained in the complement of a nonseparating plane continuum \(X\). They define the concept of an outchannel for a fixed point free map which carries the boundary of \(X\) minimally into itself and prove that such a map has a unique outchannel, and that outchannel must have variation \(1\). Also Bell's Linchpin Theorem for a foliation of a simply connected domain, by closed convex subsets, is extended to arbitrary domains in the sphere.
The authors introduce the notion of an oriented map of the plane and show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982.

Chapters

Preface

1. Introduction

1. Basic Theory

2. Preliminaries and outline of Part 1

3. Tools

4. Partitions of domains in the sphere

2. Applications of Basic Theory

5. Description of main results of Part 2

6. Outchannels and their properties

7. Fixed points