Geometric Topology in Dimensions 2 and 3

Edwin E. Moise (auth.)9780387902203, 0387902201

Geometric topology may roughly be described as the branch of the topology of manifolds which deals with questions of the existence of homeomorphisms. Only in fairly recent years has this sort of topology achieved a sufficiently high development to be given a name, but its beginnings are easy to identify. The first classic result was the SchOnflies theorem (1910), which asserts that every 1-sphere in the plane is the boundary of a 2-cell. In the next few decades, the most notable affirmative results were the “Schonflies theorem” for polyhedral 2-spheres in space, proved by J. W. Alexander [Ad, and the triangulation theorem for 2-manifolds, proved by T. Rad6 [Rd. But the most striking results of the 1920s were negative. In 1921 Louis Antoine [A ] published an extraordinary paper in which he 4 showed that a variety of plausible conjectures in the topology of 3-space were false. Thus, a (topological) Cantor set in 3-space need not have a simply connected complement; therefore a Cantor set can be imbedded in 3-space in at least two essentially different ways; a topological 2-sphere in 3-space need not be the boundary of a 3-cell; given two disjoint 2-spheres in 3-space, there is not necessarily any third 2-sphere which separates them from one another in 3-space; and so on and on. The well-known “horned sphere” of Alexander [A ] appeared soon thereafter.

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Table of contents :
Front Matter….Pages i-x
Introduction….Pages 1-8
Connectivity….Pages 9-15
Separation properties of polygons in R 2 ….Pages 16-25
The Schönflies theorem for polygons in R 2 ….Pages 26-30
The Jordan curve theorem….Pages 31-41
Piecewise linear homeomorphisms….Pages 42-45
PL approximations of homeomorphisms….Pages 46-51
Abstract complexes and PL complexes….Pages 52-57
The triangulation theorem for 2-manifolds….Pages 58-64
The Schönflies theorem….Pages 65-70
Tame imbedding in R 2 ….Pages 71-80
Isotopies….Pages 81-82
Homeomorphisms between Cantor sets….Pages 83-90
Totally disconnected compact sets in R 2 ….Pages 91-96
The fundamental group (summary)….Pages 97-100
The group of (the complement of) a link….Pages 101-111
Computations of fundamental groups….Pages 112-116
The PL Schönflies theorem in R 3 ….Pages 117-126
The Antoine set….Pages 127-133
A wild arc with a simply connected complement….Pages 134-139
A wild 2-sphere with a simply connected complement….Pages 140-146
The Euler characteristic….Pages 147-154
The classification of compact connected 2-manifolds….Pages 155-164
Triangulated 3-manifolds….Pages 165-173
Covering spaces….Pages 174-181
The Stallings proof of the loop theorem of Papakyriakopoulos….Pages 182-190
Bicollar neighborhoods; an extension of the Loop theorem….Pages 191-196
The Dehn lemma….Pages 197-200
Polygons in the boundary of a combinatorial solid torus….Pages 201-210
Limits on the Loop theorem: Stallings’s example….Pages 211-213
Polyhedral interpolation theorems….Pages 214-219
Canonical configurations….Pages 220-222
Handle decompositions of tubes….Pages 223-229
PLH approximations of homeomorphisms, for regular neighborhoods of linear graphs in R 3 ….Pages 230-238
PLH approximations of homeomorphisms, for polyhedral 3-cells….Pages 239-246
The Triangulation theorem….Pages 247-252
The Hauptvermutung ; Tame imbedding….Pages 253-255
Back Matter….Pages 256-262