Ebook: Geometric Topology in Dimensions 2 and 3

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