Geometric Topology in Dimensions 2 and 3

1. Front Matter

   Pages i-x

   PDF

2. Introduction

   • Edwin E. Moise

   Pages 1-8

3. Connectivity

   • Edwin E. Moise

   Pages 9-15

4. Separation properties of polygons in R ²

   • Edwin E. Moise

   Pages 16-25

5. The Schönflies theorem for polygons in R ²

   • Edwin E. Moise

   Pages 26-30

6. The Jordan curve theorem

   • Edwin E. Moise

   Pages 31-41

7. Piecewise linear homeomorphisms

   • Edwin E. Moise

   Pages 42-45

8. PL approximations of homeomorphisms

   • Edwin E. Moise

   Pages 46-51

9. Abstract complexes and PL complexes

   • Edwin E. Moise

   Pages 52-57

10. The triangulation theorem for 2-manifolds

    • Edwin E. Moise

    Pages 58-64

11. The Schönflies theorem

    • Edwin E. Moise

    Pages 65-70

12. Tame imbedding in R ²

    • Edwin E. Moise

    Pages 71-80

13. Isotopies

    • Edwin E. Moise

    Pages 81-82

14. Homeomorphisms between Cantor sets

    • Edwin E. Moise

    Pages 83-90

15. Totally disconnected compact sets in R ²

    • Edwin E. Moise

    Pages 91-96

16. The fundamental group (summary)

    • Edwin E. Moise

    Pages 97-100

17. The group of (the complement of) a link

    • Edwin E. Moise

    Pages 101-111

18. Computations of fundamental groups

    • Edwin E. Moise

    Pages 112-116

19. The PL Schönflies theorem in R ³

    • Edwin E. Moise

    Pages 117-126

20. The Antoine set

    • Edwin E. Moise

    Pages 127-133

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.

• Cantor
• Homeomorphism
• Manifold
• Morphism
• Topology
• theorem

• Department of Mathematics, Queens College, CUNY, Flushing, USA

  Edwin E. Moise

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