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Geometric Topology in Dimensions 2 and 3

  • Edwin E. Moise

Part of the Graduate Texts in Mathematics book series (GTM, volume 47)

Table of contents

  1. Front Matter
    Pages i-x
  2. Edwin E. Moise
    Pages 1-8
  3. Edwin E. Moise
    Pages 9-15
  4. Edwin E. Moise
    Pages 16-25
  5. Edwin E. Moise
    Pages 26-30
  6. Edwin E. Moise
    Pages 31-41
  7. Edwin E. Moise
    Pages 42-45
  8. Edwin E. Moise
    Pages 46-51
  9. Edwin E. Moise
    Pages 52-57
  10. Edwin E. Moise
    Pages 58-64
  11. Edwin E. Moise
    Pages 65-70
  12. Edwin E. Moise
    Pages 71-80
  13. Edwin E. Moise
    Pages 81-82
  14. Edwin E. Moise
    Pages 83-90
  15. Edwin E. Moise
    Pages 91-96
  16. Edwin E. Moise
    Pages 97-100
  17. Edwin E. Moise
    Pages 101-111
  18. Edwin E. Moise
    Pages 112-116
  19. Edwin E. Moise
    Pages 117-126
  20. Edwin E. Moise
    Pages 127-133
  21. Edwin E. Moise
    Pages 134-139
  22. Edwin E. Moise
    Pages 147-154
  23. Edwin E. Moise
    Pages 165-173
  24. Edwin E. Moise
    Pages 174-181
  25. Edwin E. Moise
    Pages 197-200
  26. Edwin E. Moise
    Pages 214-219
  27. Edwin E. Moise
    Pages 220-222
  28. Edwin E. Moise
    Pages 223-229
  29. Edwin E. Moise
    Pages 247-252
  30. Edwin E. Moise
    Pages 253-255
  31. Back Matter
    Pages 256-262

About this book

Introduction

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.

Keywords

Cantor Homeomorphism Manifold Morphism Topology theorem

Authors and affiliations

  • Edwin E. Moise
    • 1
  1. 1.Department of MathematicsQueens College, CUNYFlushingUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-9906-6
  • Copyright Information Springer-Verlag New York 1977
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-9908-0
  • Online ISBN 978-1-4612-9906-6
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site