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Homotopy Groups

  • Gregory L. Naber
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 25)

Abstract

The real line ℝ is not homeomorphic to the plane ℝ2, but this fact is not quite the triviality one might hope. Perhaps the most elementary proof goes as follows: Suppose there were a homeomorphism h of ℝ onto ℝ2. Select some point x 0 ∈ ℝ. The restriction of h to ℝ − {x 0} would then carry it homeomorphically onto ℝ2 − {h(x 0)}. However, \(\mathbb{R} -\{ {x}_{0}\} = (-\infty,{x}_{0}) \cup({x}_{0},\infty )\) is not connected, whereas ℝ2 − {h(x 0)} certainly is connected (indeed, pathwise connected). Since connectedness is a topological property, this cannot be and we have our contradiction.

Keywords

Topological Space Base Point Fundamental Group Homotopy Class Homotopy Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Mass.
    Massey, W.S., Algebraic Topology: An Introduction, Springer-Verlag, New York, Berlin, 1991.Google Scholar
  2. Gra.
    Gray, B., Homotopy Theory: An Introduction to Algebraic Topology, Academic Press, New York, 1975.Google Scholar
  3. Rav.
    Ravenel, D.C., Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, Orlando, 1986.Google Scholar
  4. Dug.
    Dugundji, J., Topology, Allyn and Bacon, Boston, 1966.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Gregory L. Naber
    • 1
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA

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