A Borsuk-Ulam Theorem and Index Theories

  • Jean Mawhin
  • Michel Willem
Part of the Applied Mathematical Sciences book series (AMS, volume 74)

Abstract

The (classical) Borsuk-Ulam theorem is a result which ensures that if Ω ⊂ R n is an open bounded symmetric neighborhood of the origin and if \( f:\partial \Omega \to {R^{n - 1}} \) is continuous and odd, then 0 ∈ f(∂Ω). This result can be proved using degree theory, a way of making an algebraic count of the zeros, in the closure \( \overline D \) of an open bounded set DR n , of continuous mappings g : \( \overline D \subset {R^n} \) having no zeros on ∂D. A short account of degree theory is given in Section 5.3.

Keywords

Banach Space Continuous Extension Index Theory Critical Point Theory Degree Theory 
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.

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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Jean Mawhin
    • 1
  • Michel Willem
    • 1
  1. 1.Institut de Mathematique Pure et AppliqueeLouvain-la-NeuveBelgium

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