A Borsuk-Ulam Theorem and Index Theories

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 D ⊂ R 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.