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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 20

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 20

Let \(f:S^n\to S^n\) be a continuous map from the \(n\)-sphere to itself and assume that \(n\) is even. Show that there is a point \(x\in S^n\) such that \(f(x)\in \{-x,x\}\). Does the same assertion hold if \(n\) is odd? If it does, prove it, if it does not construct a counterexample.

Raw LaTeX

Let \(f:S^n\to S^n\) be a continuous map from the \(n\)-sphere to itself and assume
that \(n\) is even. Show that there is a point \(x\in S^n\) such that
\(f(x)\in\{-x,x\}\). Does the same assertion hold if \(n\) is odd? If it does, prove it, if
it does not construct a counterexample.

Topic Tags

Fixed Point Theorems and Borsuk-Ulam