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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2005 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 2005 • Question 12

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, and if it does not, find a counterexample (for every odd \(n\)).

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,
and if it does not, find a counterexample (for every odd \(n\)).

Topic Tags

Fixed Point Theorems and Borsuk-Ulam