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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2007 • Question 14

Question

Math 6510 • Differentiable Manifolds Fall 2007 • Question 14

Let \(f,g:S^2\to S^2\) be two maps. Show that at least one of the following holds:

  • there exists \(x\in S^2\) such that \(f(x)=g(x)\),
  • there exists \(x\in S^2\) such that \(f(x)=-g(x)\), or
  • both \(f\) and \(g\) are homotopic to constant maps.

Raw LaTeX

Let \(f,g:S^2\to S^2\) be two maps. Show that at least one of the following
holds:
\begin{itemize}
\item there exists \(x\in S^2\) such that \(f(x)=g(x)\),
\item there exists \(x\in S^2\) such that \(f(x)=-g(x)\), or
\item both \(f\) and \(g\) are homotopic to constant maps.
\end{itemize}

Topic Tags

Fixed Point Theorems and Borsuk-Ulam Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds