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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2008 • Question 7

Question

Math 6510 • Differentiable Manifolds Fall 2008 • Question 7

Let \(S^2\) be the 2-sphere and \(T\) the torus.

  1. Show that every map from \(S^2\) to \(T\) is null-homotopic.
  2. Show that there are infinitely many homotopy classes of maps from \(T\) to \(S^2\).

Raw LaTeX

Let \(S^2\) be the 2-sphere and \(T\) the torus.
\begin{enumerate}
\item[(a)] Show that every map from \(S^2\) to \(T\) is null-homotopic.
\item[(b)] Show that there are infinitely many homotopy classes of maps from \(T\)
to \(S^2\).
\end{enumerate}

Topic Tags

Chain Complexes and Chain Homotopy Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Surface Topology and Surface Groups