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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2001 • Question 16

Question

Math 6510 • Differentiable Manifolds Fall 2001 • Question 16

  1. Define carefully the Hopf map \(h:S^3\to S^2\).
  2. Define what it means for a map to be a fiber bundle and sketch a proof that \(h\) is a fiber bundle.
  3. State the long exact sequence in homotopy groups associated to a fiber bundle and compute \(\pi _3(S^2)\).

Raw LaTeX

\begin{enumerate}
\item Define carefully the Hopf map \(h:S^3\to S^2\).
\item Define what it means for a map to be a fiber bundle and sketch a proof that
\(h\) is a fiber bundle.
\item State the long exact sequence in homotopy groups associated to a fiber bundle
and compute \(\pi_3(S^2)\).
\end{enumerate}

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Homotopy Groups Homotopy and Degree on Manifolds Relative Homology and Long Exact Sequences