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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2005 • Question 16

Question

Math 6510 • Differentiable Manifolds Fall 2005 • Question 16

Let \(p:S^3\to \mathbb {C}P^1=S^2\) be the map that sends \((z,w)\in S^3\subset \mathbb {C}^2\) to its equivalence class \([z:w]\in \mathbb {C}P^1\).

  1. Prove that \(p\) is a fiber bundle by explicitly writing down local trivializations.
  2. Write down the associated long exact sequence of homotopy groups.
  3. Compute \(\pi _3(S^2)\). Justify all auxiliary results you are using.

Raw LaTeX

Let \(p:S^3\to\mathbb{C}P^1=S^2\) be the map that sends
\((z,w)\in S^3\subset\mathbb{C}^2\) to its equivalence class \([z:w]\in\mathbb{C}P^1\).
\begin{enumerate}
\item Prove that \(p\) is a fiber bundle by explicitly writing down local trivializations.
\item Write down the associated long exact sequence of homotopy groups.
\item Compute \(\pi_3(S^2)\). Justify all auxiliary results you are using.
\end{enumerate}

Topic Tags

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