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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2004 • Question 20

Question

Math 6510 • Differentiable Manifolds Fall 2004 • Question 20

  1. Let \(S_g\) be the closed, orientable surface of genus \(g\). Calculate \(\pi _2(S_g)\) for all \(g\).
  2. Show that \(\pi _3(S_0)\) is non-trivial. (\(S_0\) is the 2-sphere.)

Raw LaTeX

\begin{enumerate}
\item Let \(S_g\) be the closed, orientable surface of genus \(g\). Calculate \(\pi_2(S_g)\)
for all \(g\).
\item Show that \(\pi_3(S_0)\) is non-trivial. (\(S_0\) is the 2-sphere.)
\end{enumerate}

Topic Tags

Homotopy Groups Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups