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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2018 • Question 9

Question

Math 6510 • Differentiable Manifolds Summer 2018 • Question 9

  1. State the van Kampen theorem for fundamental groups. Use it to compute, \(\pi _1(K,*)\), where \(K\) is the Klein bottle, and \(\pi _1(S_g,*)\), where \(S_g\) is the orientable surface of genus \(g\). (Here \(*\) is any base-point.)
  2. Show that \(S_g\) is indeed orientable; carefully state the results you are using.

Raw LaTeX

\begin{enumerate}
\item State the van Kampen theorem for fundamental groups. Use it to compute,
\(\pi_1(K,*)\), where \(K\) is the Klein bottle, and \(\pi_1(S_g,*)\), where \(S_g\) is
the orientable surface of genus \(g\). (Here \(*\) is any base-point.)

\item Show that \(S_g\) is indeed orientable; carefully state the results you are
using.
\end{enumerate}

Topic Tags

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