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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2016 • Question 7

Question

Math 6510 • Differentiable Manifolds Summer 2016 • Question 7

Let \(X=S^1\vee S^1\) and \(x_0\in X\) be the attaching point.

  1. Carefully state van Kampen’s Theorem and use it to compute \(\pi _1(X,x_0)\).
  2. Find two covering spaces of \(X\) with three sheets; one of which is regular (determining a normal subgroup of \(\pi _1(X,x_0)\)) and the other of which is irregular.

Raw LaTeX

Let \(X=S^1\vee S^1\) and \(x_0\in X\) be the attaching point.
\begin{enumerate}
\item[(a)] Carefully state van Kampen's Theorem and use it to compute
\(\pi_1(X,x_0)\).
\item[(b)] Find two covering spaces of \(X\) with three sheets; one of which is
regular (determining a normal subgroup of \(\pi_1(X,x_0)\)) and the other of which
is irregular.
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Covering Spaces Fundamental Groups Homotopy Groups