Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2010 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 2010 • Question 10

  1. Define a covering space \(p:\widetilde {X}\to X\).
  2. Let \(p:\widetilde {X}\to X\) be a covering space, \(\widetilde {x}_0\in \widetilde {X}\), \(x_0\in X\), and \(p(\widetilde {x}_0)=x_0\). Show that \(p_*:\pi _1(\widetilde {X},\widetilde {x}_0)\to \pi _1(X,x_0)\) is injective (carefully state the lifting property you are using).

Raw LaTeX

\begin{enumerate}
\item[(a)] Define a covering space \(p:\widetilde{X}\to X\).
\item[(b)] Let \(p:\widetilde{X}\to X\) be a covering space,
\(\widetilde{x}_0\in\widetilde{X}\), \(x_0\in X\), and \(p(\widetilde{x}_0)=x_0\).
Show that
\(p_*:\pi_1(\widetilde{X},\widetilde{x}_0)\to\pi_1(X,x_0)\) is injective
(carefully state the lifting property you are using).
\end{enumerate}

Topic Tags

Classification of Covering Spaces Covering Spaces Fundamental Groups Homotopy Groups