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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2008 • Question 11

Question

Math 6510 • Differentiable Manifolds Spring 2008 • Question 11

Let \(M\) be an open 3-manifold such that every compact set of \(M\) is contained in a solid torus.

  1. Show that \(\pi _1(M)\) is abelian.
  2. Assume that \(\pi _1(M)\) is finitely generated. Show that \(\pi _1(M)\) is trivial or \(\mathbb {Z}\).

Raw LaTeX

Let \(M\) be an open 3-manifold such that every compact set of \(M\) is contained
in a solid torus.
\begin{enumerate}
\item[(a)] Show that \(\pi_1(M)\) is abelian.
\item[(b)] Assume that \(\pi_1(M)\) is finitely generated. Show that \(\pi_1(M)\) is
trivial or \(\Z\).
\end{enumerate}

Topic Tags

Fundamental Groups Homotopy Groups Homotopy and Degree on Manifolds Surface Topology and Surface Groups