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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1998 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 1998 • Question 8

Suppose that \(K_1\supset K_2\supset K_3\supset \cdots \) is a nested sequence of compact connected subsets of \(S^2\) such that \(\pi _1(K_j)=\mathbb {Z}\) for all \(j\). Prove or provide a counterexample for each of the following statements:

  1. \(\bigcap _{j=1}^{\infty }K_j\) is connected.
  2. \(\pi _1\left (\bigcap _{j=1}^{\infty }K_j\right )\cong \mathbb {Z}\).

Raw LaTeX

Suppose that \(K_1\supset K_2\supset K_3\supset\cdots\) is a nested
sequence of compact connected subsets of \(S^2\) such that \(\pi_1(K_j)=\mathbb{Z}\) for
all \(j\). Prove or provide a counterexample for each of the following statements:
\begin{enumerate}
\item \(\bigcap_{j=1}^{\infty}K_j\) is connected.
\item \(\pi_1\left(\bigcap_{j=1}^{\infty}K_j\right)\cong\mathbb{Z}\).
\end{enumerate}

Topic Tags

Fundamental Groups Homotopy Groups Mayer-Vietoris Sequences Relative Homology and Long Exact Sequences