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Question

Math 6510 • Differentiable Manifolds

Summer 2018 • Question 7

Question

Math 6510 • Differentiable Manifolds Summer 2018 • Question 7

Let \(X\) be a path-connected topological space.

  1. Let \(x_0,x_1\) be points in \(X\), and let \(\alpha :I\to X\) be any path from \(x_0\) to \(x_1\). Explain how \(\alpha \) induces an isomorphism of groups \(\alpha _*:\pi _1(X,x_0)\xrightarrow {\sim }\pi _1(X,x_1)\).

  2. Give an example where \(x_1=x_0\) (so \(\alpha \) is a loop at \(x_0\)), but the map \(\alpha _*:\pi _1(X,x_0)\to \pi _1(X,x_0)\) is not the identity map.
  3. Give an example of a space \(X\) with \(H_1(X;\mathbb {Z})=0\) but \(\pi _1(X,x_0)\ne 0\). State carefully the results you are using.

Raw LaTeX

Let \(X\) be a path-connected topological space.
\begin{enumerate}
\item Let \(x_0,x_1\) be points in \(X\), and let \(\alpha:I\to X\) be any path from
\(x_0\) to \(x_1\). Explain how \(\alpha\) induces an isomorphism of groups
\(\alpha_*:\pi_1(X,x_0)\xrightarrow{\sim}\pi_1(X,x_1)\).

\newpage

\item Give an example where \(x_1=x_0\) (so \(\alpha\) is a loop at \(x_0\)), but the
map \(\alpha_*:\pi_1(X,x_0)\to\pi_1(X,x_0)\) is not the identity map.

\item Give an example of a space \(X\) with \(H_1(X;\Z)=0\) but
\(\pi_1(X,x_0)\ne0\). State carefully the results you are using.
\end{enumerate}

Topic Tags

Fundamental Groups Homotopy Groups Point-Set Topology and Separation Theorems