Exercise 54.5

Question

Consider the covering map $p 	imes p\colon \mathbb{R} 	imes \mathbb{R} 	o S^1 	imes S^1$ of Example $4$ of $\S53$. Consider the path
\begin{equation*}
  f(t) = (\cos 2\pi t,\sin 2\pi t) 	imes (\cos 4\pi t,\sin 4\pi t)
\end{equation*}
in $S^1 	imes S^1$. Sketch what $f$ looks like when $S^1 	imes S^1$ is identified with the doughnut surface $D$. Find a lifting $	ilde{f}$ of $f$ to $\mathbb{R} 	imes \mathbb{R}$, and sketch it.

Amit Awasthi · Accepted Answer

\begin{proof}[Solution]
  We have the family of liftings $	ilde{f}_n(t) = (n+t/2,n+t)$, where $n \in
  \mathbb{Z}$. We omit the sketches.
\end{proof}

Exercise 54.5

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Exercise 54.5

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