Exercise 54.4

Question

Consider the covering map $p\colon \mathbb{R} 	imes \mathbb{R}_+ 	o \mathbb{R}^2 - 0$ of Example $6$ of $\S53$. Find liftings of the paths
\begin{align*}
  f(t) &= (2-t,0),\
  g(t) &= ((1+t)\cos 2\pi t,(1+t)\sin 2\pi t),\
  h(t) &= f * g.
\end{align*}
Sketch these paths and their liftings.

Amit Awasthi · Accepted Answer

\begin{proof}[Solution]
  We see first that the covering map $p$ is the mapping
  \begin{equation*}
    (x,s) \mapsto ((\cos 2\pi x,\sin 2\pi x),s) \mapsto s(\cos 2\pi x,\sin 2\pi x).
  \end{equation*}
  Thus, we have the family of liftings
  \begin{align*}
    	ilde{f}_n(t) &= (n,2-t)\
    	ilde{g}_n(t) &= (t+n,1+t)\
    	ilde{h}_n(t) &= \begin{cases}
      (n,2-2t) & 	ext{if}~t \in [0,1/2]\
      (2t+n-1,2t) & 	ext{if}~t \in [1/2,1]
    \end{cases}
  \end{align*}
  where $n \in \mathbb{Z}$. We omit the sketches.
\end{proof}

Exercise 54.4

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Exercise 54.4

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