Math 6510 • Differentiable Manifolds

Let \(f\) be the homeomorphism of the annulus \(S^1\times[0,1]\) given by
\[
  f(z,s)=(ze^{2\pi is},s)
\]
where we view \(S^1\) as the set of unit norm complex numbers.
\begin{enumerate}
\item[(a)] Construct an explicit isotopy (i.e. homotopy through homeomorphisms)
between \(f\) and the identity that does not move the points of \(S^1\times\{0\}\).
\item[(b)] Prove that there is no homotopy between \(f\) and the identity that does
not move points on both \(S^1\times\{0\}\) and \(S^1\times\{1\}\).
\end{enumerate}