Any orientation-preserving automorphism of the annulus is isotopic to the identity

Question

How can I prove that an orientation-preserving self-homeomorphism of the annulus $[0,1]\times S^1$ that preserves each boundary component is isotopic to the identity?

Answer 1

It is false. The mapping class group of the annulus is isomorphic to $\mathbb{Z}$ and it is generated by a Dehn twist around the core of the annulus, which is to say the map defined as $$ T(t,\theta)=(t,\theta +2\pi t). $$ You can find a proof in Farb and Margalit's book A Primer on Mapping Class Group.