## mapping class group of an annulus

What's the mapping class group of an annulus? here the homeomorphism doesn't need to fix boundary component pointwise,but rather setwise. I don't have source in my hand, so I have to ask here.

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If it has to preserve the boundary components (but can rotate them), then it is trivial. If it can flip the boundary components (but again can rotate them), then it is Z/2Z. See Farb-Margalit's book "A primer on mapping class groups" (available on either of their webpages) for details. – Andy Putman Sep 22 2010 at 23:29