# Pure Mapping class group and mapping class group

Hi, everyone.

I am not sure it is proper to ask the following question on here.

Let $S$ be a genus $g\geq 1$ surface with 2-puncture, i.e. genus $g$ closed surface with 2 points removed. And there is a compact surface $S_{1}\subset S$ such that $S-S_{1}$ consists of 2 once-punctured disk.

Now in Farb and Margalit' book "A primer on mapping class groups ", they defined the $Mod(S_{1})$ and $PMod(S)$.

My first question is

Is it true that we can treat $PMod(S)$ as a subgroup of $Mod(S_{1})$?

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Look up the section covering configuration spaces, the associated fibre bundles and the homotopy long exact sequence. There's all the information there in the book to answer your question yourself. –  Ryan Budney Dec 14 '12 at 3:29
It is quotient of a subgroup (by the center). If this is the first question, I think you forgot to write the second one. –  Misha Dec 14 '12 at 3:34