"A Primer on Mapping Class Groups" wrote



> 
> Let $\mathrm{Homeo}\_+(S, \partial S)$ denote the group of orientation-preserving
> homeomorphisms of $S$ that restrict to the identity on $\partial S$. $\mathrm{Mod}(S)$ is the
> group of isotopy classes of elements of $\mathrm{Homeo}\_+(S, \partial S)$, where isotopies
> are required to fix the boundary pointwise.
> 
> 
> 


and



> 
> Let $\mathrm{PMod}(S\_{g,n})$ denote the pure mapping class group of $S\_{g,n}$, which is
> defined to be the subgroup of $\mathrm{Mod}(S\_{g,n})$ consisting of elements that
> fix each puncture individually.
> 
> 
> 


What is the difference between the two definitions? Doesn't "restrict to the identity on $\partial S$" mean for for all $x\in \partial S$ we have $\phi(x)=\mathrm{id}(x)=x$, which implies $\phi$ fixes each puncture individually?



Mapping class group and pure mapping class group
I do not recall the conventions adopted in the Primer, but there is a wide difference between boundary components (which will be embedded circles or lines) and punctures (which are “missing points” and are typically given by specifying a “punctured disk”). Under most conventions punctures are not “part” of the boundary (whatever that might mean…)


I greatly prefer, when possible, to avoid discussions of punctures and instead use the formalism of “marked points”. So, suppose that $S$ is a connected compact surface. Suppose that $Z \subset S$ is a finite set disjoint from the boundary. Then a mapping class of $(S, Z)$ is any homeomorphism of $S$ that permutes the points of $Z$. A pure mapping class of the pair $(S,Z)$ is required to fix each component of $\partial S$ setwise and also fix $Z$ pointwise.


There are further variations (if $Z$ is allowed to meet the boundary, if we fix the boundary pointwise, and so on). It really depends on what you want to do.


