# Mapping class group and pure mapping class group

"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?

• Your last sentence sounds right but I don't think the reverse implication holds. Mar 1, 2022 at 16:05
• I think $S_g$ here means the closed surface of genus $g$, i.e., $\partial S_g=\varnothing$. Mar 1, 2022 at 18:01

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.
In fact this is the only difference between the two groups if I remember correctly. If $$(S,\partial S)$$ is the surface with boundary obtained by deleting open discs around each puncture of $$S_{g,n}$$, the inclusion $$(S, \partial S)\to S_{g,n}$$ induces a map $$\text{Mod}(S,\partial S)\to \text{PMod}(S_{g,n})$$. This map is surjective with kernel the free abelian group generated by the Dehn twists I describe above.