$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\PMod{PMod}\DeclareMathOperator\Homeo{Homeo}$I am very confused about the definition of mapping class group and pure mapping class group (and their generating sets).

In *Primer on Mapping class groups*:

$\Mod(S)$ is the group of isotopy classes of elements of $\Homeo^+(S, \partial S)$, where isotopies are required to fix the boundary pointwise

and

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

**Question 1**: From these two definitions, is it true that for compact surfaces with boundary (no puncture), the mapping class group is the same as the pure mapping class group?

Generating sets of mapping class group:

THEOREM 4.1

For $g\ge 0$, the mapping class group $\Mod(S_g)$ is generated by finitely many Dehn twists about nonseparating simple closed curves.

Corollary 4.15

For any $g$, $n\ge 0$, the group $\Mod(S_{g,n})$ is generated by a finite number of Dehn twists and half-twists.

Corollary 4.16

Let $S$ be any surface of genus $g\ge 2$. The group $\PMod(S)$ is generated by finitely many Dehn twists about nonseparating simple closed curves in $S$.

Assume that Question 1 is has a positive answer, meaning that for compact surfaces with boundary, the mapping class group is the same as the pure mapping class group. Then we ask

**Question 2** For compact surfaces with boundary, is the mapping class group generated by a finite number of Dehn twists?

**Question 3:** If Question 2 has a positive answer, what makes the difference between punctures and boundary, why don't we define

$\Mod(S)$ is the group of isotopy classes of elements of $\Homeo^+(S, \partial S)$

and

$\PMod(S_{g,n})$ the pure mapping class group of $S_{g,n}$ ($n$ is the number of punctures and boundary components) is the subgroup of $\Mod(S_{g,n})$ consisting of elements that fix each puncture and fix each boundary component setwise (send each puncture/or boundary component to itself)?

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