Definition. Let $\Sigma_{g, n}^{m}$ denote a topological surface with genus $g \geqq 0, n \geqq 0$ punctures, and $m \geqq 0$ boundary components, i.e. such that filling in the $n$ punctures gives a compact surface with $m$ boundary components homeomorphic to circles. The associated pure mapping class group, denoted $\Gamma_{g, n}^{m}$, is defined to be the group of classes of orientation-preserving diffeomorphisms of $\Sigma_{g, n}^{m}$ fixing the boundaries pointwise.
what is mean of pointwise in above definition ? we have in real analysis pointwise convergence.
Generally, if $X$ is a set (or a set with structure, such as a punctured Riemann surface) and $A$ is a subset of $X$ (such as a boundary component), a mapping $f:X \to X$ preserves $A$ pointwise if $f(x) = x$ for every $x$ in $A$. By contrast, we say $f$ preserves $A$ (as a set) simply to mean $f(A) \subseteq A$.