Mapping class group

Given a Riemann surface $F_{g,n}$ of genus $g$ with $n$ boundary components, the mapping class group $\Gamma_{g,n}$ is the group of diffeomorphisms of $F_{g,n}$ which fix the boundary pointwise, modulo the subgroup of diffeomorphisms isotopic to the identity. Equivalently, it is the group of connected components $\pi_0 \mathrm{Diff}(F_{g,n})$.