Linearity of some low-complexity mapping class groups

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus $g$ with $b$ boundary components and $n$ punctures, we show that these groups are isomorphic to some groups related to the braid groups and the Artin group of type $D_4$ in the cases when $g=0$ with $b$ and $n$ arbitrary, and when $g=1$ and $b+n$ is at most $3$. As a corollary, we conclude that the pure mapping class groups are linear in these cases.