Pure mapping class group of the three-punctured sphere ($\mathrm{PMod}(S_{0,3})=1$)

The pure mapping class group of the three-punctured sphere is trivial. The three simple arcs joining distinct puncture pairs form an ideal triangulation; a pure homeomorphism can be isotoped to fix these arcs, and the Alexander trick on the two complementary discs finishes the isotopy to the identity.