Pure mapping class group ($\mathrm{PMod}(S)$)

The pure mapping class group consists of mapping classes that fix every puncture individually.

Pushing a distinguished puncture once around a loop $\gamma$ defines a mapping class. For a simple loop, the push is the product of opposite Dehn twists about the two boundary components of a thin annular neighborhood of $\gamma$.

For a finite-type surface $S$ of negative Euler characteristic and the surface $S^{\ast }$ obtained by adding a puncture, forgetting that puncture gives

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.

The Birman exact sequence and $\mathrm{PMod}(S_{0,3})=1$ identify $\mathrm{PMod}(S_{0,4})$ with ${\pi }_1(S_{0,3})$, the free group of rank two.

Forgetting the fifth puncture gives a split Birman exact sequenceLifts of two free generators generate a copy of $F_2$ meeting the point-pushing kernel $F_3$ trivially.