Alexander method

Work on a compact oriented surface, possibly with finitely many marked points, and fix its boundary pointwise during isotopies. Suppose an Alexander system cuts the surface into discs, each containing at most one marked point. This is the filling hypothesis needed here.

A homeomorphism preserving every member up to isotopy can be adjusted by a simultaneous isotopy to preserve its structure graph of an Alexander system. If the induced graph action fixes every vertex and every oriented edge, the remaining maps on the complementary discs are isotopic to the identity by the Alexander trick. Thus the homeomorphism is isotopic to the identity. Since there are only finitely many automorphisms of the finite oriented graph, the pointwise curve-and-arc-class stabilizer is finite.^[ref]

The strong filling condition matters for a surface with boundary. A boundary-parallel Dehn twist can fix every interior curve class while remaining nontrivial relative to the boundary. A system cutting all the way into discs detects this missing information.

A homeomorphism of a closed disc that fixes its boundary pointwise is isotopic relative to the boundary to the identity. Radially interpolate the action toward the center after identifying the disc with the unit ball.

The center of $\mathrm{Mod}(T^2)$ has order two. A central class commutes with twists about two curves intersecting once, hence preserves both curve classes. The Alexander method reduces it to the identity or the elliptic involution $\iota :x\mapsto -x$, and both are central.