= Alexander method
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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.<A primer on mapping class groups (Version 5.0) by Benson Farb and Dan Margalit>[§2.3, Proposition2.8]{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.
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