= Solution
Define the <arc complex> $\mathcal A(S)$ as follows. Its vertices are isotopy classes, relative to the punctures, of unoriented <essential proper arc on a punctured surface>[essential simple proper arcs]. A set of $k+1$ vertices spans a $k$-dimensional <simplex> when the classes have representatives with pairwise disjoint interiors. Every orientation-preserving homeomorphism sends such representatives to such representatives, and isotopic homeomorphisms induce the same permutation of classes. Hence
$$
\operatorname{Mod}(S)\curvearrowright\mathcal A(S)
$$
by simplicial automorphisms.
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