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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 4 / a / Solution

A proper arc on a punctured surface approaches punctures at both ends. It is simple when its interior is embedded, and essential when it cannot be isotoped, relative to its ends, into a puncture.


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