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

Define the arc complex $\mathcal{A}(S)$ as follows. Its vertices are isotopy classes, relative to the punctures, of unoriented 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. Henceby simplicial automorphisms.