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

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.

Fix punctures $p,q,r$. There is one isotopy class ${\ell }_p$ of essential arc from $p$ back to $p$: by the Jordan curve theorem, it separates $q$ from $r$, and any two such arcs are isotopic relative to the punctures.

Altogether $\mathcal{A}(S_{0,3})$ has six vertices:The joining arcs $a_{pq},a_{pr},a_{qr}$ span a triangle. For each puncture $p$, the three verticesspan another triangle. These are all the maximal simplices: ${\ell }_p$ must cross $a_{qr}$, and returning arcs based at distinct punctures cannot be made disjoint.

Every orientation-preserving permutation of the three punctures is realizable, while a homeomorphism acting trivially on the punctures is isotopic to the identity. ThusIt permutes the labels in the displayed description. There are two vertex orbits, the three joining arcs and the three returning arcs, as summarized by the arc complex of the three-punctured sphere.

Choose an essential returning arc $\alpha$ based at $p$ and an essential simple closed curve $\gamma$ with $i(\alpha ,\gamma )>0$. The iteratesare again simple proper arcs based at $p$. Their geometric intersection number with a fixed transverse arc grows linearly with $|n|$, so they represent infinitely many isotopy classes. This is the standard Dehn twist construction.

There are exactly two $\mathrm{Mod}(S_{0,4})$-orbits of vertices. Equality or inequality of the two endpoints is preserved by every homeomorphism. Conversely, a homeomorphism can send any ordered configuration of punctures and complementary discs of an arc to any other of the same endpoint type. Thus all arcs joining distinct punctures lie in one orbit, and all arcs returning to one puncture lie in the other. These are the arc-complex vertex orbits of the four-punctured sphere.