Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 156 / 4

The curve complex $\mathcal{C}(S)$ has one vertex for each isotopy class of essential simple closed curves, and a set of vertices spans a simplex when it has pairwise disjoint representatives.

To prove connectedness, put two curves $\alpha ,\beta$ in minimal position and induct on $i(\alpha ,\beta )$. If it is zero, their vertices are equal or joined by an edge. If it is positive, surgery of $\alpha$ along an outermost segment of $\beta$ produces a boundary component $\gamma$ of a regular neighborhood of the surgery. At least one choice is essential; it is disjoint from $\beta$ and satisfiesThe induction hypothesis gives a path from $\alpha$ to $\gamma$, and the edge from $\gamma$ to $\beta$ completes it. Thus $\mathcal{C}(S)$ is connected. This is the connectedness of the curve complex.

Curves ${\alpha }_1,\dots ,{\alpha }_n$ in minimal position fill a surface $S$ when every essential simple closed curve has positive geometric intersection with at least one ${\alpha }_i$. On a closed surface this is equivalent to every component ofbeing a disc: a non-disc complementary component contains an essential curve, while any curve disjoint from the collection lies in such a component.

Write the genus-two surface aswhere $\alpha$ is separating and each $T_i$ is a one-holed torus. In each $T_i$, choose two disjoint essential proper arcs $p_i,q_i$ from the boundary to itself that form a cut system, so $T_i\setminus (p_i\cup q_i)$ is a disc. Arrange their four endpoints on each copy of $\alpha$ and glue the boundaries so thatjoin cyclically into one simple closed curve $\beta$. After smoothing at the four gluing points, $i(\alpha ,\beta )=4$. Cutting along $\alpha$ and these four arcs leaves one disc from each $T_i$, so $S\setminus (\alpha \cup \beta )$ consists of two discs. Hence $(\alpha ,\beta )$ is the filling pair on a closed genus-two surface.

If their curve-complex distance were at most two, there would be an essential curve $\gamma$ disjoint from both: it would be the intermediate vertex of a length-two path. This contradicts filling. Therefore