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

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.