This note is filed under “things I know nothing about”. If you want to learn from someone who knows something, I recommend “Notes on the complex of curves” by Saul Schleimer.
The curve complex is an infinite simplicial complex associated to a (topological) surface. It is already interesting enough to look at the 1-skeleton of this complex, which is simply an infinite graph. I will try to say something about this graph using as an example the closed surface of genus 2, known as a double torus. (Some call it a pretzel, but most pretzels I’ve seen have genus 3.)
Imagine a simple closed curve on this surface. If can be continuously shrunk to a point, it is of no interest to us, because it does not see the topology of the surface. Let us consider only essential curves: those that cannot be shrunk to a point. We are also not interested in the exact position of the curve, and so consider two curves and equivalent (isotopic) if it’s possible to slide into along the surface. For example, A and B are equivalent, but B and C are not, and neither are C and D.
These equivalence classes will be the vertices of our graph. There are infinitely many vertices, because one can construct complicated curves by, say, spiraling about one handle a bunch of times, then switching to the other, and then coming back. To define a graph we must know when to draw an edge between two vertices. The answer is: when they can be represented by curves that do not intersect. So, the vertices corresponding to A, C, and D would all be connected to one another. Let us denote this graph by , the 1-skeleton of the curve complex .
The first non-obvious fact about is that it is connected. Think about what this means: now matter how crazily intertwined two curves and are, one can find a finite sequence such that and can slide out of each other’s way.
The second non-obvious fact is that is an unbounded metric space. The metric is defined as on any connected graph: the distance between two vertices is the minimal number of edges one must travel to get from to . So, neighboring vertices are at distance 1. If the distance between two curves and is 3 or more, then any other curve must intersect either or . This is expressed by saying that and fill the surface; there is no room for anyone else. Therefore, if we cut the surface along , it falls apart into simply connected pieces, i.e., disks. Note that if the curves were smooth, so are the disks, because we do not create corners in this process.
To exhibit a pair of curves at distance 4 (and to prove that the example works) is a serious undertaking. So it should not come as a surprise that the large-scale geometry of the curve complex is a rich and challenging subject of study. It is a Gromov hyperbolic space, so there is a well-defined boundary at infinity which can be equipped with one of (equivalent) visual metrics. The topology of this boundary remains to be understood.
One last remark. Some vertices in the complex correspond to curves that bound a disk inside of the surface. Let be the union of such vertices. This set is not invariant under homeomorphisms of the surface; it depends on how it sits inside of ; or, in other words, on how the surface is “filled in” to become a 3-dimensional handlebody . This makes it possible to quantify the complexity of a homeomorphism : if the distance between and is large, then the homeomorphism is rather complicated.