Curve complex

In mathematics, the curve complex is a simplicial complex "C"("S") associated to a finite-type surface "S", which encodes the combinatorics of simple closed curves on "S". The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups. It was introduced by W.J.Harvey in 1978.
Curve complexes.
Definition.
Let formula_1 be a finite type connected oriented surface. More specifically, let formula_2 be a connected oriented surface of genus formula_3 with formula_4 boundary components and formula_5 punctures.
The "curve complex" formula_6 is the simplicial complex defined as follows:
Examples.
For surfaces of small complexity (essentially the torus, punctured torus, and four-holed sphere), with the definition above the curve complex has infinitely many connected components. One can give an alternate and more useful definition by joining vertices if the corresponding curves have minimal intersection number. With this alternate definition, the resulting complex is isomorphic to the Farey graph.
Geometry of the curve complex.
Basic properties.
If formula_1 is a compact surface of genus formula_11 with formula_12 boundary components the dimension of formula_6 is equal to formula_14. In what follows, we will assume that formula_15. The complex of curves is never locally finite (i.e. every vertex has infinitely many neighbors). A result of Harer asserts that formula_6 is in fact homotopically equivalent to a wedge sum of spheres.
Intersection numbers and distance on "C"("S").
The combinatorial distance on the 1-skeleton of formula_6 is related to the intersection number between simple closed curves on a surface, which is the smallest number of intersections of two curves in the isotopy classes. For example
for any two nondisjoint simple closed curves formula_19. One can compare in the other direction but the results are much more subtle (for example there is no uniform lower bound even for a given surface) and harder to prove.
Hyperbolicity.
It was proved by Masur and Minsky that the complex of curves is a Gromov hyperbolic space. Later work by various authors gave alternate proofs of this fact and better information on the hyperbolicity.
Relation with the mapping class group and Teichmüller space.
Action of the mapping class group.
The mapping class group of formula_1 acts on the complex formula_6 in the natural way: it acts on the vertices by formula_22 and this extends to an action on the full complex. This action allows to prove many interesting properties of the mapping class groups.
While the mapping class group itself is not a hyperbolic group, the fact that formula_6 is hyperbolic still has implications for its structure and geometry.
Comparison with Teichmüller space.
There is a natural map from Teichmüller space to the curve complex, which takes a marked hyperbolic structures to the collection of closed curves realising the smallest possible length (the systole). It allows to read off certain geometric properties of the latter, in particular it explains the empirical fact that while Teichmüller space itself is not hyperbolic it retains certain features of hyperbolicity.
Applications to 3-dimensional topology.
Heegaard splittings.
A simplex in formula_6 determines a "filling" of formula_1 to a handlebody. Choosing two simplices in formula_6 thus determines a Heegaard splitting of a three-manifold, with the additional data of an Heegaard diagram (a maximal system of disjoint simple closed curves bounding disks for each of the two handlebodies). Some properties of Heegaard splittings can be read very efficiently off the relative positions of the simplices:
In general the minimal distance between simplices representing diagram for the splitting can give information on the topology and geometry (in the sense of the geometrisation conjecture of the manifold) and vice versa. A guiding principle is that the minimal distance of a Heegaard splitting is a measure of the complexity of the manifold.
Kleinian groups.
As a special case of the philosophy of the previous paragraph, the geometry of the curve complex is an important tool to link combinatorial and geometric properties of hyperbolic 3-manifolds, and hence it is a useful tool in the study of Kleinian groups. For example, it has been used in the proof of the ending lamination conjecture.
Random manifolds.
A possible model for random 3-manifolds is to take random Heegaard splittings. The proof that this model is hyperbolic almost surely (in a certain sense) uses the geometry of the complex of curves.