In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the curve complex is a
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
''C''(''S'') associated to a finite-type
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
''S'', which encodes the combinatorics of
simple closed curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
s on ''S''. The curve complex turned out to be a fundamental tool in the study of the geometry of the
Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
, of
mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mot ...
s and of
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s. It was introduced by W.J.Harvey in 1978.
Curve complexes
Definition
Let
be a finite type connected oriented surface. More specifically, let
be a connected oriented surface of genus
with
boundary components and
punctures.
The ''curve complex''
is the simplicial complex defined as follows:
*The vertices are the free
homotopy class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
es of essential (neither homotopically trivial nor
peripheral
A peripheral or peripheral device is an auxiliary device used to put information into and get information out of a computer. The term ''peripheral device'' refers to all hardware components that are attached to a computer and are controlled by the ...
) simple closed curves on
;
*If
represent distinct vertices of
, they span a simplex if and only if they can be homotoped to be pairwise disjoint.
Examples
For surfaces of small complexity (essentially the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
, 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
is a compact surface of genus
with
boundary components the dimension of
is equal to
. In what follows, we will assume that
. The complex of curves is never locally finite (i.e. every vertex has infinitely many neighbors). A result of Harer asserts that
is in fact
homotopically equivalent to a
wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...
of spheres.
Intersection numbers and distance on ''C''(''S'')
The combinatorial distance on the 1-skeleton of
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
. 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 In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properti ...
. 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
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mot ...
of
acts on the complex
in the natural way: it acts on the vertices by
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
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
, the fact that
is hyperbolic still has implications for its structure and geometry.
Comparison with Teichmüller space
There is a natural map from
Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
to the curve complex, which takes a marked hyperbolic structures to the collection of closed curves realising the smallest possible length (the
systole
Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ''sun ...
). 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
determines a "filling" of
to a handlebody. Choosing two simplices in
thus determines a
Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Definitions
Let ''V'' and ''W'' be handlebodies of genus ''g'', an ...
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:
*the splitting is reducible if and only if it has a diagram represented by simplices which have a common vertex;
*the splitting is weakly reducible if and only if it has a diagram represented by simplices which are linked by an edge.
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
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensi ...
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 In hyperbolic geometry, the ending lamination theorem, originally conjectured by , states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geod ...
.
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.
Notes
References
*Harvey, W. J. (1981). "Boundary Structure of the Modular Group". ''Riemann Surfaces and Related Topics. Proceedings of the 1978 Stony Brook Conference ''. 1981.
*
*
*
*
*
*
Benson Farb
Benson Stanley Farb (born October 25, 1967) is an American mathematician at the University of Chicago. His research fields include geometric group theory and low-dimensional topology.
Early life
A native of Norristown, Pennsylvania, Farb earne ...
and Dan Margalit, ''A primer on mapping class groups''. Princeton Mathematical Series, 49.
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financial su ...
, Princeton, NJ, 2012. {{ISBN, 978-0-691-14794-9
Topology
Geometric group theory