In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.
Contents
- History
- Mapping class group of orientable surfaces
- The mapping class groups of the sphere and the torus
- Mapping class group of surfaces with boundary and punctures
- Mapping class group of an annulus
- Braid groups and mapping class groups
- The DehnNielsenBaer theorem
- The Birman exact sequence
- Dehn twists
- The NielsenThurston classification
- Pseudo Anosov diffeomorphisms
- Action on Teichmller space
- Action on the curve complex
- Pants complex
- Markings complex
- The DehnLickorish theorem
- Finite presentability
- Other systems of generators
- Cohomology of the mapping class group
- The Torelli subgroup
- Residual finiteness and finite index subgroups
- Finite subgroups
- General facts on subgroups
- Linear representations
- References
The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory.
The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups.
History
The mapping class group appeared in the first half of the twentieth century. Its origins lie in the study of the topology of hyperbolic surfaces, and especially in the study of the intersections of closed curves on these surfaces. The earliest contributors were Max Dehn and Jakob Nielsen: Dehn proved finite generation of the group, and Nielsen gave a classification of mapping classes and proved that all automorphisms of the fundamental group of a surface can be represented by homeomorphisms (the Dehn–Nielse–Baer theorem).
The Dehn–Nielsen theory was reinterpreted in the mid-seventies by Thurston who gave the subject a more geometric flavour and used this work to great effect in his program for the study of three-manifolds.
More recently the mapping class group has been by itself a central topic in geometric group theory, where it provides a testing ground for various conjectures and techniques.
Mapping class group of orientable surfaces
Let
is a distance inducing the compact-open topology on
This is a countable group.
If we modify the definition to include all homeomorphisms we obtain the extended mapping class group
This definition can also be made in the differentiable category: if we replace all instances of "homeomorphism" above with "diffeomorphism" we obtain the same group, that is the inclusion
The mapping class groups of the sphere and the torus
Suppose that
The mapping class group of the torus
Mapping class group of surfaces with boundary and punctures
In the case where
A surface with punctures is a compact surface with a finite number of points removed ("punctures"). The mapping class group of such a surface is defined as above (note that the mapping classes are allowed to permute the punctures, but not the boundary components.
Mapping class group of an annulus
Any annulus is homeomorphic to the subset
which is the identity on both boundary components
Braid groups and mapping class groups
Braid groups can be defined as the mapping class groups of a disc with punctures. More precisely, the braid group on n strands is naturally isomorphic to the mapping class group of a disc with n punctures.
The Dehn–Nielsen–Baer theorem
If
The image of the mapping class group is an index 2 subgroup of the outer automorphism group, which can be characterised by its action on homology.
The conclusion of the theorem does not hold when
The Birman exact sequence
This is a exact sequence relating the mapping class group of surfaces with the same genus and boundary but different number of punctures. It is a fundamental tool which allows to use recursive arguments in the study of mapping class groups. It was proven by Joan Birman in 1969. The exact statement is as follows.
LetIn the case where
Dehn twists
If
In the mapping class group of the torus identified with
corresponds to the Dehn twist about a horizontal curve in the torus.
The Nielsen–Thurston classification
There is a classification of the mapping classes on a surface, originally due to Nielsen and rediscovered by Thurston, which can be stated as follows. An element
The main content of the theorem is that a mapping class which is neither of finite order must be pseudo-Anosov, which can be defined explicitly by dynamical properties.
Pseudo-Anosov diffeomorphisms
The study of pseudo-Anosov diffeomorphisms of a surface is fundamental. They are the most interesting diffeomorphisms, since finite-order mapping classes are isotopic to isometries and thus well-understood, and the study of reducible classes indeed essentially reduces to the study of mapping classes on smaller surfaces which may themselves be either finite order or pseudo-Anosov.
Pseudo-Anosov mapping classes are "generic" in the mapping class group in various ways. For example, a random walk on the mapping class group will end on a pseudo-Anosov element with a probability tending to 1 as the number of steps grows.
Action on Teichmüller space
Given a punctured surface
This action has many interesting properties; for example it is properly discontinuous (though not free). It is compatible with various geometric structures (metric or complex) with which
The action extends to the Thurston boundary of Teichmüller space, and the Nielsen-Thurston classification of mapping classes can be seen in the dynamical properties of the action on Teichmüller space together with its Thurston boundary. Namely:
Action on the curve complex
The curve complex of a surface
This action, together with combinatorial and geometric properties of the curve complex, can be used to prove various properties of the mapping class group. In particular, it explains some of the hyperbolic properties of the mapping class group: while as mentioned in the previous section the mapping class group is not an hyperbolic group it has some properties reminiscent of those.
Pants complex
The pants complex of a compact surface
Markings complex
The stabilisers of the mapping class group's action on the curve and pants complexes are quite large. The markings complex is a complex whose vertices are markings of
A marking is determined by a pants decomposition
The Dehn–Lickorish theorem
The mapping class group is generated by the subset of Dehn twists about all simple closed curves on the surface. The Dehn–Lickorish theorem states that it is sufficient to select a finite number of those to generate the mapping class group. This generalises the fact that
In particular the mapping class group of a surface is a finitely generated group.
The least possible cardinality of Dehn twists generating the mapping class group of a closed surface of genus
Finite presentability
It is possible to prove that all relations between the Dehn twists in a generating set for the mapping class group can be written as combinations of a finite number among them. This means that the mapping class group of a surface is a finitely presented group.
One way to prove this theorem is to deduce it from the properties of the action of the mapping class group on the pants complex: the stabiliser of a vertex is seen to be finitely presented, and the action is cofinite. Since the complex is connected and simply connected it follows that the mapping class group must be finitely generated. There are other ways of getting finite presentations, but in practice the only one to yield explicit relations for all geni is that described in this paragraph with a slightly different complex instead of the curve complex, called the cut system complex.
An example of a relation between Dehn twists occurring in this presentation is the lantern relation.
Other systems of generators
There are other interesting systems of generators for the mapping class group besides Dehn twists. For example
Cohomology of the mapping class group
If
The first homology of the mapping class group is finite and it follows that the first cohomology group is finite as well.
The Torelli subgroup
As singular homology is functorial, the mapping class group
This map is in fact a surjection with image equal to the integer points
The kernel of the morphism
Residual finiteness and finite-index subgroups
An example of application of the Torelli subgroup is the following result:
The mapping class group is residually finite.The proof proceeds first by using residual finiteness of the linear group
An interesting class of finite-index subgroups is given by the kernels of the morphisms:
The kernel of
Finite subgroups
The mapping class group has only finitely many classes of finite groups, as follwos from the fact that the finite-index subgroup
A bound on the order of finite subgroups can also be obtained through geometric means. The solution to the Nielsen realisation problem implies that any such group is realised as the group of isometries of an hyperbolic surface of genus
General facts on subgroups
The mapping class groups satisfy the Tits alternative: that is, any subgroup of it either contains a non-abelian free subgroup or it is virtually solvable (in fact abelian).
Any subgroup which is not reducible (that is it does not preserve a set of isotopy class of disjoint simple closed curves) must contain a pseudo-Anosov element.
Linear representations
It is an open question whether the mapping class group is a linear group or not. Besides the symplectic representation on homology explained above there are other interesting finite-dimensional linear representations arising from topological quantum field theory. The images of these representations are contained in arithmetic groups which are not symplectic, and this allows to construct many more finite quotients of
In the other direction there is a lower bound for the dimension of a (putative) faithful representation, which has to be at least