Mapping class group of a surface

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.
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–Nielsen–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.
Definition and examples.
Mapping class group of orientable surfaces.
Let formula_1 be a connected, closed, orientable surface and formula_2 the group of orientation-preserving, or positive, homeomorphisms of formula_3. This group has a natural topology, the compact-open topology. It can be defined easily by a distance function: if we are given a metric formula_4 on formula_3 inducing its topology then the function defined by
is a distance inducing the compact-open topology on formula_2. The connected component of the identity for this topology is denoted formula_8. By definition it is equal to the homeomorphisms of formula_3 which are isotopic to the identity. It is a normal subgroup of the group of positive homeomorphisms, and the mapping class group of formula_3 is the group
This is a countable group.
If we modify the definition to include all homeomorphisms we obtain the "extended mapping class group" formula_12, which contains the mapping class group as a subgroup of index 2.
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 formula_13 induces an isomorphism between the quotients by their respective identity components.
The mapping class groups of the sphere and the torus.
Suppose that formula_3 is the unit sphere in formula_15. Then any homeomorphism of formula_3 is isotopic to either the identity or to the restriction to formula_3 of the symmetry in the plane formula_18. The latter is not orientation-preserving and we see that the mapping class group of the sphere is trivial, and its extended mapping class group is formula_19, the cyclic group of order 2.
The mapping class group of the torus formula_20 is naturally identified with the modular group formula_21. It is easy to construct a morphism formula_22: every formula_23 induces a diffeomorphism of formula_24 via formula_25. The action of diffeomorphisms on the first homology group of formula_24 gives a left-inverse formula_27 to the morphism formula_28 (proving in particular that it is injective) and it can be checked that formula_27 is injective, so that formula_30 are inverse isomorphisms between formula_31 and formula_21. In the same way, the extended mapping class group of formula_24 is formula_34.
Mapping class group of surfaces with boundary and punctures.
In the case where formula_3 is a compact surface with a non-empty boundary formula_36 then the definition of the mapping class group needs to be more precise. The group formula_37 of homeomorphisms relative to the boundary is the subgroup of formula_2 which restrict to the identity on the boundary, and the subgroup formula_39 is the connected component of the identity. The mapping class group is then defined as
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 formula_41 of formula_42. One can define a diffeomorphism formula_43 by the following formula:
which is the identity on both boundary components formula_45. The mapping class group of formula_46 is then generated by the class of formula_43.
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 formula_3 is closed and formula_49 is a homeomorphism of formula_3 then we can define an automorphism formula_51 of the fundamental group formula_52 as follows: fix a path formula_53 between formula_54 and formula_55 and for a loop formula_56 based at formula_54 representing an element formula_58 define formula_59 to be the element of the fundamental group associated to the loop formula_60. This automorphism depends on the choice of formula_53, but only up to conjugation. Thus we get a well-defined map from formula_62 to the outer automorphism group formula_63. This map is a morphism and its kernel is exactly the subgroup formula_8. The Dehn–Nielsen–Baer theorem states that it is in addition surjective. In particular, it implies that:
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 formula_3 has a non-empty boundary (except in a finite number of cases). In this case the fundamental group is a free group and the outer automorphism group Out(Fn) is strictly larger than the image of the mapping class group via the morphism defined in the previous paragraph. The image is exactly those outer automorphisms which preserve each conjugacy class in the fundamental group corresponding to a boundary component.
The Birman exact sequence.
This is an exact sequence relating the mapping class group of surfaces with the same genus and boundary but a 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.
In the case where formula_3 itself has punctures the mapping class group formula_72 must be replaced by the finite-index subgroup of mapping classes fixing formula_73.
Elements of the mapping class group.
Dehn twists.
If formula_74 is an oriented simple closed curve on formula_3 and one chooses a closed tubular neighbourhood formula_46 then there is a homeomorphism formula_49 from formula_46 to the canonical annulus formula_79 defined above, sending formula_74 to a circle with the counterclockwise orientation. This is used to define a homeomorphism formula_81 of formula_1 as follows: on formula_83 it is the identity, and on formula_46 it is equal to formula_85. The class of formula_81 in the mapping class group formula_87 does not depend on the choice of formula_49 made above, and the resulting element is called the "Dehn twist" about formula_74. If formula_74 is not null-homotopic this mapping class is nontrivial, and more generally the Dehn twists defined by two non-homotopic curves are distinct elements in the mapping class group.
In the mapping class group of the torus identified with formula_21 the Dehn twists correspond to unipotent matrices. For example, the matrix
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 formula_93 is either:
The main content of the theorem is that a mapping class which is neither of finite order nor reducible 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.
Actions of the mapping class group.
Action on Teichmüller space.
Given a punctured surface formula_3 (usually without boundary) the Teichmüller space formula_99 is the space of marked complex (equivalently, conformal or complete hyperbolic) structures on formula_3. These are represented by pairs formula_101 where formula_102 is a Riemann surface and formula_103 a homeomorphism, modulo a suitable equivalence relation. There is an obvious action of the group formula_2 on such pairs, which descends to an action of formula_87 on Teichmüller space.
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 formula_99 can be endowed. In particular, the Teichmüller metric can be used to establish some large-scale properties of the mapping class group, for example that the maximal quasi-isometrically embedded flats in formula_87 are of dimension formula_108.
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 formula_3 is a complex whose vertices are isotopy classes of simple closed curves on formula_3. The action of the mapping class groups formula_87 on the vertices carries over to the full complex. The action is not properly discontinuous (the stabiliser of a simple closed curve is an infinite group).
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.
Other complexes with a mapping class group action.
Pants complex.
The pants complex of a compact surface formula_3 is a complex whose vertices are the pants decompositions of formula_3 (isotopy classes of maximal systems of disjoint simple closed curves). The action of formula_87 extends to an action on this complex. This complex is quasi-isometric to Teichmüller space endowed with the Weil–Petersson metric.
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 formula_3, which are acted upon by, and have trivial stabilisers in, the mapping class group formula_87. It is (in opposition to the curve or pants complex) a locally finite complex which is quasi-isometric to the mapping class group.
A marking is determined by a pants decomposition formula_119 and a collection of transverse curves formula_120 such that every one of the formula_121 intersects at most one of the formula_122, and this "minimally" (this is a technical condition which can be stated as follows: if formula_123 are contained in a subsurface homeomorphic to a torus then they intersect once, and if the surface is a four-holed sphere they intersect twice). Two distinct markings are joined by an edge if they differ by an "elementary move", and the full complex is obtained by adding all possible higher-dimensional simplices.
Generators and relations for mapping class groups.
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 formula_21 is generated by the matrices
In particular, the mapping class group of a surface is a finitely generated group.
The smallest number of Dehn twists that can generate the mapping class group of a closed surface of genus formula_126 is formula_127; this was proven later by Humphries.
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, formula_87 can be generated by two elements or by involutions.
Cohomology of the mapping class group.
If formula_3 is a surface of genus formula_97 with formula_131 boundary components and formula_132 punctures then the virtual cohomological dimension of formula_87 is equal to formula_134.
The first homology of the mapping class group is finite and it follows that the first cohomology group is finite as well.
Subgroups of the mapping class groups.
The Torelli subgroup.
As singular homology is functorial, the mapping class group formula_135 acts by automorphisms on the first homology group formula_136. This is a free abelian group of rank formula_137 if formula_3 is closed of genus formula_97. This action thus gives a linear representation formula_140.
This map is in fact a surjection with image equal to the integer points formula_141 of the symplectic group. This comes from the fact that the intersection number of closed curves induces a symplectic form on the first homology, which is preserved by the action of the mapping class group. The surjectivity is proven by showing that the images of Dehn twists generate formula_141.
The kernel of the morphism formula_143 is called the "Torelli group" of formula_3. It is a finitely generated, torsion-free subgroup and its study is of fundamental importance for its bearing on both the structure of the mapping class group itself (since the arithmetic group formula_141 is comparatively very well understood, a lot of facts about formula_87 boil down to a statement about its Torelli subgroup) and applications to 3-dimensional topology and algebraic geometry.
Residual finiteness and finite-index subgroups.
An example of application of the Torelli subgroup is the following result:
The proof proceeds first by using residual finiteness of the linear group formula_141, and then, for any nontrivial element of the Torelli group, constructing by geometric means subgroups of finite index which does not contain it.
An interesting class of finite-index subgroups is given by the kernels of the morphisms:
The kernel of formula_149 is usually called a "congruence subgroup" of formula_87. It is a torsion-free group for all formula_151 (this follows easily from a classical result of Minkowski on linear groups and the fact that the Torelli group is torsion-free).
Finite subgroups.
The mapping class group has only finitely many classes of finite groups, as follows from the fact that the finite-index subgroup formula_152 is torsion-free, as discussed in the previous paragraph. Moreover, this also implies that any finite subgroup of formula_87 is a subgroup of the finite group formula_154.
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 formula_97. Hurwitz's bound then implies that the maximal order is equal to formula_156.
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 formula_87.
In the other direction there is a lower bound for the dimension of a (putative) faithful representation, which has to be at least formula_158.