# Mapping class group

Last updated

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.

## Motivation

Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous maps with continuous inverses: functions which stretch and deform the space continuously without breaking or glueing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets K into open subsets U as K and U range throughout our original topological space, completed with their finite intersections (which must be open by definition of topology) and arbitrary unions (again which must be open). This gives a notion of continuity on the space of functions, so that we can consider continuous deformation of the homeomorphisms themselves: called homotopies. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the group structure from the functional composition group structure already present on the space of homeomorphisms.

## Definition

The term mapping class group has a flexible usage. Most often it is used in the context of a manifold M. The mapping class group of M is interpreted as the group of isotopy classes of automorphisms of M. So if M is a topological manifold, the mapping class group is the group of isotopy classes of homeomorphisms of M. If M is a smooth manifold, the mapping class group is the group of isotopy classes of diffeomorphisms of M. Whenever the group of automorphisms of an object X has a natural topology, the mapping class group of X is defined as ${\displaystyle \operatorname {Aut} (X)/\operatorname {Aut} _{0}(X)}$, where ${\displaystyle \operatorname {Aut} _{0}(X)}$ is the path-component of the identity in ${\displaystyle \operatorname {Aut} (X)}$. (Notice that in the compact-open topology, path components and isotopy classes coincide, i.e., two maps f and g are in the same path-component iff they are isotopic). For topological spaces, this is usually the compact-open topology. In the low-dimensional topology literature, the mapping class group of X is usually denoted MCG(X), although it is also frequently denoted ${\displaystyle \pi _{0}(\operatorname {Aut} (X))}$, where one substitutes for Aut the appropriate group for the category to which X belongs. Here ${\displaystyle \pi _{0}}$ denotes the 0-th homotopy group of a space.

So in general, there is a short exact sequence of groups:

${\displaystyle 1\rightarrow \operatorname {Aut} _{0}(X)\rightarrow \operatorname {Aut} (X)\rightarrow \operatorname {MCG} (X)\rightarrow 1.}$

Frequently this sequence is not split. [1]

If working in the homotopy category, the mapping class group of X is the group of homotopy classes of homotopy equivalences of X.

There are many subgroups of mapping class groups that are frequently studied. If M is an oriented manifold, ${\displaystyle \operatorname {Aut} (M)}$ would be the orientation-preserving automorphisms of M and so the mapping class group of M (as an oriented manifold) would be index two in the mapping class group of M (as an unoriented manifold) provided M admits an orientation-reversing automorphism. Similarly, the subgroup that acts as the identity on all the homology groups of M is called the Torelli group of M.

## Examples

### Sphere

In any category (smooth, PL, topological, homotopy) [2]

${\displaystyle \operatorname {MCG} (S^{2})\simeq \mathbb {Z} /2\mathbb {Z} ,}$

corresponding to maps of degree  ±1.

### Torus

In the homotopy category

${\displaystyle \operatorname {MCG} (\mathbf {T} ^{n})\simeq \operatorname {SL} (n,\mathbb {Z} ).}$

This is because the n-dimensional torus ${\displaystyle \mathbf {T} ^{n}=(S^{1})^{n}}$ is an Eilenberg–MacLane space.

For other categories if ${\displaystyle n\geq 5}$, [3] one has the following split-exact sequences:

${\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}$

In the PL-category

${\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\oplus {\binom {n}{2}}\mathbb {Z} _{2}\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}$

(⊕ representing direct sum). In the smooth category

${\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\oplus {\binom {n}{2}}\mathbb {Z} _{2}\oplus \sum _{i=0}^{n}{\binom {n}{i}}\Gamma _{i+1}\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}$

where ${\displaystyle \Gamma _{i}}$ are the Kervaire–Milnor finite abelian groups of homotopy spheres and ${\displaystyle \mathbb {Z} _{2}}$ is the group of order 2.

### Surfaces

The mapping class groups of surfaces have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of ${\displaystyle \operatorname {MCG} (\mathbf {T} ^{2})}$ above), since they act on Teichmüller space and the quotient is the moduli space of Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to hyperbolic groups and to higher rank linear groups[ citation needed ]. They have many applications in Thurston's theory of geometric three-manifolds (for example, to surface bundles). The elements of this group have also been studied by themselves: an important result is the Nielsen–Thurston classification theorem, and a generating family for the group is given by Dehn twists which are in a sense the "simplest" mapping classes. Every finite group is a subgroup of the mapping class group of a closed, orientable surface,; [4] in fact one can realize any finite group as the group of isometries of some compact Riemann surface (which immediately implies that it injects in the mapping class group of the underlying topological surface).

#### Non-orientable surfaces

Some non-orientable surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the real projective plane ${\displaystyle \mathbf {P} ^{2}(\mathbb {R} )}$ is isotopic to the identity:

${\displaystyle \operatorname {MCG} (\mathbf {P} ^{2}(\mathbb {R} ))=1.}$

The mapping class group of the Klein bottle K is:

${\displaystyle \operatorname {MCG} (K)=\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}.}$

The four elements are the identity, a Dehn twist on a two-sided curve which does not bound a Möbius strip, the y-homeomorphism of Lickorish, and the product of the twist and the y-homeomorphism. It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity.

We also remark that the closed genus three non-orientable surface N3 (the connected sum of three projective planes) has:

${\displaystyle \operatorname {MCG} (N_{3})=\operatorname {GL} (2,\mathbb {Z} ).}$

This is because the surface N has a unique class of one-sided curves such that, when N is cut open along such a curve C, the resulting surface ${\displaystyle N\setminus C}$ is a torus with a disk removed. As an unoriented surface, its mapping class group is ${\displaystyle \operatorname {GL} (2,\mathbb {Z} )}$. (Lemma 2.1 [5] ).

### 3-Manifolds

Mapping class groups of 3-manifolds have received considerable study as well, and are closely related to mapping class groups of 2-manifolds. For example, any finite group can be realized as the mapping class group (and also the isometry group) of a compact hyperbolic 3-manifold. [6]

## Mapping class groups of pairs

Given a pair of spaces (X,A) the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of (X,A) is defined as an automorphism of X that preserves A, i.e. f: XX is invertible and f(A) = A.

If KS3 is a knot or a link, the symmetry group of the knot (resp. link) is defined to be the mapping class group of the pair (S3, K). The symmetry group of a hyperbolic knot is known to be dihedral or cyclic, moreover every dihedral and cyclic group can be realized as symmetry groups of knots. The symmetry group of a torus knot is known to be of order two Z2.

## Torelli group

Notice that there is an induced action of the mapping class group on the homology (and cohomology) of the space X. This is because (co)homology is functorial and Homeo0 acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy). The kernel of this action is the Torelli group, named after the Torelli theorem.

In the case of orientable surfaces, this is the action on first cohomology H1(Σ) ≅ Z2g. Orientation-preserving maps are precisely those that act trivially on top cohomology H2(Σ) ≅ Z. H1(Σ) has a symplectic structure, coming from the cup product; since these maps are automorphisms, and maps preserve the cup product, the mapping class group acts as symplectic automorphisms, and indeed all symplectic automorphisms are realized, yielding the short exact sequence:

${\displaystyle 1\to \operatorname {Tor} (\Sigma )\to \operatorname {MCG} (\Sigma )\to \operatorname {Sp} (H^{1}(\Sigma ))\cong \operatorname {Sp} _{2g}(\mathbf {Z} )\to 1}$

One can extend this to

${\displaystyle 1\to \operatorname {Tor} (\Sigma )\to \operatorname {MCG} ^{*}(\Sigma )\to \operatorname {Sp} ^{\pm }(H^{1}(\Sigma ))\cong \operatorname {Sp} _{2g}^{\pm }(\mathbf {Z} )\to 1}$

The symplectic group is well understood. Hence understanding the algebraic structure of the mapping class group often reduces to questions about the Torelli group.

Note that for the torus (genus 1) the map to the symplectic group is an isomorphism, and the Torelli group vanishes.

## Stable mapping class group

One can embed the surface ${\displaystyle \Sigma _{g,1}}$ of genus g and 1 boundary component into ${\displaystyle \Sigma _{g+1,1}}$ by attaching an additional hole on the end (i.e., gluing together ${\displaystyle \Sigma _{g,1}}$ and ${\displaystyle \Sigma _{1,2}}$), and thus automorphisms of the small surface fixing the boundary extend to the larger surface. Taking the direct limit of these groups and inclusions yields the stable mapping class group, whose rational cohomology ring was conjectured by David Mumford (one of conjectures called the Mumford conjectures). The integral (not just rational) cohomology ring was computed in 2002 by Ib Madsen and Michael Weiss, proving Mumford's conjecture.

## Related Research Articles

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape".

In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of a normal vector allows one to use the right-hand rule to define a "clockwise" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a "clockwise" orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure in the space cannot be moved continuously on that surface and back to it starting point so that it looks like its own mirror image.

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In mathematics, specifically algebraic topology, a covering map is a continuous function from a topological space to a topological space such that each point in has an open neighbourhood evenly covered by . In this case, is called a covering space and the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.

In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group. These concepts are named after the mathematician J. H. C. Whitehead.

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface.

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.

In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Homeomorphism groups are topological invariants in the sense that the homeomorphism groups of homeomorphic topological spaces are isomorphic as groups.

In mathematics, the Teichmüller space of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from to itself.

In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function T : VW that is:

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 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.

In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem is also often stated as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction.

This is a glossary of properties and concepts in algebraic topology in mathematics.

In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics, they are used to describe assignments of a collection of points to positions in a topological space. More specifically, configuration spaces in mathematics are particular examples of configuration spaces in physics in the particular case of several non-colliding particles.

In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the general linear group of X, the group of invertible linear transformations from X to itself.

## References

1. Morita, Shigeyuki (1987). "Characteristic classes of surface bundles". Inventiones Mathematicae . 90 (3): 551–577. doi:10.1007/bf01389178. MR   0914849.
2. Earle, Clifford J.; Eells, James (1967), "The diffeomorphism group of a compact Riemann surface", Bulletin of the American Mathematical Society , 73: 557–559, doi:, MR   0212840
3. MR 0520490 (80f:57014) Hatcher, A. E. Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 3–21, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. (Reviewer: Gerald A. Anderson) 57R52
4. Greenberg, Leon (1974), "Maximal groups and signatures", Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Annals of Mathematics Studies, 79, Princeton, N.J.: Princeton University Press, pp. 207–226, MR   0379835
5. Scharlemann, Martin (1982). "The complex of curves on nonorientable surfaces". Journal of the London Mathematical Society . Series 2. 25 (1): 171–184.
6. S. Kojima, Topology and its Applications, Volume 29, Issue 3, August 1988, Pages 297–307