# Mapping class group

﻿
Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of '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 functions with continuous inverses: functions which stretch and deform the space continuously without puncturing or breaking the space. It can be seen fairly easily that this space 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 automorphisms.

## 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 Aut(X) / Aut0(X) where Aut0(X) is the path-component of the identity in Aut(X). 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 π0Aut(X) where one substitutes Aut for the appropriate notion of automorphism for whichever category X is an object of. π0 denotes the 0-th homotopy group of a space.

So in general, there is a short-exact sequence of groups: $1 \rightarrow {\rm Aut}_0(X) \rightarrow {\rm Aut}(X) \rightarrow {\rm MCG}(X) \rightarrow 1.$

Frequently this sequence is not split.

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, 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 trivially on the homology of M is called the Torelli group of M, one could think of this as the mapping class group of a homologically-marked surface.

## Examples

### Sphere

In any category (smooth, PL, topological, homotopy) ${\rm MCG}(S^2) \simeq {\mathbb Z}/2{\mathbb Z},$

corresponding to maps of degree $\pm 1$.

### Torus

In the homotopy category ${\rm MCG}(T^n) \simeq {\rm GL}(n, {\mathbb Z}).$

This is because Tn = (S1)n is an Eilenberg-MacLane space.

Provided that $n \geq 5$, there are split-exact sequences: $0\to \mathbb Z_2^\infty\to MCG(T^n) \to GL(n,\mathbb Z)\to 0$

In the PL-category $0\to \mathbb Z_2^\infty\oplus\binom n2\mathbb Z_2\to MCG (T^n)\to GL(n,\mathbb Z)\to 0$

In the smooth category $0\to \mathbb Z_2^\infty\oplus\binom n2\mathbb Z_2\oplus\sum_{i=0}^n\binom n i\Gamma_{i+1}\to MCG(T^n)\to GL(n,\mathbb Z)\to 0$

where Γi are Kervaire-Milnor finite abelian groups of homotopy spheres, $\mathbb Z_2$ is the group of order 2.

### Surfaces

The mapping class groups of surfaces have been heavily studied, and are called Teichmüller modular groups. (Note the special case of MCG(T2) above.) This is perhaps due to their strange similarity to higher rank linear groups as well as many applications, via surface bundles, in Thurston's theory of geometric three-manifolds. For more information on this topic see the Nielsen–Thurston classification theorem and the article on Dehn twists. Every finite group is a subgroup of the mapping class group of a closed, orientable surface, moreover one can realize any finite group as the group of isometries of some compact Riemann surface.

#### Non-orientable surfaces

Some non-orientable surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the real projective plane ${\mathbb RP}^2$ is isotopic to the identity: ${\rm MCG}({\mathbb RP}^2) = 1.$

The mapping class group of the Klein bottle K is: ${\rm MCG}(K)={\mathbb Z}/2{\mathbb Z}\oplus{\mathbb Z}/2{\mathbb Z}.$

The four elements are the identity, a Dehn twist on the 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 has: ${\rm MCG(N_3)} = {\rm GL}(2, {\mathbb Z}).$

This is because the surface has a unique one-sided curve that, when cut open, yields a once-holed torus. This is discussed in a paper of Martin Scharlemann.

### 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.

## 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. $f : X \to X$ is invertible and f(A) = A.

### Symmetry group of knot and links

If $K\subset S^3$ 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 dihedra 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 $\mathbb Z_2$.

## 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.

In the case of orientable surfaces, this is the action on first cohomology $H^1(\Sigma)\cong \mathbb{Z}^{2g}$. Orientation-preserving maps are precisely those that act trivially on top cohomology $H^2(\Sigma) \cong \mathbb{Z}$. $H^1(\Sigma)$ 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: $1 \to \mbox{Tor}(\Sigma) \to \mbox{MCG}(\Sigma) \to \mbox{Sp}(H^1(\Sigma)) \cong \mbox{Sp}_{2g}(\mathbb{Z}) \to 1$

One can extend this to $1 \to \mbox{Tor}(\Sigma) \to \mbox{MCG}^*(\Sigma) \to \mbox{Sp}^{\pm}(H^1(\Sigma)) \cong \mbox{Sp}^{\pm}_{2g}(\mathbb{Z}) \to 1$

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 Σg,1 of genus g and 1 boundary component into Σg + 1,1 by attaching an additional hole on the end (i.e, gluing together Σg,1 and Σ1,2), 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 Madsen and Weiss, proving Mumford's conjecture.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• mapping class group — noun The group of self homeomorphisms of a topological space, modulo homotopy …   Wiktionary

• Ideal class group — In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by a certain group known as an ideal class group (or class group). If… …   Wikipedia

• Group theory — is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… …   Wikipedia

• Group Destination Address — are broadcast address in class D domain. Multicast IP address are Class D IP address, from 224.0.0.0 to 239.255.255.255. They are also referred to as Group Destination Address (GDA). There is a mapping between IP address and MAC address for GDA.… …   Wikipedia

• Mapping cone — In mathematics, especially homotopy theory, the mapping cone is a construction Cf of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated Cf. Contents 1 Definition 1.1 Example of circle …   Wikipedia

• Group ring — This page discusses the algebraic group ring of a discrete group; for the case of a topological group see group algebra, and for a general group see Group Hopf algebra. In algebra, a group ring is a free module and at the same time a ring,… …   Wikipedia

• Mapping of Unicode characters — Unicode’s Universal Character Set has a potential capacity to support over 1 million characters. Each UCS character is mapped to a code point which is an integer between 0 and 1,114,111 used to represent each character within the internal logic… …   Wikipedia

• Group scheme — In mathematics, a group scheme is a group object in the category of schemes. That is, it is a scheme G with the equivalent properties* there is a group law expressible as a multiplication μ and inversion map ι on G ; or * G is a functor (as in… …   Wikipedia

• Class (computer programming) — In object oriented programming, a class is a construct that is used as a blueprint to create instances of itself – referred to as class instances, class objects, instance objects or simply objects. A class defines constituent members which enable …   Wikipedia

• Group Policy — Local Security Policy editor in Windows 7 Group Policy is a feature of the Microsoft Windows NT family of operating systems. Group Policy is a set of rules that control the working environment of user accounts and computer accounts. Group Policy… …   Wikipedia