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 $\backslash operatorname(X)/\backslash operatorname\_0(X)$, where $\backslash operatorname\_0(X)$ is the path-component of the identity in $\backslash operatorname(X)$. (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 $\backslash pi\_0(\backslash operatorname(X))$, where one substitutes for Aut the appropriate group for the category to which ''X'' belongs. Here $\backslash pi\_0$ denotes the 0-th homotopy group of a space.
So in general, there is a short exact sequence of groups:
:$1\; \backslash rightarrow\; \backslash operatorname\_0(X)\; \backslash rightarrow\; \backslash operatorname(X)\; \backslash rightarrow\; \backslash operatorname(X)\; \backslash rightarrow\; 1.$
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, $\backslash operatorname(M)$ 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)
:$\backslash operatorname(S^2)\; \backslash simeq\; \backslash Z/2\backslash Z,$
corresponding to maps of degree ±1.

** Torus **

In the homotopy category
:$\backslash operatorname(\backslash mathbf^n)\; \backslash simeq\; \backslash operatorname(n,\backslash Z).$
This is because the n-dimensional torus $\backslash mathbf^n\; =\; (S^1)^n$ is an Eilenberg–MacLane space.
For other categories if $n\backslash ge\; 5$, one has the following split-exact sequences:
In the category of topological spaces
:$0\backslash to\; \backslash Z\_2^\backslash infty\backslash to\; \backslash operatorname(\backslash mathbf^n)\; \backslash to\; \backslash operatorname(n,\backslash Z)\backslash to\; 0$
In the PL-category
:$0\backslash to\; \backslash Z\_2^\backslash infty\backslash oplus\backslash binom\; n2\backslash Z\_2\backslash to\; \backslash operatorname(\backslash mathbf^n)\backslash to\; \backslash operatorname(n,\backslash Z)\backslash to\; 0$
(⊕ representing direct sum).
In the smooth category
:$0\backslash to\; \backslash Z\_2^\backslash infty\backslash oplus\backslash binom\; n2\backslash Z\_2\backslash oplus\backslash sum\_^n\backslash binom\; n\; i\backslash Gamma\_\backslash to\; \backslash operatorname(\backslash mathbf^n)\backslash to\; \backslash operatorname(n,\backslash Z)\backslash to\; 0$
where $\backslash Gamma\_i$ are the Kervaire–Milnor finite abelian groups of homotopy spheres and $\backslash Z\_2$ 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 $\backslash operatorname(\backslash mathbf^2)$ 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. 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,; 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 $\backslash mathbf^2(\backslash R)$ is isotopic to the identity:
:$\backslash operatorname(\backslash mathbf^2(\backslash R))\; =\; 1.$
The mapping class group of the Klein bottle ''K'' is:
:$\backslash operatorname(K)=\; \backslash Z\_2\; \backslash oplus\; \backslash Z\_2.$
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 ''N''_{3} (the connected sum of three projective planes) has:
:$\backslash operatorname(N\_3)\; =\; \backslash operatorname(2,\backslash Z).$
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 $N\backslash setminus\; C$ is ''a torus with a disk removed''. As an unoriented surface, its mapping class group is $\backslash operatorname(2,\backslash Z)$. (Lemma 2.1).

** 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.S. Kojima, ''Topology and its Applications'', Volume 29, Issue 3, August 1988, Pages 297–307

** 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'' → ''X'' is invertible and ''f(A)'' = ''A''.

** Symmetry group of knot and links **

If ''K'' ⊂ 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 (S^{3}, ''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 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 Homeo_{0} 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 ''H''^{1}(Σ) ≅ Z^{2''g''}. Orientation-preserving maps are precisely those that act trivially on top cohomology ''H''^{2}(Σ) ≅ Z. ''H''^{1}(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:
:$1\; \backslash to\; \backslash operatorname(\backslash Sigma)\; \backslash to\; \backslash operatorname(\backslash Sigma)\; \backslash to\; \backslash operatorname(H^1(\backslash Sigma))\; \backslash cong\; \backslash operatorname\_(\backslash mathbf)\; \backslash to\; 1$
One can extend this to
:$1\; \backslash to\; \backslash operatorname(\backslash Sigma)\; \backslash to\; \backslash operatorname^*(\backslash Sigma)\; \backslash to\; \backslash operatorname^(H^1(\backslash Sigma))\; \backslash cong\; \backslash operatorname^\_(\backslash mathbf)\; \backslash to\; 1$
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 $\backslash Sigma\_$ of genus ''g'' and 1 boundary component into $\backslash Sigma\_$ by attaching an additional hole on the end (i.e., gluing together $\backslash Sigma\_$ and $\backslash Sigma\_$), 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.

See also

*Braid groups, the mapping class groups of punctured discs *Homotopy groups *Homeotopy groups *Lantern relation

References

* *''Automorphisms of surfaces after Nielsen and Thurston'' by Andrew Casson and Steve Bleiler. * "Mapping Class Groups" by Nikolai V. Ivanov in the ''Handbook of Geometric Topology''.

''A Primer on Mapping Class Groups''

by Benson Farb and Dan Margalit * * * *

** Stable mapping class group **

The stable moduli space of Riemann surfaces: Mumford's conjecture

by Ib Madsen and Michael S. Weiss, 2002 *:Published as

The stable moduli space of Riemann surfaces: Mumford's conjecture

by Ib Madsen and Michael S. Weiss, 2007, Annals of Mathematics

** External links **

many references {{DEFAULTSORT:Mapping Class Group Category:Geometric topology Category:Homeomorphisms

Examples

See also

*Braid groups, the mapping class groups of punctured discs *Homotopy groups *Homeotopy groups *Lantern relation

References

* *''Automorphisms of surfaces after Nielsen and Thurston'' by Andrew Casson and Steve Bleiler. * "Mapping Class Groups" by Nikolai V. Ivanov in the ''Handbook of Geometric Topology''.

''A Primer on Mapping Class Groups''

by Benson Farb and Dan Margalit * * * *

The stable moduli space of Riemann surfaces: Mumford's conjecture

by Ib Madsen and Michael S. Weiss, 2002 *:Published as

The stable moduli space of Riemann surfaces: Mumford's conjecture

by Ib Madsen and Michael S. Weiss, 2007, Annals of Mathematics

many references {{DEFAULTSORT:Mapping Class Group Category:Geometric topology Category:Homeomorphisms