Editing Mapping class group (section)

==Examples==

=== Sphere ===
In any category (smooth, PL, topological, homotopy)<ref>{{citation| mr=0212840|last1=Earle|first1= Clifford J.|author1-link=Clifford John Earle Jr.| last2= Eells|first2= James|author2-link=James Eells| 
title=The diffeomorphism group of a compact Riemann surface|
journal=[[Bulletin of the American Mathematical Society]] | volume=73|year=1967|issue=4 |pages=557–559|doi=10.1090/S0002-9904-1967-11746-4|doi-access=free}}</ref>
:<math>\operatorname{MCG}(S^2) \simeq \Z/2\Z,</math>
corresponding to maps of [[Degree of a continuous mapping|degree]]&nbsp;±1.

=== Torus ===
In the [[homotopy category]]
:<math> \operatorname{MCG}(\mathbf{T}^n) \simeq \operatorname{GL}(n,\Z). </math>
This is because the [[Torus#n-dimensional torus|n-dimensional torus]] <math>\mathbf{T}^n = (S^1)^n</math> is an [[Eilenberg–MacLane space]].

For other categories if <math>n\ge  5</math>,<ref>{{cite book |first=A.E. |last=Hatcher |chapter=Concordance spaces, higher simple-homotopy theory, and applications |chapter-url={{GBurl|6hsDCAAAQBAJ|p=3}} |title=Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1 |series=Proceedings of Symposia in Pure Mathematics |publisher= |location= |date=1978 |volume=32 |issue=1 |isbn=978-0-8218-9320-3 |pages=3–21 |doi=10.1090/pspum/032.1/520490 |mr=0520490}}</ref> one has the following split-exact sequences:

In the [[category of topological spaces]]
:<math>0\to \Z_2^\infty\to \operatorname{MCG}(\mathbf{T}^n) \to \operatorname{GL}(n,\Z)\to 0</math>
In the [[Piecewise linear manifold|PL-category]]
:<math>0\to \Z_2^\infty\oplus\binom n2\Z_2\to \operatorname{MCG}(\mathbf{T}^n)\to \operatorname{GL}(n,\Z)\to 0</math>
(⊕ representing [[direct sum]]).
In the [[Smooth manifold|smooth category]]
:<math>0\to \Z_2^\infty\oplus\binom n2\Z_2\oplus\sum_{i=0}^n\binom n i\Gamma_{i+1}\to \operatorname{MCG}(\mathbf{T}^n)\to \operatorname{GL}(n,\Z)\to 0</math>

where <math>\Gamma_i</math> are the Kervaire–Milnor finite abelian groups of [[homotopy sphere]]s and <math>\Z_2</math> is the group of order 2.

=== Surfaces ===
{{Main article | Mapping class group of a surface}}

The mapping class groups of [[Surface (topology)|surface]]s have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of <math>\operatorname{MCG}(\mathbf{T}^2)</math> 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 group]]s and to higher rank linear groups{{citation needed|date=July 2016}}. They have many applications in [[William Thurston|Thurston]]'s theory of geometric [[three-manifold]]s (for example, to [[surface bundle]]s).  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 twist]]s 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;<ref>{{cite book |first=Leon |last=Greenberg |chapter=Maximal groups and signatures |chapter-url={{GBurl|EFbQCwAAQBAJ|p=207}} |title=Discontinuous Groups and Riemann Surfaces: Proceedings of the 1973 Conference at the University of Maryland |publisher=Princeton University Press |series=Annals of Mathematics Studies |volume=79 |date=1974 |isbn=978-1-4008-8164-2 |pages=207–226 |mr=0379835}}</ref> 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 [[orientability|non-orientable]] surfaces have mapping class groups with simple presentations.  For example, every homeomorphism of the [[real projective plane]] <math>\mathbf{P}^2(\R)</math> is isotopic to the identity:

:<math> \operatorname{MCG}(\mathbf{P}^2(\R)) = 1. </math>

The mapping class group of the [[Klein bottle]] ''K'' is:

:<math> \operatorname{MCG}(K)= \Z_2 \oplus  \Z_2.</math>

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 (mathematics)|genus]] three non-orientable surface ''N''<sub>3</sub> (the connected sum of three projective planes) has:

:<math> \operatorname{MCG}(N_3) = \operatorname{GL}(2,\Z). </math>

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 <math>N\setminus C</math> is ''a torus with a disk removed''. As an unoriented surface, its mapping class group is <math>\operatorname{GL}(2,\Z)</math>. (Lemma 2.1<ref>{{cite journal |first=Martin |last=Scharlemann |title=The complex of curves on nonorientable surfaces |journal=Journal of the London Mathematical Society |volume=s2-25 |issue=1 |pages=171–184 |date=February 1982 |doi=10.1112/jlms/s2-25.1.171 |citeseerx=10.1.1.591.2588}}</ref>).

=== 3-Manifolds ===
Mapping class groups of [[3-manifold]]s 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.<ref>{{cite journal |first=S. |last=Kojima |title=Isometry transformations of hyperbolic 3-manifolds |journal=Topology and Its Applications |volume=29 |issue=3 |pages=297–307 |date=August 1988 |doi=10.1016/0166-8641(88)90027-2 |url=|doi-access= }}</ref>