Editing Mapping class group (section)

== 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 [[ambient isotopy|isotopy classes]] of [[automorphism]]s of ''M''.  So if ''M'' is a [[topological manifold]], the mapping class group is the group of isotopy classes of [[Homeomorphism group|homeomorphisms]] of ''M''.  If ''M'' is a [[smooth manifold]], the mapping class group is the group of isotopy classes of [[diffeomorphism]]s of ''M''.  Whenever the group of automorphisms of an object ''X'' has a natural [[topological space|topology]], the mapping class group of ''X'' is defined as <math>\operatorname{Aut}(X)/\operatorname{Aut}_0(X)</math>, where <math>\operatorname{Aut}_0(X)</math> is the [[connected space|path-component]] of the identity in <math>\operatorname{Aut}(X)</math>. (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{{Citation needed|date=October 2021}}).  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 <math>\pi_0(\operatorname{Aut}(X))</math>, where one substitutes for Aut the appropriate group for the [[category theory|category]] to which ''X'' belongs. Here <math>\pi_0</math> denotes the 0-th [[homotopy group]] of a space.

So in general, there is a [[Exact sequence#Short exact sequence|short]] [[exact sequence]] of groups:

:<math>1 \rightarrow \operatorname{Aut}_0(X) \rightarrow \operatorname{Aut}(X) \rightarrow \operatorname{MCG}(X) \rightarrow 1.</math>

Frequently this sequence is not [[split exact sequence|split]].<ref>
{{cite journal | last=Morita | first=Shigeyuki | title=Characteristic classes of surface bundles | journal=[[Inventiones Mathematicae]] | volume=90 | issue=3 | year=1987 | doi=10.1007/bf01389178 | pages=551–577 | bibcode=1987InMat..90..551M | mr=0914849| url=http://projecteuclid.org/euclid.bams/1183552184 }}
</ref>

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

There are many [[subgroup]]s of mapping class groups that are frequently studied.  If ''M'' is an oriented manifold, <math>\operatorname{Aut}(M)</math> 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 (mathematics)|homology groups]] of ''M'' is called the '''Torelli group''' of ''M''.