Editing Homeomorphism group (section)

==Mapping class group==
{{Main|Mapping class group}}
In [[geometric topology]] especially, one considers the [[quotient group]] obtained by quotienting out by [[Homotopy#Isotopy|isotopy]], called the [[mapping class group]]:
:<math>{\rm MCG}(X) = {\rm Homeo}(X) / {\rm Homeo}_0(X)</math>.
The MCG can also be interpreted as the 0th [[homotopy group]], <math>{\rm MCG}(X) = \pi_0({\rm Homeo}(X))</math>.
This yields the [[short exact sequence]]:
:<math>1 \rightarrow {\rm Homeo}_0(X) \rightarrow {\rm Homeo}(X) \rightarrow {\rm MCG}(X) \rightarrow 1.</math>

In some applications, particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically trivial homeomorphisms, and then (at times) the extension.