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In [[mathematics]], particularly [[topology]], the '''homeomorphism group''' of a [[topological space]] is the [[group (mathematics)|group]] consisting of all [[homeomorphism]]s from the space to itself with [[function composition]] as the group [[binary operation|operation]]. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of [[automorphism group]]s. Homeomorphism groups are [[topological invariant]]s in the sense that the homeomorphism groups of homeomorphic topological spaces are [[Group isomorphism|isomorphic as groups]].
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==Properties and Examples==
There is a natural [[group action]] of the homeomorphism group of a space on that space. If this action is [[transitive group action|transitive]], then the space is said to be [[Homogeneous space|homogeneous]].
 
===Topology===
{{Expand section|date=March 2009}}
As with other sets of maps between topological spaces, the homeomorphism group can be given a topology, such as the [[compact-open topology]] (in the case of regular, locally compact spaces), making it into a [[topological group]].
 
In the category of topological spaces with homeomorphisms, object groups are exactly homeomorphism groups.
 
==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.
 
==See also==
* [[Mapping class group]]
 
==References==
*{{Springer|id=H/h047610|title=homeomorphism group}}
 
{{DEFAULTSORT:Homeomorphism Group}}
[[Category:Group theory]]
[[Category:Topology]]
[[Category:Topological groups]]

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