Editing Homeomorphism group

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]]. They are important to the theory of topological spaces, generally exemplary of [[automorphism group]]s and [[Topological property|topologically invariant]] in the [[group isomorphism]] sense.

==Properties and examples==
There is a natural [[Group action (mathematics)|group action]] of the homeomorphism group of a space on that space. Let <math>X</math> be a topological space and denote the homeomorphism group of <math>X</math> by <math>G</math>. The action is defined as follows:

<math>\begin{align}
    G\times X &\longrightarrow X\\
    (\varphi, x) &\longmapsto \varphi(x)
\end{align}</math>

This is a group action since for all <math>\varphi,\psi\in G</math>,

<math>\varphi\cdot(\psi\cdot x)=\varphi(\psi(x))=(\varphi\circ\psi)(x)</math>,

where <math>\cdot</math> denotes the group action, and the [[identity element]] of <math>G</math> (which is the [[identity function]] on <math>X</math>) sends points to themselves. 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 space|regular]], [[locally compact space]]s the group multiplication is then continuous.

If the space is [[Compact space|compact]] and [[Hausdorff space|Hausdorff]], the inversion is continuous as well and <math>\operatorname{Homeo}(X)</math> becomes a [[topological group]]. If <math>X</math> is Hausdorff, locally compact,  and [[Locally connected space|locally connected]] this holds as well.<ref name="vu.nl">{{citation
 | last = Dijkstra | first = Jan J.
 | doi = 10.2307/30037630
 | issue = 10
 | journal = [[The American Mathematical Monthly]]
 | mr = 2186833
 | pages = 910–912
 | title = On homeomorphism groups and the compact-open topology
 | url = https://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf
 | volume = 112
 | year = 2005| jstor = 30037630
 }}</ref> 
Some locally compact separable metric spaces exhibit an inversion map that is not continuous, resulting in <math>\text{Homeo}(X)</math> not forming a topological group.<ref name="vu.nl"/>

==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==
{{Reflist}}
*{{Springer|id=H/h047610|title=homeomorphism group}}

{{DEFAULTSORT:Homeomorphism Group}}
[[Category:Group theory]]
[[Category:Topology]]
[[Category:Topological groups]]