Homeomorphism group

In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Homeomorphism groups are topological invariants in the sense that the homeomorphism groups of homeomorphic topological spaces are isomorphic as groups.
Properties and Examples.
There is a natural group action of the homeomorphism group of a space on that space. Let formula_1 be a topological space and denote the homeomorphism group of formula_1 by formula_3. The action is defined as follows:
formula_4
This is a group action since for all formula_5, 
formula_6
where formula_7 denotes the group action, and the identity element of formula_3 (which is the identity function on formula_1) sends points to themselves. If this action is transitive, then the space is said to be homogeneous.
Topology.
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 the group multiplication is then continuous.
If the space is compact and Hausdorff, the inversion is continuous as well and formula_10 becomes a topological group as one can easily show. 
If formula_1 is Hausdorff, locally compact and locally connected this holds as well. 
However there are locally compact separable metric spaces for which the inversion map is not continuous and formula_10 therefore not a topological group.
In the category of topological spaces with homeomorphisms, group objects are exactly homeomorphism groups.
Mapping class group.
In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group:
The MCG can also be interpreted as the 0th homotopy group, formula_14.
This yields the short exact sequence:
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.