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. If this action is transitive, then the space is said to be homogeneous.

Topology
As with other sets of maps, the homeomorphism group can be given a topology, such as the compact-open topology, making it into a topological group.

Mapping class group
In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group:
 * $${\rm MCG}(X) = {\rm Homeo}(X) / {\rm Homeo}_0(X)$$

The MCG can also be interpreted as the 0th homotopy group, $${\rm MCG}(X) = \pi_0({\rm Homeo}(X))$$. This yields the short exact sequence:
 * $$1 \rightarrow {\rm Homeo}_0(X) \rightarrow {\rm Homeo}(X) \rightarrow {\rm MCG}(X) \rightarrow 1.$$

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.