, particularly topology
, the homeomorphism group of a topological space
is the group
consisting of all homeomorphism
s 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 group
s. Homeomorphism groups are topological invariant
s 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
be a topological space and denote the homeomorphism group of
. The action is defined as follows:
This is a group action since for all
denotes the group action, and the identity element
(which is the identity function
) sends points to themselves. If this action is transitive
, then the space is said to be homogeneous
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
becomes a topological group
as one can easily show.
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
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
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.
* Mapping class group