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 $X$ be a topological space and denote the homeomorphism group of $X$ by $G$. The action is defined as follows: $\backslash begin\; G\backslash times\; X\; \&\backslash longrightarrow\; X\backslash \backslash \; (\backslash varphi,\; x)\; \&\backslash longmapsto\; \backslash varphi(x)\; \backslash end$ This is a group action since for all $\backslash varphi,\backslash psi\backslash in\; G$, $\backslash varphi\backslash cdot(\backslash psi\backslash cdot\; x)=\backslash varphi(\backslash psi(x))=(\backslash varphi\backslash circ\backslash psi)(x)$ where $\backslash cdot$ denotes the group action, and the identity element of $G$ (which is the identity function on $X$) 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 $Homeo(X)$ becomes a topological group as one can easily show. If $X$ is Hausdorff, locally compact and locally connected this holds as well.http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf However there are locally compact separable metric spaces for which the inversion map is not continuous and $Homeo(X)$ 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: :$(X)\; =\; (X)\; /\; \_0(X)$ The MCG can also be interpreted as the 0th homotopy group, $(X)\; =\; \backslash pi\_0((X))$. This yields the short exact sequence: :$1\; \backslash rightarrow\; \_0(X)\; \backslash rightarrow\; (X)\; \backslash rightarrow\; (X)\; \backslash rightarrow\; 1.$ 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.

See also

* Mapping class group

References

* {{DEFAULTSORT:Homeomorphism Group Category:Group theory Category:Topology Category:Topological groups

Properties and Examples

There is a natural group action of the homeomorphism group of a space on that space. Let $X$ be a topological space and denote the homeomorphism group of $X$ by $G$. The action is defined as follows: $\backslash begin\; G\backslash times\; X\; \&\backslash longrightarrow\; X\backslash \backslash \; (\backslash varphi,\; x)\; \&\backslash longmapsto\; \backslash varphi(x)\; \backslash end$ This is a group action since for all $\backslash varphi,\backslash psi\backslash in\; G$, $\backslash varphi\backslash cdot(\backslash psi\backslash cdot\; x)=\backslash varphi(\backslash psi(x))=(\backslash varphi\backslash circ\backslash psi)(x)$ where $\backslash cdot$ denotes the group action, and the identity element of $G$ (which is the identity function on $X$) 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 $Homeo(X)$ becomes a topological group as one can easily show. If $X$ is Hausdorff, locally compact and locally connected this holds as well.http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf However there are locally compact separable metric spaces for which the inversion map is not continuous and $Homeo(X)$ 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: :$(X)\; =\; (X)\; /\; \_0(X)$ The MCG can also be interpreted as the 0th homotopy group, $(X)\; =\; \backslash pi\_0((X))$. This yields the short exact sequence: :$1\; \backslash rightarrow\; \_0(X)\; \backslash rightarrow\; (X)\; \backslash rightarrow\; (X)\; \backslash rightarrow\; 1.$ 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.

See also

* Mapping class group

References

* {{DEFAULTSORT:Homeomorphism Group Category:Group theory Category:Topology Category:Topological groups