Homeomorphism group

There is a natural group action of the homeomorphism group of a space on that space. Let ${\displaystyle X}$ be a topological space and denote the homeomorphism group of ${\displaystyle X}$ by ${\displaystyle G}$. The action is defined as follows:

${\displaystyle {\begin{aligned}G\times X&\longrightarrow X\\(\varphi ,x)&\longmapsto \varphi (x)\end{aligned}}}$

This is a group action since for all ${\displaystyle \varphi ,\psi \in G}$,

${\displaystyle \varphi \cdot (\psi \cdot x)=\varphi (\psi (x))=(\varphi \circ \psi )(x)}$

where ${\displaystyle \cdot }$ denotes the group action, and the identity element of ${\displaystyle G}$ (which is the identity function on ${\displaystyle X}$) sends points to themselves. If this action is transitive, then the space is said to be homogeneous.

In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group:

${\displaystyle {\rm {MCG}}(X)={\rm {Homeo}}(X)/{\rm {Homeo}}_{0}(X)}$

The MCG can also be interpreted as the 0th homotopy group, ${\displaystyle {\rm {MCG}}(X)=\pi _{0}({\rm {Homeo}}(X))}$. This yields the short exact sequence:

${\displaystyle 1\rightarrow {\rm {Homeo}}_{0}(X)\rightarrow {\rm {Homeo}}(X)\rightarrow {\rm {MCG}}(X)\rightarrow 1.}$

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.

• Mapping class group

• "homeomorphism group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]