Homeotopy

The homotopy group functors ${\displaystyle \pi _{k}}$ assign to each path-connected topological space ${\displaystyle X}$ the group ${\displaystyle \pi _{k}(X)}$ of homotopy classes of continuous maps ${\displaystyle S^{k}\to X.}$

Another construction on a space ${\displaystyle X}$ is the group of all self-homeomorphisms ${\displaystyle X\to X}$, denoted ${\displaystyle {\rm {Homeo}}(X).}$ If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that ${\displaystyle {\rm {Homeo}}(X)}$ will in fact be a topological group under the compact-open topology.

Under the above assumptions, the homeotopy groups for ${\displaystyle X}$ are defined to be:

${\displaystyle HME_{k}(X)=\pi _{k}({\rm {Homeo}}(X)).}$

Thus ${\displaystyle HME_{0}(X)=\pi _{0}({\rm {Homeo}}(X))=MCG^{*}(X)}$ is the mapping class group for ${\displaystyle X.}$ In other words, the mapping class group is the set of connected components of ${\displaystyle {\rm {Homeo}}(X)}$ as specified by the functor ${\displaystyle \pi _{0}.}$

According to the Dehn-Nielsen theorem, if ${\displaystyle X}$ is a closed surface then ${\displaystyle HME_{0}(X)={\rm {Out}}(\pi _{1}(X)),}$ i.e., the zeroth homotopy group of the automorphisms of a space is the same as the outer automorphism group of its fundamental group.