Homeotopy

Not to be confused with homotopy.

In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.

Definition

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

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

Under the above assumptions, the homeotopy groups for $X$ are defined to be:

$HM{E}_k(X)={\pi }_k(\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(X)).$

Thus $HM{E}_0(X)={\pi }_0(\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(X))=MC{G}^{*}(X)$ is the mapping class group for $X.$ In other words, the mapping class group is the set of connected components of $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(X)$ as specified by the functor ${\pi }_0.$

Example

According to the Dehn-Nielsen theorem, if $X$ is a closed surface then $HM{E}_0(X)=\mathrm{O}\mathrm{u}\mathrm{t}({\pi }_1(X)),$ i.e., the zeroth homotopy group of the automorphisms of a space is the same as the outer automorphism group of its fundamental group.