Homeotopy

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 ${{\rm {Homeo}}}(X).$ If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that ${{\rm {Homeo}}}(X)$ 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:

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

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

Example

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

References

• G.S. McCarty. Homeotopy groups. Trans. A.M.S. 106(1963)293-304.
• R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610.