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 ${\rm {Homeo}}(X).$ 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 mapping class group for $X.$ In other words, the 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 $HME_{0}(X)={\rm {Out}}(\pi _{1}(X)),$ i.e., the zeroth homotopy group of the automorphisms of a space is the same as the outer automorphism group of its fundamental group.

References

• McCarty, G.S. (1963). "Homeotopy groups" (PDF). Transactions of the American Mathematical Society. 106 (2): 293–304. doi:10.1090/S0002-9947-1963-0145531-9. JSTOR 1993771.
• Arens, R. (1946). "Topologies for homeomorphism groups". American Journal of Mathematics. 68 (4): 593–610. doi:10.2307/2371787. JSTOR 2371787.