Homeotopy

:Be careful not to confuse "homeotopy" 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 $S^k\; o\; X.$

Another construction on a space $X$ is the group of all self-homeomorphisms $X\; o\; X$, denoted $\{\; m\; Homeo\}(X).$ If "X" is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that $\{\; m\; Homeo\}(X)$ 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:

:$HME\_k(X)=pi\_k(\{\; m\; Homeo\}(X)).$

Thus $HME\_0(X)=pi\_0(\{\; m\; Homeo\}(X))=MCG^*(X)$ is the extended mapping class group for $X.$ In other words, the extended mapping class group is the set of connected components of $\{\; m\; Homeo\}(X)$ as specified by the functor $pi\_0.$

Example

According to the Dehn-Nielsen theorem, if $X$ is a closed surface then $HME\_0(X)=\{\; m\; Out\}(pi\_1(X)),$ 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.