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

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

$HME\_k(X)=\backslash pi\_k(\{\backslash rm\; Homeo\}(X)).$

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

Example

---

If $X$ a surface then $HME\_0(X)=\{\backslash rm\; Out\}(\backslash pi\_1(X)),$ 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.

Algebraic topology