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 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.