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 assign to each path-connected topological space the group of homotopy classes of continuous maps
Another construction on a space is the group of all self-homeomorphisms, denoted If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that will in fact be a topological group under the compact-open topology.
Under the above assumptions, the homeotopy groups for are defined to be:
Thus is the mapping class group for In other words, the mapping class group is the set of connected components of as specified by the functor

Example

According to the Dehn-Nielsen theorem, if is a closed surface then i.e., the zeroth homotopy group of the automorphisms of a space is the same as the outer automorphism group of its fundamental group.