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 formula_1 assign to each path-connected topological space formula_2 the group formula_3 of homotopy classes of continuous maps formula_4
Another construction on a space formula_2 is the group of all self-homeomorphisms formula_6, denoted formula_7 If "X" is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that formula_8 will in fact be a topological group under the compact-open topology.
Under the above assumptions, the homeotopy groups for formula_2 are defined to be:
Thus formula_11 is the mapping class group for formula_12 In other words, the mapping class group is the set of connected components of formula_8 as specified by the functor formula_14
Example.
According to the Dehn-Nielsen theorem, if formula_2 is a closed surface then formula_16 the outer automorphism group of its fundamental group.