In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphismss of that space.

Definition

The homotopy group functors assign to each path-connected of homotopy classes

is the group of all self-homeomorphisms If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens will in fact be a topological group under the compact-open topology.

Under the above assumptions, the homeotopy are defined to be:

is the mapping class group

Example

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

References