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.
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 extended mapping class group for In other words, the extended mapping class group is the set of connected components of as specified by the functor
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