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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


According to the Dehn-Nielsen theorem, if is a closed surface then the outer automorphism group of its fundamental group.


  • G.S. McCarty. Homeotopy groups. Trans. A.M.S. 106(1963)293-304.
  • R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610.