Homeotopy

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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.

Definition[edit]

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

Example[edit]

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

References[edit]

  • 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.