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

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

Example

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According to the Dehn-Nielsen theorem, if   is a closed surface then   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

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  • McCarty, G.S. (1963). "Homeotopy groups" (PDF). Transactions of the American Mathematical Society. 106 (2): 293–304. doi:10.1090/S0002-9947-1963-0145531-9. JSTOR 1993771.
  • Arens, R. (1946). "Topologies for homeomorphism groups". American Journal of Mathematics. 68 (4): 593–610. doi:10.2307/2371787. JSTOR 2371787.