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Homeotopy
From Wikipedia, the free encyclopedia
- Be careful not to confuse "homeotopy" with homotopy.
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 πk assign to each path-connected topological space X the group πk(X) of homotopy classes of continuous maps 
Another construction on a space X is the group of all self-homeomorphisms 
, denoted Homeo(X). If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that Homeo(X) will in fact be a topological group under the compact-open topology.
Under the above assumptions, the homeotopy groups for X are defined to be:
- HMEk(X) = πk(Homeo(X)).
Thus HME0(X) = π0(Homeo(X)) = MCG * (X) is the extended mapping class group for X. In other words, the extended mapping class group is the set of connected components of Homeo(X) as specified by the functor π0.
If X a surface then HME0(X) = Out(π1(X)), 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.