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
of homotopy classes of continuous maps
Another construction on a space
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
is the mapping class group for
In other words, the mapping class group is the set of connected components of
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.
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