Homeotopy

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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