# 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.

## Definition

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

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

## References

- 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.