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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 $\pi_k$ assign to each path-connected topological space $X$ the group $\pi_k\left(X\right)$ of homotopy classes of continuous maps $S^k\to X.$
Another construction on a space $X$ is the group of all self-homeomorphisms $X \to X$, denoted $\left(X\right).$ If ''X'' is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that $\left(X\right)$ 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:
:$HME_k\left(X\right)=\pi_k\left(\left(X\right)\right).$
Thus $HME_0\left(X\right)=\pi_0\left(\left(X\right)\right)=MCG^$
*(X) is the extended mapping class group for $X.$ In other words, the extended mapping class group is the set of connected components of $\left(X\right)$ as specified by the functor $\pi_0.$

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