Homeotopy

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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.

[edit] Definition

The homotopy group functors $π k$ assign to each path-connected topological space $X$ the group $π k (X)$ 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 $Homeo(X).$ 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:

H M E k (X) = π k (Homeo(X)).

Thus $H M E 0 (X) = π 0 (Homeo(X)) = M C G * (X)$ 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 $π 0 .$