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

Another construction on a space X is the group of all self-homeomorphisms , 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:

HME_k(X) = π_k(Homeo(X)).

Thus HME₀(X) = π₀(Homeo(X)) = 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 Homeo(X) as specified by the functor π₀.

[edit] Example

According to the Dehn-Nielsen theorem, if X is a closed surface then HME₀(X) = Out(π₁(X)), the outer automorphism group of its fundamental group.

[edit] 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.