Homeotopy

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Not to be confused with homotopy. ‹See Tfd›

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[edit]

The homotopy group functors \pi_k assign to each path-connected topological space X the group \pi_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 {\rm Homeo}(X). If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that {\rm 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)=\pi_k({\rm Homeo}(X)).

Thus HME_0(X)=\pi_0({\rm 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 {\rm Homeo}(X) as specified by the functor \pi_0.

Example[edit]

According to the Dehn-Nielsen theorem, if X is a closed surface then HME_0(X)={\rm Out}(\pi_1(X)), the outer automorphism group of its fundamental group.

References[edit]

  • 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.