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 π k {\displaystyle \pi _{k}} assign to each path-connected topological space X {\displaystyle X} the group π k ( X ) {\displaystyle \pi _{k}(X)} of homotopy classes of continuous maps S k → X . {\displaystyle S^{k}\to X.}
Another construction on a space X {\displaystyle X} is the group of all self-homeomorphisms X → X {\displaystyle X\to X} , denoted H o m e o ( X ) . {\displaystyle {\rm {Homeo}}(X).} If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that H o m e o ( X ) {\displaystyle {\rm {Homeo}}(X)} will in fact be a topological group under the compact-open topology. Under the above assumptions, the homeotopy groups for X {\displaystyle X} are defined to be:
H M E k ( X ) = π k ( H o m e o ( X ) ) . {\displaystyle HME_{k}(X)=\pi _{k}({\rm {Homeo}}(X)).}
Thus H M E 0 ( X ) = π 0 ( H o m e o ( X ) ) = M C G ∗ ( X ) {\displaystyle HME_{0}(X)=\pi _{0}({\rm {Homeo}}(X))=MCG^{*}(X)} is the mapping class group for X . {\displaystyle X.} In other words, the mapping class group is the set of connected components of H o m e o ( X ) {\displaystyle {\rm {Homeo}}(X)} as specified by the functor π 0 . {\displaystyle \pi _{0}.}
Example According to the Dehn-Nielsen theorem, if X {\displaystyle X} is a closed surface then H M E 0 ( X ) = O u t ( π 1 ( X ) ) , {\displaystyle HME_{0}(X)={\rm {Out}}(\pi _{1}(X)),} i.e., the zeroth homotopy group of the automorphisms of a space is the same as the outer automorphism group of its fundamental group.
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