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.
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 → X .
Another construction on a space X is the group of all self-homeomorphisms X → X , denoted H o m e o ( 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 ) 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 ( H o m e o ( X ) ) . Thus H M E 0 ( X ) = π 0 ( H o m e o ( 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 H o m e o ( X ) as specified by the functor π 0 .
According to the Dehn-Nielsen theorem, if X is a closed surface then H M E 0 ( X ) = O u t ( π 1 ( X ) ) , the outer automorphism group of its fundamental group.
