Jump to content

Homeotopy

From Wikipedia, the free encyclopedia

This is an old revision of this page, as edited by Mathstudent123 (talk | contribs) at 20:54, 22 May 2019 (Definition: Removed nonstandard notation. Oriented mapping class group is Homeo^+/~ in other articles as well.). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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 assign to each path-connected topological space the group of homotopy classes of continuous maps

Another construction on a space is the group of all self-homeomorphisms , denoted If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that will in fact be a topological group under the compact-open topology.

Under the above assumptions, the homeotopy groups for are defined to be:

Thus is the mapping class group for In other words, the mapping class group is the set of connected components of as specified by the functor

Example

According to the Dehn-Nielsen theorem, if is a closed surface then the outer automorphism group of its fundamental group.

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.