Let X be a topological Hausdorff space. Let Homeo(X) be the group of homeomorphisms XX, which can be also turn into a topological spaceMathworldPlanetmath by means of the compact-open topologyMathworldPlanetmath. And let πk be the k-th homotopy groupMathworldPlanetmath functor.

Then the k-th homeotopy is defined as:


that is, the group of homotopy classes of maps SkHomeo(X). Which is different from πk(X), the group of homotopy classes of maps SkX.

One important result for any low dimensional topologist is that for a surface F


which is the F’s extended mapping class group.


G.S. McCarty, Homeotopy groups, Trans. A.M.S. 106(1963)293-304.

Title homeotopy
Canonical name Homeotopy
Date of creation 2013-03-22 15:41:54
Last modified on 2013-03-22 15:41:54
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 17
Author juanman (12619)
Entry type Definition
Classification msc 20F38
Synonym mapping class group
Related topic isotopyMathworldPlanetmath
Related topic group
Related topic homeomorphism
Related topic Group
Related topic Isotopy
Related topic Homeomorphism