homeotopy
Let X be a topological Hausdorff space. Let Homeo(X) be the group of homeomorphisms X→X, which can be also turn into a topological space by means of the compact-open topology. And let πk be the k-th homotopy group functor.
Then the k-th homeotopy is defined as:
ℋk(X)=πk(Homeo(X)) |
that is, the group of homotopy classes of maps Sk→Homeo(X). Which is different from πk(X), the group of homotopy classes of maps Sk→X.
One important result for any low dimensional topologist is that for a surface F
ℋ0(F)=Out(π1(F)) |
which is the F’s extended mapping class group.
Reference
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 | isotopy |
Related topic | group |
Related topic | homeomorphism |
Related topic | Group |
Related topic | Isotopy |
Related topic | Homeomorphism |