homeotopy

Let $X$ be a topological Hausdorff space. Let $\mathrm{Homeo}(X)$ be the group of homeomorphisms $X\to X$, which can be also turn into a topological space by means of the compact-open topology. And let ${\pi }_k$ be the k-th homotopy group functor.

Then the k-th homeotopy is defined as:

$${\mathscr{H}}_k(X)={\pi }_k(\mathrm{Homeo}(X))$$

that is, the group of homotopy classes of maps $S^k\to \mathrm{Homeo}(X)$. Which is different from ${\pi }_k(X)$, the group of homotopy classes of maps $S^k\to X$.

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

$${\mathscr{H}}_0(F)=\mathrm{Out}({\pi }_1(F))$$

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