# homeotopy

Let $X$ be a topological Hausdorff space. Let ${\rm Homeo}(X)$ 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:

 ${\cal{H}}_{k}(X)=\pi_{k}({\rm Homeo}(X))$

that is, the group of homotopy classes of maps $S^{k}\to{\rm Homeo}(X)$. 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 $F$

 ${\cal{H}}_{0}(F)={\rm Out}(\pi_{1}(F))$

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