homeotopyHomeotopy2006-02-20 23:40:482006-12-24 16:21:46Definitionhomotopy group functor% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $X$ 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 {\bf 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.
{\bf Reference}
G.S. McCarty, {\it Homeotopy groups}, Trans. A.M.S. 106(1963)293-304.