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.