Homeotopy groups for nonsingular foliations of the plane

Abstract

We consider a special class of nonsingular oriented foliations $F$ on noncompact surfaces $\Sigma$ whose spaces of leaves have the structure similar to the structure of rooted trees of finite diameter. Let $H^+(F)$ be the group of all homeomorphisms of $\Sigma$ mapping the leaves onto leaves and preserving their orientations. Also let $K$ be the group of homeomorphisms of the quotient space $\Sigma /F$ induced by $H^+(F)$. By $H^+_0(F)$ and $K_0$ we denote the corresponding subgroups formed by the homeomorphisms isotopic to identity mappings. Our main result establishes the isomorphism between the homeotopy groups $\pi_0 H^+(F) = H^+(F)/H^+ _0 (F)$ and $\pi_ 0K = K/K_0$.