Foliations with non-compact leaves on surfaces

Homeotopy Groups for Nonsingular Foliations of the Plane

“…Since ∂ − S i is glued to the boundary component J i × {1} by an affine homeomorphism, and the formulas for G are affine for each fixed t and y, it follows that those formulas agree on J i × {1} and ∂ − S i , c.f. [8]. This implies that G is a continuous map.…”

“…Moreover, suppose ω = X p(γ) ∼ Y q(γ) is a leaf such that ∂ − S λ = X p(γ) and ∂ + S λ ′ = Y p(γ) , see Figure 2.1(a). Then the topological structure of the foliation F near ω is "similar" to the structure of F near "internal" leaves of strips and such a leaf is non-special as well, see [8,Lemma 3.2].…”

“…It follows from [8,Theorem 3.7] that every striped surface (in the sense of (2.1), see Remark 2.3) is foliated homeomorphic either to C or to a reduced surface.…”

“…Kaplan in [5,6] has also mentioned that a non-singular foliation on the plane is glued of countably many strips along open boundary intervals and such that each strip has a foliation by parallel lines. In a recent paper S. Maksymenko and E. Polulyah [8] studied non-singular foliations F on arbitrary non-compact surfaces Σ glued from strips in a similar way. They proved contractibility of the connected components of groups H(F ) of homeomorphisms of Σ mapping leaves onto leaves.…”