[Paper Review] Characterization of striped surfaces

Sergiy Maksymenko, Eugene Polulyakh|arXiv (Cornell University)|Oct 16, 2017

Mathematical Dynamics and Fractals7 references3 citations

TL;DR

This paper provides a complete characterization of striped foliated surfaces—two-dimensional manifolds foliated by non-compact, closed leaves—by showing that such surfaces admit a decomposition into open strips $\mathbb{R} \times (0,1)$ if and only if the family of singular leaves (where the leaf space fails to be locally Hausdorff) is locally finite. The result generalizes earlier work on special leaves and establishes a topological criterion for the existence of a striped structure via the finiteness of singularities in neighborhoods.

ABSTRACT

Let $Z$ be a non-compact two-dimensional manifold and $Δ$ be a one-dimensional foliation of $Z$ such that $\partial Z$ consists of leaves of $Δ$ and each leaf of $Δ$ is a non-compact closed subset of $Z$. We obtain a characterization of a subclass of such foliated surfaces $(Z,Δ)$ glued from open strips $\mathbb{R} imes(0,1)$ with boundary leaves along some of their boundary intervals.

Motivation & Objective

• To generalize the characterization of striped foliated surfaces beyond the class of surfaces with only special leaves.
• To define and analyze a broader class of singular leaves corresponding to non-Hausdorff points in the leaf space.
• To establish a necessary and sufficient condition for a foliated surface to admit a striped structure.
• To unify and extend previous results on foliated surfaces with strip decompositions, particularly those involving locally trivial fibrations.

Proposed method

• Introduce the concept of singular leaves as those for which the leaf space lacks a neighborhood homeomorphic to $[0,1]$ with its interior.
• Use the quotient topology on the leaf space $Y = Z/\Delta$ to define the Hausdorff closure and identify singular points.
• Prove that the family of singular leaves is locally finite if and only if the foliated surface admits a striped atlas.
• Apply a technical result on cutting along isolated leaves to analyze the structure of the surface near singularities.
• Use reduced atlases and local foliated charts to analyze the behavior of curves intersecting singular leaves.
• Leverage topological properties of the leaf space, including the behavior of continuous maps from intervals to the leaf space, to derive contradictions when singularities accumulate.

Experimental results

Research questions

• RQ1 Under what topological conditions does a foliated surface admit a decomposition into open strips $\mathbb{R} \times (0,1)$?
• RQ2 How does the concept of singular leaves generalize the earlier notion of special leaves in the context of striped foliated surfaces?
• RQ3 What is the relationship between the local finiteness of singular leaves and the existence of a striped atlas?
• RQ4 Can the failure of the leaf space to be Hausdorff be characterized in terms of the structure of the surface and its foliation?
• RQ5 What topological obstructions arise when singular leaves accumulate in a neighborhood?

Key findings

• A foliated surface $(Z, \Delta)$ admits a striped structure if and only if the family of singular leaves is locally finite.
• The class of singular leaves generalizes the earlier notion of special leaves, and the new characterization applies to a broader class of foliated surfaces.
• The leaf space $Z/\Delta$ fails to be Hausdorff precisely at points corresponding to singular leaves, and this failure is controlled by local finiteness.
• If the interior of $Z$ is neither a cylinder nor a Möbius band, then any simple closed curve transverse to the foliation must intersect the set of singular leaves.
• The assumption that the family of singular leaves is locally finite prevents accumulation of singularities, which would otherwise contradict the existence of a strip decomposition.