[Paper Review] Topological equivalence to a projection

Volodymyr Sharko, Yuliia Soroka|arXiv (Cornell University)|Jul 14, 2016

Digital Filter Design and Implementation4 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for a continuous real-valued function on the plane to be topologically equivalent to a coordinate projection: if its level sets form a regular family of curves and each level set is path-connected, then the function is topologically equivalent to the projection onto the y-coordinate. The result relies on the topological structure of level sets and the existence of cross-sections, leading to a global trivialization of the function via a homeomorphism to ℝ×(a,b).

ABSTRACT

We present a necessary and sufficient condition for the topological equivalence of a continuous function on a plane to a projection onto one of coordinates.

Motivation & Objective

• To characterize when a continuous function f: ℝ² → ℝ is topologically equivalent to the projection (x,y) ↦ y.
• To identify topological conditions on the level sets of f that ensure global trivialization via a homeomorphism.
• To generalize the classification of real-valued functions on surfaces by focusing on the absence of critical points and regularity of level curves.
• To provide a topological criterion that ensures the image of f is an open interval and f admits a global coordinate system.

Proposed method

• Define topological equivalence via homeomorphisms h: ℝ² → ℝ² and k: ℝ → ℝ such that k∘f = g∘h.
• Introduce the concept of a regular family of curves: locally parallel curves with r-neighbourhoods satisfying a product structure.
• Use cross-sections—arcs meeting each curve in the family at most once—to establish monotonicity of f along such arcs.
• Prove that f restricted to any cross-section is strictly monotone, implying injectivity on such arcs.
• Construct a global homeomorphism φ: ℝ×(a,b) → ℝ² such that f∘φ(x,y) = y, using countable partitioning of the image into intervals.
• Glue local trivializations φ_k over [c_k, c_{k+1}] using transition maps to define a global diffeomorphism-like structure in the topological category.

Experimental results

Research questions

• RQ1 Under what topological conditions on the level sets of a continuous function f: ℝ² → ℝ is f topologically equivalent to the projection onto the y-coordinate?
• RQ2 How does the regularity of a family of level curves relate to the global trivialization of the function?
• RQ3 Can the absence of critical points be replaced by weaker topological conditions on level sets to ensure topological equivalence to a projection?
• RQ4 What role do cross-sections play in establishing the global structure of f when level sets are path-connected and regular?
• RQ5 Is the image of f necessarily an open interval under the given conditions, and how does this affect the construction of the global trivializing homeomorphism?

Key findings

• If all level sets of f are path-connected and the family of level curves is regular, then f(ℝ²) is an open interval (a,b), possibly extended to ±∞.
• There exists a homeomorphism φ: ℝ×(a,b) → ℝ² such that f∘φ(x,y) = y, proving that f is topologically equivalent to the projection (x,y) ↦ y.
• Each cross-section relative to the level set family is strictly monotone under f, ensuring that f is injective along such arcs.
• The construction of the global trivializing homeomorphism relies on partitioning the image of f into a countable, nested sequence of intervals (c_k, c_{k+1}) with local trivializations.
• The transition between local trivializations is well-defined and continuous, ensuring the global map φ is a homeomorphism.
• The result generalizes previous classifications of functions on surfaces by showing that regularity and path-connectedness of level sets suffice for topological equivalence to a projection, even without differentiability.